Recent zbMATH articles in MSC 37https://zbmath.org/atom/cc/372022-11-17T18:59:28.764376ZUnknown authorWerkzeugTricky math, but trippy graphics: the quixotic search for the ``3D Mandelbrot''https://zbmath.org/1496.000432022-11-17T18:59:28.764376Z"Merow, Sophia D."https://zbmath.org/authors/?q=ai:merow.sophia-d(no abstract)Mathematical modeling. A dynamical systems approach to analyze practical problems in STEM disciplineshttps://zbmath.org/1496.000552022-11-17T18:59:28.764376Z"Palacios, Antonio"https://zbmath.org/authors/?q=ai:palacios.antonioPublisher's description: This book provides qualitative and quantitative methods to analyze and better understand phenomena that change in space and time. An innovative approach is to incorporate ideas and methods from dynamical systems and equivariant bifurcation theory to model, analyze and predict the behavior of mathematical models. In addition, real-life data is incorporated in the derivation of certain models. For instance, the model for a fluxgate magnetometer includes experiments in support of the model.
The book is intended for interdisciplinary scientists in STEM fields, who might be interested in learning the skills to derive a mathematical representation for explaining the evolution of a real system. Overall, the book could be adapted in undergraduate- and postgraduate-level courses, with students from various STEM fields, including: mathematics, physics, engineering and biology.Quasi-invariant measures for continuous group actionshttps://zbmath.org/1496.031422022-11-17T18:59:28.764376Z"Kechris, Alexander S."https://zbmath.org/authors/?q=ai:kechris.alexander-sSummary: The class of ergodic, invariant probability Borel measure for the shift action of a countable group is a \(G_\delta\) set in the compact, metrizable space of probability Borel measures. We study in this paper the descriptive complexity of the class of ergodic, quasi-invariant probability Borel measures and show that for any infinite countable group \(\Gamma\) it is \(\boldsymbol{\Pi^0_3}\)-hard, for the group \(\mathbb{Z}\) it is \(\boldsymbol{\Pi^0_3}\)-complete, while for the free group \(\mathbb{F}_\infty\) with infinite, countably many generators it is \(\boldsymbol{\Pi^0_\alpha}\)-complete, for some ordinal \(\alpha\) with \(3\leq\alpha\leq\omega+2\). The exact value of this ordinal is unknown.
For the entire collection see [Zbl 1454.03009].Isomorphism and weak conjugacy of free Bernoulli subflowshttps://zbmath.org/1496.031842022-11-17T18:59:28.764376Z"Clemens, John D."https://zbmath.org/authors/?q=ai:clemens.john-danielSummary: We show that for \(G\) a countable group which is not locally finite, the isomorphism of free \(G\)-subflows is bi-reducible with with the universal countable Borel equivalence relation \(E_{\infty}\), a result obtained independently by Gao-Jackson-Seward. We also show that the same result holds for the relation of weak conjugacy and for any intermediate Borel equivalence relation.
For the entire collection see [Zbl 1454.03009].Order of torsion for reduction of linearly independent points for a family of Drinfeld moduleshttps://zbmath.org/1496.110872022-11-17T18:59:28.764376Z"Ghioca, Dragos"https://zbmath.org/authors/?q=ai:ghioca.dragos"Shparlinski, Igor E."https://zbmath.org/authors/?q=ai:shparlinski.igor-eSummary: Let \(q\) be a power of the prime number \(p\), let \(K = \mathbb{F}_q(t)\), and let \(r \geqslant 2\) be an integer. For points \(\mathbf{a}, \mathbf{b} \in K\) which are \(\mathbb{F}_q\)-linearly independent, we show that there exist positive constants \(N_0\) and \(c_0\) such that for each integer \(\ell \geqslant N_0\) and for each generator \(\tau\) of \(\mathbb{F}_{q^\ell} / \mathbb{F}_q\), we have that for all except \(N_0\) values \(\lambda \in \overline{\mathbb{F}_q} \), the corresponding specializations \(\mathbf{a}(\tau)\) and \(\mathbf{b}(\tau)\) cannot have orders of degrees less than \(c_0 \log \log \ell\) as torsion points for the Drinfeld module \(\Phi^{( \tau , \lambda )} : \mathbb{F}_q [T] \longrightarrow \operatorname{End}_{\overline{\mathbb{F}_q}}( \mathbb{G}_a)\) (where \(\mathbb{G}_a\) is the additive group scheme), given by \(\Phi_T^{( \tau , \lambda )}(x) = \tau x + \lambda x^q + x^{q^r} \).Hausdorff measure of sets of Dirichlet non-improvable affine formshttps://zbmath.org/1496.110962022-11-17T18:59:28.764376Z"Kim, Taehyeong"https://zbmath.org/authors/?q=ai:kim.taehyeong"Kim, Wooyeon"https://zbmath.org/authors/?q=ai:kim.wooyeonBased on authors' abstract: For a decreasing real valued function \(\psi\), a pair \((A, b)\) of a real \(m\times n\) matrix \(A\) and \(b\in\mathbb R^m\) is said to be \(\psi\)-Dirichlet improvable if there exist \(\mathbf{p}\in\mathbb Z^m\) and \(\mathbf{q}\in\mathbb Z^n\) such that \[ \|A\mathbf{q}+\mathbf{b}-\mathbf{p}\|^m<\psi(T)\quad\hbox{and}\quad \|q\|^n<T \] for all sufficiently large \(T\), where \(\|\cdot\|\) stands for the supremum norm of \(\mathbb R^k\) given by \(\|\mathbf{x}\|=\max_{1\le i\le k}|x_i|\). \textit{D. Kleinbock} and \textit{N. Wadleigh} [Compos. Math. 155, No. 7, 1402--1423 (2019; Zbl 1429.11124)] established the zero-one law for the Lebesgue measure of the \(\psi\)-Dirichlet non-improvable set. In this paper, the authors prove a similar criterion for the Hausdorff measure of the \(\psi\)-Dirichlet non-improvable set. Also, we extend this result to the singly metric case that \(\mathbf{b}\) is fixed. As an application, by considering \(\psi_a\) with \(a=\frac{m w}{n}\), we compute the Hausdorff dimension of the set of pairs \((A,\mathbf{b})\) with uniform Diophantine exponents \(\hat{w}(A,\mathbf{b})\le w\).
Reviewer: Takao Komatsu (Hangzhou)Higher arithmetic degrees of dominant rational self-mapshttps://zbmath.org/1496.140212022-11-17T18:59:28.764376Z"Dang, Nguyen-Bac"https://zbmath.org/authors/?q=ai:dang.nguyen-bac"Ghioca, Dragos"https://zbmath.org/authors/?q=ai:ghioca.dragos"Hu, Fei"https://zbmath.org/authors/?q=ai:hu.fei"Lesieutre, John"https://zbmath.org/authors/?q=ai:lesieutre.john"Satriano, Matthew"https://zbmath.org/authors/?q=ai:satriano.matthewThe paper under review formulates an analogue, for higher dimensional cycles, of a conjecture of \textit{S. Kawaguchi} and \textit{J. H. Silverman} [J. Reine Angew. Math. 713, 21--48 (2016; Zbl 1393.37115)] on the growth rate of heights of rational points in the forward orbit of a dominant rational self-map of an algebraic variety defined over \(\overline{\mathbb{Q}}\). The authors also formulate some conjectures relating the arithmetic degree to the dynamical degree.
Specifically, let \(f\colon X\dashrightarrow X\) be a dominant rational map defined over \(\overline{\mathbb{Q}}\). The arithmetic degree of a \(\overline{\mathbb{Q}}\)-point \(P\) is the limit
\[
\alpha_f(P)=\lim_{n\to\infty} h^+(f^n(P))^{1/n}
\]
where \(h^+\) is \(\max\{h,1\}\) fro some Weil height function \(h\) on \(X\). Kawaguchi and Silverman conjecture that if the forward orbit of \(P\) is Zariski dense, then
\[
\alpha_f(P) = \lim_{n\to\infty} ((f^n)^*H\cdot H^{d-1})^{1/n}
\]
where \(d=\dim X\).
The authors define the \(k\)th arithmetic degree of \(f\) by
\[
\alpha_{k}(f) = \limsup_{n\to\infty} \widehat{\deg}_{k}(f^n)^{1/n}
\]
where \(\widehat{\deg}\) is the \(k\)th degree of \(f\) with respect to some arithmetically ample line bundle. The authors prove that the sequence \(\{\alpha_k(f)\}\) is log concave, and that if \(f\) is birational, then \(\alpha_k(f)=\alpha_{d+1-k}(f^{-1})\). They also prove that
\[
\alpha_1(f)=\lim_{n\to\infty} \widehat{\deg}_1(f^n)^{1/n}
\]
and that this limit is independent of the choice of integral or birational model of \(X\), and of the choice of arithmetically ample line bundle used in the definition of \(\widehat{\deg}\).
If the \(k\)th dynamical degree of \(f\) is given by
\[
\lambda_k(f) = \lim_{n\to\infty} \deg_k(f^n)^{1/n}
\]
then the authors also prove that
\[
\alpha_k(f)\geq\max\{\lambda_k(f),\lambda_{k-1}(f)\}
\]
and that
\[
\alpha_1(f)=\lambda_1(f).
\]
Reviewer: David McKinnon (Waterloo)Endomorphisms of regular rooted trees induced by the action of polynomials on the ring \(\mathbb{Z}_d\) of \(d\)-adic integershttps://zbmath.org/1496.200472022-11-17T18:59:28.764376Z"Ahmed, Elsayed"https://zbmath.org/authors/?q=ai:ahmed.elsayed-a"Savchuk, Dmytro"https://zbmath.org/authors/?q=ai:savchuk.dmytro-mA descriptive construction of trees and Stallings' theoremhttps://zbmath.org/1496.200492022-11-17T18:59:28.764376Z"Tserunyan, Anush"https://zbmath.org/authors/?q=ai:tserunyan.anushSummary: We give a descriptive construction of trees for multi-ended graphs, which yields yet another proof of Stallings' theorem on ends of groups. Even though our proof is, in principle, not very different from already existing proofs and it draws ideas from [\textit{B. Krön}, Groups Complex. Cryptol. 2, No. 2, 213--221 (2010; Zbl 1222.20018)] it is written in a way that easily adapts to the setting of countable Borel equivalence relations, leading to a free decomposition result and a sufficient condition for treeability.
For the entire collection see [Zbl 1454.03009].Ledrappier-Young formulae for a family of nonlinear attractorshttps://zbmath.org/1496.280072022-11-17T18:59:28.764376Z"Jurga, Natalia"https://zbmath.org/authors/?q=ai:jurga.natalia"Lee, Lawrence D."https://zbmath.org/authors/?q=ai:lee.lawrence-dSummary: We study a natural class of invariant measures supported on the attractors of a family of nonlinear, non-conformal iterated function systems introduced by Falconer, Fraser and Lee. These are pushforward quasi-Bernoulli measures, a class which includes the well-known class of Gibbs measures for Hölder continuous potentials. We show that these measures are exact dimensional and that their exact dimensions satisfy a Ledrappier-Young formula.Recurrence in the dynamics of meromorphic correspondences and holomorphic semigroupshttps://zbmath.org/1496.320172022-11-17T18:59:28.764376Z"Londhe, Mayuresh"https://zbmath.org/authors/?q=ai:londhe.mayureshSummary: This paper studies recurrence phenomena in iterative holomorphic dynamics of certain multi-valued maps. In particular, we prove an analogue of the Poincaré recurrence theorem for meromorphic correspondences with respect to certain dynamically interesting measures associated with them. Meromorphic correspondences present a significant measure-theoretic obstacle: the image of a Borel set under a meromorphic correspondence need not be Borel. We manage this issue using the Measurable Projection Theorem, which is an aspect of descriptive set theory. We also prove a result concerning invariance properties of the supports of the measures mentioned.Global surfaces of Section with positive genus for dynamically convex Reeb flowshttps://zbmath.org/1496.320422022-11-17T18:59:28.764376Z"Hryniewicz, Umberto L."https://zbmath.org/authors/?q=ai:hryniewicz.umberto-l"Salomão, Pedro A. S."https://zbmath.org/authors/?q=ai:salomao.pedro-a-s"Siefring, Richard"https://zbmath.org/authors/?q=ai:siefring.richardSummary: We establish some new existence results for global surfaces of section of dynamically convex Reeb flows on the three-sphere. These sections often have genus, and are the result of a combination of pseudoholomorphic methods with some elementary ergodic methods.Chow's theorem for real analytic Levi-flat hypersurfaceshttps://zbmath.org/1496.320542022-11-17T18:59:28.764376Z"Fernández-Pérez, Arturo"https://zbmath.org/authors/?q=ai:fernandez-perez.arturo"Mol, Rogério"https://zbmath.org/authors/?q=ai:mol.rogerio-s"Rosas, Rudy"https://zbmath.org/authors/?q=ai:rosas.rudySummary: In this article we provide a version of Chow's theorem for real analytic Levi-flat hypersurfaces in the complex projective space \(\mathbb{P}^n\), \(n\geq 2\). More specifically, we prove that a real analytic Levi-flat hypersurface \(M\subset\mathbb{P}^n\), with singular set of real dimension at most \(2n-4\) and whose Levi leaves are contained in algebraic hypersurfaces, is tangent to the levels of a rational function in \(\mathbb{P}^n\). As a consequence, \(M\) is a semialgebraic set. We also prove that a Levi foliation on \(\mathbb{P}^n\) -- a singular real analytic foliation whose leaves are immersed complex manifolds of codimension one -- satisfying similar conditions -- singular set of real dimension at most \(2n-4\) and all leaves algebraic -- is defined by the level sets of a rational function.Basic theory of differential equations with fractional substantial derivative in Banach spaceshttps://zbmath.org/1496.340242022-11-17T18:59:28.764376Z"Wang, Yejuan"https://zbmath.org/authors/?q=ai:wang.yejuan"Wang, Yaxiong"https://zbmath.org/authors/?q=ai:wang.yaxiong.1Summary: The theory of local existence, extremal solutions and global existence of solutions are established for an abstract fractional differential equation with fractional substantial derivative in a Banach space by assuming that the nonlinear term is weakly continuous in some sense. We then apply the theory results to a first order lattice system with fractional substantial derivative, the existence of a compact global attractor is established. In addition, the global attractor is a singleton is also proved under extra Lipschitz conditions.Embedded solitons in second-harmonic-generating latticeshttps://zbmath.org/1496.340302022-11-17T18:59:28.764376Z"Susanto, Hadi"https://zbmath.org/authors/?q=ai:susanto.hadi"Malomed, Boris A."https://zbmath.org/authors/?q=ai:malomed.boris-aSummary: Embedded solitons are exceptional modes in nonlinear-wave systems with the propagation constant falling in the system's propagation band. An especially challenging topic is seeking for such modes in nonlinear dynamical lattices (discrete systems). We address this problem for a system of coupled discrete equations modeling the light propagation in an array of tunnel-coupled waveguides with a combination of intrinsic quadratic (second-harmonic-generating) and cubic nonlinearities. Solutions for discrete embedded solitons (DESs) are constructed by means of two analytical approximations, adjusted, severally, for broad and narrow DESs, and in a systematic form with the help of numerical calculations. DESs of several types, including ones with identical and opposite signs of their fundamental-frequency and second-harmonic components, are produced. In the most relevant case of narrow DESs, the analytical approximation produces very accurate results, in comparison with the numerical findings. In this case, the DES branch extends from the propagation band into a semi-infinite gap as a family of regular discrete solitons. The study of stability of DESs demonstrates that, in addition to ones featuring the well-known property of semi-stability, there are linearly stable DESs which are genuinely robust modes.Weak and strong singularities problems to Liénard equationhttps://zbmath.org/1496.340472022-11-17T18:59:28.764376Z"Xin, Yun"https://zbmath.org/authors/?q=ai:xin.yun"Hu, Guixin"https://zbmath.org/authors/?q=ai:hu.guixinSummary: This paper is devoted to an investigation of the existence of a positive periodic solution for the following singular Liénard equation:
\[
x^{\prime\prime}+f\bigl(x(t)\bigr)x^\prime(t)+a(t)x= \frac{b(t)}{x^{\alpha}}+e(t),
\] where the external force \(e(t)\) may change sign, \( \alpha\) is a constant and \(\alpha >0\). The novelty of the present article is that for the first time we show that weak and strong singularities enables the achievement of a new existence criterion of positive periodic solution through an application of the Manásevich-Mawhin continuation theorem. Recent results in the literature are generalized and significantly improved, and we give the existence interval of periodic solution of this equation. At last, two examples and numerical solution (phase portraits and time portraits of periodic solutions of the example) are given to show applications of the theorem.Periodic orbits bifurcating from a Hopf equilibrium of 2-dimensional polynomial Kolmogorov systems of arbitrary degreehttps://zbmath.org/1496.340642022-11-17T18:59:28.764376Z"Djedid, Djamila"https://zbmath.org/authors/?q=ai:djedid.djamila"Llibre, Jaume"https://zbmath.org/authors/?q=ai:llibre.jaume"Makhlouf, Amar"https://zbmath.org/authors/?q=ai:makhlouf.amarSummary: A Hopf equilibrium of a differential system in \(\mathbb{R}^2\) is an equilibrium point whose linear part has eigenvalues \(\pm\omega i\) with \(\omega\neq 0\), where \(i=\sqrt{-1}\). We provide necessary and sufficient conditions for the existence of a limit cycle bifurcating from a Hopf equilibrium of 2-dimensional polynomial Kolmogorov systems of arbitrary degree. We provide an estimation of the bifurcating small limit cycle and also characterize the stability of this limit cycle.The center conditions for a perturbed cubic center via the fourth-order Melnikov functionhttps://zbmath.org/1496.340652022-11-17T18:59:28.764376Z"Asheghi, Rasoul"https://zbmath.org/authors/?q=ai:asheghi.rasoulSummary: In this paper, we first consider a cubic integrable system under general quadratic perturbations. We then study the Melnikov functions of the perturbed system up to the fourth order. Our studies show that the first four Melnikov functions are sufficient to obtain the center conditions for the perturbed system.On generation of a limit cycle from a separatrix loop of a sewn saddle-nodehttps://zbmath.org/1496.340662022-11-17T18:59:28.764376Z"Roĭtenberg, Vladimir Shleĭmovich"https://zbmath.org/authors/?q=ai:roitenberg.vladimir-shleimovichSummary: The article considers dynamical systems on the plane, defined by continuous piecewise smooth vector fields. Such systems are used as mathematical models of real processes with switching. An important task is to find the conditions for the generation of periodic trajectories when the parameters change. The paper describes the bifurcation of the birth of a periodic trajectory from the loop of the separatrix of a sewn saddle-node -- an analogue of the classical bifurcation of the separatrix loop of a saddle-node of a smooth dynamical system. Consider a one-parameter family \(\{ X_\varepsilon \}\) of continuous piecewise-smooth vector fields on the plane. Let \(z^0\) be a point on the switching line. Let's choose the local coordinates \(x,y\) in which \(z^0\) has zero coordinates, and the switching line is given by the equation \(y = 0\). Let the vector field \(X_0\) in a semi-neighborhood \(y \ge 0 (y \le 0)\) coincide with a smooth vector field \(X_0^+ (X_0^- )\), for which the point \(z^0\) is a stable rough node (rough saddle), and the proper subspaces of the matrix of the linear part of the field in \(z^0\) do not coincide with the straight line \(y = 0\). The singular point \(z^0\) is called a sewn saddle-node. There is a single trajectory \(L_0\) that is \(\alpha \)-limit to \(z^0 \) -- the outgoing separatrix of the point \(z^0 \). It is assumed that \(L_0\) is also \(\omega \)-limit to \(z^0\), and enters \(z^0\) in the leading direction of the node of the field \(X_0^+ \). For generic family, when the parameter \(\varepsilon\) changes, the sewn saddle-node either splits into a rough node and a rough saddle, or disappears. In the paper it is proved that in the latter case the only periodic trajectory of the field \(X_\varepsilon\) is generated from the contour \(L_0 \cup \{ z^0 \} \) -- a stable limit cycle.Dynamical robustness in a heterogeneous network of globally coupled nonlinear oscillatorshttps://zbmath.org/1496.340692022-11-17T18:59:28.764376Z"Gowthaman, I."https://zbmath.org/authors/?q=ai:gowthaman.i"Singh, Uday"https://zbmath.org/authors/?q=ai:singh.uday-pratap|singh.uday-narayan|singh.uday-partap|singh.uday-prasad"Chandrasekar, V. K."https://zbmath.org/authors/?q=ai:chandrasekar.v-k"Senthilkumar, D. V."https://zbmath.org/authors/?q=ai:senthilkumar.dharmapuri-vijayanSummary: Deterioration or failure of even a fraction of the microscopic constituents of a large class of networks leads to the loss of the macroscopic activity of the network as a whole. We deduce the evolution equation of two macroscopic order parameters from a globally coupled network of heterogeneous oscillators following the self-consistent field approach under the strong coupling limit. The macroscopic order parameter is used to classify the stable nontrivial steady state and the macroscopic oscillatory state of the network. We examine the dynamical robustness of the network by including a limiting factor that limits the degree of diffusion and a self-feedback factor in the network in addition to the heterogeneity of the network. The heterogeneity is introduced using the parameter specifying the distance from the Hopf bifurcation, which is drawn from a random statistical distribution. We also deduce the critical stability curves from the evolution equation of the macroscopic order parameters demarcating the stable nontrivial steady state and the macroscopic oscillatory state in the system parameter space. We show that a large heterogeneity and a large self-feedback factor facilitates the onset of the stable macroscopic oscillatory state by destabilizing the aging transition state, whereas limiting the degree of the diffusion favors the sustained macroscopic oscillation of the heterogeneous network.Emergence of extreme events in coupled systems with time-dependent interactionshttps://zbmath.org/1496.340702022-11-17T18:59:28.764376Z"Kumarasamy, Suresh"https://zbmath.org/authors/?q=ai:kumarasamy.suresh"Srinivasan, Sabarathinam"https://zbmath.org/authors/?q=ai:srinivasan.sabarathinam"Gogoi, Pragjyotish Bhuyan"https://zbmath.org/authors/?q=ai:gogoi.pragjyotish-bhuyan"Prasad, Awadhesh"https://zbmath.org/authors/?q=ai:prasad.awadheshDynamics of two or more nonlinear interacting systems being a subject of this paper is a very interesting topic. Usually a general solution to equations of motion of such a system cannot be found in symbolic form and so numerical methods are applied. In this work, the system of two Stuart-Landau (SL) oscillators in the presence of time-dependent interactions is studied and Mathematica software is used to find numerical solutions to the equations of motion for some chosen values of the system parameters. The authors consider two cases of time-varying coupling strengths. In the first case parameter determining the interactions is a periodic function of time and so the interaction depends on time explicitly. In the second case parameter determining the interactions depends on the distance or coordinates which are functions of time and so such a case is considered as ``implicit time dependence''. Note that in the first case interaction of two SL oscillators depends on the parameter $d=1+\gamma$ that is a function of time because $\gamma=f\cos [[(\Omega t)]]$. As the parameter \(d\) appears in the denominator of the term describing this interaction becomes very large for $f=0.999$ when $\cos [[(\Omega t)]]=-1$ or $\gamma=-0.999$. So it is quite natural that large amplitude oscillations may occur that are considered as extreme events, and this results are confirmed by the corresponding numerical solutions of the equations of motion. At the same time the authors introduce the ``quasi-stable equilibrium points'' a sense of which is not clear. It seems such equilibrium points are determined as solutions of the equations which are obtained if the right-hand sides of equations (3) are equated to zero. But these equations contain function of time $\gamma=f\cos [[(\Omega t)]]$ and so the corresponding ``equilibrium points'' are functions of time, as well. The question arises what is the sense of such solutions and what does it mean stability of such ``equilibrium points''? The authors equate the Jacobian matrix (5) to zero and state that ``solving this equation, we get the stability of the equilibrium points in the systems''. This statement is very strange because the Jacobian matrix is a function of gamma and so it is a function of time. So it would be good to give a definition of stability which is used to define the corresponding criteriums of stability. This can help to understand the results and to reproduce all the calculations if a reader will be interested in their checking. Summarizing, one can state that the problem is very interesting but the results obtained are very questionable and must be checked carefully.
Reviewer: Alexander Prokopenya (Warszawa)On the existence of periodic solutions to one class of systems of nonlinear differential equationshttps://zbmath.org/1496.340762022-11-17T18:59:28.764376Z"Demidenko, G. V."https://zbmath.org/authors/?q=ai:demidenko.gennadii-vThe author investigates the existence of periodic solutions for a class of nonlinear ordinary differential equations. The exponential dichotomy of the linear part is also considered. Stability criteria for the periodic state are also proposed.
Reviewer: Gani T. Stamov (Sliven)Subharmonic solutions in reversible non-autonomous differential equationshttps://zbmath.org/1496.340772022-11-17T18:59:28.764376Z"Eze, Izuchukwu"https://zbmath.org/authors/?q=ai:eze.izuchukwu"García-Azpeitia, Carlos"https://zbmath.org/authors/?q=ai:garcia-azpeitia.carlos"Krawcewicz, Wieslaw"https://zbmath.org/authors/?q=ai:krawcewicz.wieslaw-z"Lv, Yanli"https://zbmath.org/authors/?q=ai:lv.yanliLet \(p>0\) be a fixed number. The authors are interested in subharmonic solutions of the system
\[
\ddot{u}(t) = f(t,u(t)),\ u(t)\in V
\]
where \(f(t,u)\) is a continuous map, \(p\)-periodic with respect to the temporal variable. More precisely, let \(V := \mathbb{R}^k\) and let \(p = 2 \pi\) without loss of generality. Assume that \(f: \mathbb{R}\times V \rightarrow V\) is a continuous function satisfying the following symmetry conditions:
\begin{enumerate}
\item[(\(S_1\))] For all \(t \in \mathbb{R}\) and \(x \in V\) we have \(f(t+2\pi,x) = f(t,x)\) (\textit{dihedral symmetry});
\item[(\(S_2\))] For all \(t \in \mathbb{R}\) and \(x \in V\) we have \(f(-t,x) = f(t,x)\) (\textit{time-reversibility});
\item[(\(S_3\))] For all \(t \in \mathbb{R}\) and \(x \in V\) we have \(f(t,-x) = -f(t,x)\) (\textit{antipodal \(\mathbb{Z}_2\)-symmetry}).
\end{enumerate}
The symmetric properties of the system of study allow reformulation of the problem of existence of the subharmonic \(2\pi m\)-periodic solutions as a question about the operator equation \(\mathcal{F}(u)=0\) with \(D_m\times \mathbb{Z}_2\)-symmetries in the functional space \(\mathcal{E} := C_{2\pi m}^2(\mathbb{R};V)\). The authors introduce an additional symmetry to the system of study before proving several results on the existence and multiplicity of subharmonic solutions. Namely, let \(\Gamma\) be a finite group then
\begin{enumerate}
\item[(\(S_4\))] For all \(t \in \mathbb{R}\), \(x \in V\), and \(\sigma \in \Gamma\), we have \(f(t,\sigma x) = \sigma f(t,x)\) (\textit{\(\Gamma\)-equivariant}).
\end{enumerate}
The last condition allows for the restatement of the original problem as the \(G\)-equivariant operator equation with respect to the full group
\[
G := \Gamma \times D_m \times \mathbb{Z}_2.
\]
If the isotropy group \(G_u\) of a solution \(u\) satisfies \(\{ e \} \times \mathbb{Z}_m \times\{ 1 \} \nleq G_u\), then \(u\) is a subharmonic solution.
The authors prove several novel results in the paper. Most notably Theorems 2.6 and 2.10. The main technical tool is Brower \(\textbf{G}\)-equivariant degree theory. Given a group \(G\) corresponding \(\textbf{G}\)-equivariant Brower degree is computed using the computer algebra system GAP. In addition, the authors discuss the bifurcation problem of subharmonic solutions in the case of a system depending on an extra parameter \(\alpha\). The paper is clear and easy to follow.
Reviewer: Predrag Punosevac (Pittsburgh)Turnpike properties of solutions of a differential inclusion with a Lyapunov function. IIhttps://zbmath.org/1496.340932022-11-17T18:59:28.764376Z"Zaslavski, Alexander J."https://zbmath.org/authors/?q=ai:zaslavski.alexander-jSummary: We study the turnpike phenomenon for approximate solutions of optimal problems governed by a differential inclusion with a Lyapunov function. This differential inclusion generates a dynamical system which has a prototype in mathematical economics. In our previous research we obtained turnpike results for a collection of approximate optimal trajectories with a fixed initial point. In the present paper, under a certain assumption, we extend these results for all approximate optimal trajectories.
For Part I see [ibid. 7, No. 3, 1085--1102 (2022; Zbl 1490.34061)].Variation of constants formula and exponential dichotomy for nonautonomous non-densely defined Cauchy problemshttps://zbmath.org/1496.340962022-11-17T18:59:28.764376Z"Magal, Pierre"https://zbmath.org/authors/?q=ai:magal.pierre"Seydi, Ousmane"https://zbmath.org/authors/?q=ai:seydi.ousmaneLet \(X\) be a Banach space and \(A:D(A)\to X\) be a linear operator with possibly non-dense domain. Denote \(\overline{D(A)}=X_0\). Let \(\{B(t)\}_{t\in\mathbb{R}}\subset \mathcal L(X_0,X)\) be a locally bounded and strongly continuous family of linear operators.
Assume that \(\exists \omega\in\mathbb{R}\) and \(M\geq1\) s.t. \((\omega,+\infty)\subset\rho(A)\),
\[
\|(\lambda I-A)^{-k}\|_{\mathcal{L}(X_0,X)}\leq M(\lambda-\omega)^{-k}\quad\forall \lambda>\omega,k\geq1
\]
and \(\lim_{\lambda\to\infty}(\lambda-A)^{-1}x=0\;\forall x\in X\). Suppose that for each \(\tau>0\) and \(f\in C([0,\tau],X)\) the equation \(u'_f=Au_f+f\) has a unique mild solution \(u_f\in C([0,\tau],X_0)\) with \(u_f(0)=0\). Suppose also that \(\sup_{t\in[-n,n]}\|B(t)\|_{\mathcal{L}(X_0,X)}<+\infty\) for all integer \(n\geq1\).
In \(X\) consider the differential non-homogeneous equation
\[
\frac{du(t)}{dt}=(A+B(t))u(t)+f(t),\quad t\geq t_0,\quad u(t_0)=x_0\in X_0.
\]
Then for each \(t_0\), \(x_0\in X_0\) and \(f\in C([t_0,+\infty],X)\) the equation has a unique mild solution
\[
u(t)=U_B(t,t_0)x_0+\lim_{\lambda\to+\infty} \int_{t_0}^tU_{B}(t,s)\lambda(\lambda I-A)^{-1}f(s)ds.
\]
Here \(U_B(t,s)\) is an evolution family for the related homogeneous equation.
If in addition \(\sup_{\mathbb{R}}\|B(t)\|_{\mathcal{L}(X_0,X)}<+\infty\), then the evolution \(U_B\) has an exponantial dichotomy. A related representation for the solution \(u\) is obtained.
Applications to PDEs with non-local conditions are given.
Reviewer: Nikita V. Artamonov (Moskva)Finite-time synchronization of nonlinear fractional chaotic systems with stochastic actuator faultshttps://zbmath.org/1496.340982022-11-17T18:59:28.764376Z"Sweetha, S."https://zbmath.org/authors/?q=ai:sweetha.s"Sakthivel, R."https://zbmath.org/authors/?q=ai:sakthivel.rathinasamy"Harshavarthini, S."https://zbmath.org/authors/?q=ai:harshavarthini.sSummary: This paper states with the objective of investigating the synchronization problem of nonlinear delayed fractional-order chaotic systems in conjunction with quantization, actuator faults, randomly occurring parametric uncertainties and exogenous disturbances. Moreover, the actuator faults are randomly occurring at any instant of time. The resultant random variables obeying Bernoulli distribution are introduced to account stochastic behavior. In spite of ensuring the robust performance, the finite-time synchronization of the addressed system is achieved and satisfies passive disturbance attenuation level by developing robust quantized stochastic reliable control protocol. As a consequence, the fast synchronization of the considered system is ensured in a finite time period. Owing to this perspective, the desired controller gain matrices can be obtained by solving developed linear matrix inequality. Further, the effectiveness of the theoretical result developed in this paper is validated via numerical simulation.Rigorous verification of Hopf bifurcations in functional differential equations of mixed typehttps://zbmath.org/1496.341032022-11-17T18:59:28.764376Z"Church, Kevin E. M."https://zbmath.org/authors/?q=ai:church.kevin-e-m"Lessard, Jean-Philippe"https://zbmath.org/authors/?q=ai:lessard.jean-philippeThe paper is concerned with the development of a numerical method to prove the existence of Hopf bifurcations in simple functional differential equations of mixed type, otherwise known as advance-delay equations, or sometimes as forward-backward equations. The question of interest is to consider the properties of the eigenvalues, and use is made of the Newton-Kantorovich theorem. The authors prove the existence of Hopf bifurcations in the Lasota-Wazewska-Czyzewska model and the existence of periodic traveling waves in the Fisher equation with nonlocal reaction. The overall objective of the work is to `develop numerical methods which can lead to computer-assisted proofs of existence of different type of dynamical objects arising in the study of differential equations.' Consequently, there is a section that discusses computer-assisted proofs of some of the theorems presented and links are provided to the relevant code.
Reviewer: Neville Ford (Chester)Spectral instability of peakons in the \(b\)-family of the Camassa-Holm equationshttps://zbmath.org/1496.350572022-11-17T18:59:28.764376Z"Lafortune, Stéphane"https://zbmath.org/authors/?q=ai:lafortune.stephane"Pelinovsky, Dmitry E."https://zbmath.org/authors/?q=ai:pelinovsky.dmitry-eDynamical behavior in a reaction-diffusion system with prey-taxishttps://zbmath.org/1496.350582022-11-17T18:59:28.764376Z"Song, Yingwei"https://zbmath.org/authors/?q=ai:song.yingwei"Zhang, Tie"https://zbmath.org/authors/?q=ai:zhang.tie"Li, Jinpeng"https://zbmath.org/authors/?q=ai:li.jinpengSummary: In this article, we study a diffusive predator-prey system with prey-taxis under homogeneous Neumann boundary conditions. We establish the existence and boundedness of nonnegative global solutions. Through comparison with the system without prey-taxis, we find that the positive constant equilibrium remains stable for positive prey-taxis, while negative prey-taxis makes it unstable.Asymptotic \(H^2\) regularity of a stochastic reaction-diffusion equationhttps://zbmath.org/1496.350732022-11-17T18:59:28.764376Z"Cui, Hongyong"https://zbmath.org/authors/?q=ai:cui.hongyong"Li, Yangrong"https://zbmath.org/authors/?q=ai:li.yangrongSummary: In this paper we study the asymptotic dynamics for the weak solutions of the following stochastic reaction-diffusion equation defined on a bounded smooth domain \(\mathcal{O} \subset\mathbb{R}^N\), \(N \leqslant 3\), with Dirichlet boundary condition:
\[
\mathrm{d} u +(-\Delta u + u^3 -\beta u) \mathrm{d}t = g(x) \mathrm{d}t+h(x) \mathrm{d} W, \quad u|_{t = 0} = u_0\in H: = L^2 (\mathcal{O}),
\]
where \(\beta >0\), \(g\in H\), and \(W\) a scalar and two-sided Wiener process with a regular perturbation intensity \(h\). We first construct an \(H^2\) tempered random absorbing set of the system, and then prove an \((H,H^2)\)-smoothing property and conclude that the random attractor of the system is in fact a finite-dimensional tempered random set in \(H^2\) and pullback attracts tempered random sets in \(H\) under the topology of \(H^2\). The main technique we shall employ is comparing the regularity of the stochastic equation to that of the corresponding deterministic equation for which the asymptotic \(H^2\) regularity is already known.Multi-valued random dynamics of stochastic wave equations with infinite delayshttps://zbmath.org/1496.350932022-11-17T18:59:28.764376Z"Wang, Jingyu"https://zbmath.org/authors/?q=ai:wang.jingyu"Wang, Yejuan"https://zbmath.org/authors/?q=ai:wang.yejuan"Caraballo, Tomás"https://zbmath.org/authors/?q=ai:caraballo.tomasSummary: This paper is devoted to the asymptotic behavior of solutions to a non-autonomous stochastic wave equation with infinite delays and additive white noise. The nonlinear terms of the equation are not expected to be Lipschitz continuous, but only satisfy continuity assumptions along with growth conditions, under which the uniqueness of the solutions may not hold. Using the theory of multi-valued non-autonomous random dynamical systems, we prove the existence and measurability of a compact global pullback attractor.Sufficient conditions for the continuity of inertial manifolds for singularly perturbed problemshttps://zbmath.org/1496.350972022-11-17T18:59:28.764376Z"Bonfoh, Ahmed"https://zbmath.org/authors/?q=ai:bonfoh.ahmed-sSummary: We consider a nonlinear evolution equation in the form
\[
\mathrm{U_t + A_\varepsilon U + N_\varepsilon G_\varepsilon (U)} = 0,
\tag{\(\mathrm{E}_{\varepsilon}\)}
\]
together with its singular limit problem as \(\varepsilon\to 0\)
\[
U_t+ A U+ \mathrm{N} G(U) = 0,
\tag{E}
\]
where \(\varepsilon\in (0,1]\) (possibly \(\varepsilon = 0\)), \(\mathrm{A}_\varepsilon\) and \(\mathrm{A}\) are linear closed (possibly) unbounded operators, \(\mathrm{N}_\varepsilon\) and \(\mathrm{N}\) are linear (possibly) unbounded operators, \(\mathrm{G}_\varepsilon\) and \(\mathrm{G}\) are nonlinear functions. We give sufficient conditions on \(\mathrm{A}_\varepsilon\), \(\mathrm{N}_\varepsilon\) and \(\mathrm{G}_\varepsilon\) (and also \(\mathrm{A}, \mathrm{N}\) and \(\mathrm{G})\) that guarantee not only the existence of an inertial manifold of dimension independent of \(\varepsilon\) for \((E_\varepsilon)\) on a Banach space \(\mathcal{H}\), but also the Hölder continuity, lower and upper semicontinuity at \(\varepsilon = 0\) of the intersection of the inertial manifold with a bounded absorbing set. Applications to higher order viscous Cahn-Hilliard-Oono equations, the hyperbolic type equations and the phase-field systems, subject to either Neumann or Dirichlet boundary conditions (BC) (in which case \(\Omega\subset \mathbb{R}^d\) is a bounded domain with smooth boundary) or periodic BC (in which case \(\Omega = \Pi_{i = 1}^d (0,L_i), \, L_i>0)\), \(d = 1\), 2 or 3, are considered. These three classes of dissipative equations read
\[
\phi_t+N(\varepsilon \phi_t+N^{\alpha+1} \phi +N\phi + g(\phi))+\sigma\phi = 0,\quad\alpha\in\mathbb{N},\\
\tag{\(\mathrm{P}_\varepsilon\)}
\]
\[
\varepsilon \phi_{tt}+\phi_t+N^\alpha(N \phi + g(\phi))+ \sigma\phi = 0,\quad\alpha = 0, 1,\\
\tag{\(\mathrm{H}_\varepsilon\)}
\]
and
\[
\begin{cases}
\phi_t+N^\alpha (N \phi + g(\phi)-u)+\sigma\phi = 0,\\
\varepsilon u_t+\phi_t+N u = 0,
\end{cases}
\alpha = 0, 1
\tag{\(\mathrm{S}_\varepsilon\)}
\]
respectively, where \(\sigma\ge 0\) and the Laplace operator is defined as
\[
N = -\Delta:\mathscr{D}(N) = \{\psi\in H^2(\Omega),\,\psi \text{ subject to the BC}\}\to \dot L^2(\Omega) \text{ or }L^2(\Omega).
\]
We assume that, for a given real number \(\mathfrak{c}_1>0,\) there exists a positive integer \(n = n(\mathfrak{c}_1)\) such that \(\lambda_{n+1}-\lambda_n>\mathfrak{c}_1\), where \(\{\lambda_k\}_{k\in\mathbb{N}^*}\) are the eigenvalues of \(N\). There exists a real number \(\mathscr{M}>0\) such that the nonlinear function \(g: V_j\to V_j\) satisfies the conditions \(\|g(\psi)\|_j\le\mathscr{M}\) and \(\|g(\psi)-g(\varphi)\|_{V_j}\le\mathscr{M}\|\psi-\varphi\|_{V_j}\), \(\forall\psi\), \(\varphi\in V_j\), where \(V_j = \mathscr{D}(N^{j/2})\), \(j = 1\) for Problems \((P_\epsilon)\) and \((S_\epsilon)\) and \(j = 0\), \(2\alpha\) for Problem \((H_\epsilon)\). We further require \(g\in{\mathcal C}^1(V_j, V_j)\), \(\|g'(\psi)\varphi\|_j\le\mathscr{M}\|\varphi\|_j\) for Problems \((H_\epsilon)\) and \((S_\epsilon)\).Qualitative and quantitative analysis of nonlinear dynamics by the complete discrimination system for polynomial methodhttps://zbmath.org/1496.351462022-11-17T18:59:28.764376Z"Kai, Yue"https://zbmath.org/authors/?q=ai:kai.yue"Chen, Shuangqing"https://zbmath.org/authors/?q=ai:chen.shuangqing"Zheng, Bailin"https://zbmath.org/authors/?q=ai:zheng.bailin"Zhang, Kai"https://zbmath.org/authors/?q=ai:zhang.kai"Yang, Nan"https://zbmath.org/authors/?q=ai:yang.nan"Xu, Wenlong"https://zbmath.org/authors/?q=ai:xu.wenlongSummary: We show that via the complete discrimination system for polynomial method, the bifurcation, critical condition and topological properties of the nonlinear dynamics can be seen very easily. Concrete example of perturbed Gardner's equation with high order dispersion also verifies our conclusion. The results indicate that the complete discrimination system for polynomial method can not only be used to get quantitative results such as the classification of the traveling wave solutions, but also to conduct qualitative analysis for the nonlinear differential equations.Dynamics of solitons with periodic loops intrinsic localized modes and modulational instability in a quantum 2D nonlinear square Klein-Gordon chainhttps://zbmath.org/1496.351482022-11-17T18:59:28.764376Z"Djoufack, Z. I."https://zbmath.org/authors/?q=ai:djoufack.z-i"Kenmogne, Fabien"https://zbmath.org/authors/?q=ai:kenmogne.fabien"Nguenang, J. P."https://zbmath.org/authors/?q=ai:nguenang.jean-pierre"Kenfack-Jiotsa, A."https://zbmath.org/authors/?q=ai:kenfack-jiotsa.aurelienSummary: We investigate the effects of dispersion and nonlinearity interactions in view of generation of nonlinear intrinsic localized modes (ILMs) and modulational instability (MI) of a quantum 2D Klein-Gordon chain. For this, using Glauber's coherent state representation and multiple-scale method, we find that the system supports nonlinear excitations referred to bright, dark radial symmetric and bright, dark bilateral symmetric line non-traveling and traveling ILMs. We also find that moving bright and radial solution are evolving like localized periodic loops, which is very interesting in this work. By means of linear stability approach, the instability criteria, threshold amplitude, instability areas showing the generation of bright ILMs predicted to be unstable and growth rate of unstable modes in the system are obtained. Numerical investigations are performed to support the analytical analysis, and an excellent agreement with the analytical results has been found.The initial-boundary value problems of the new two-component generalized Sasa-Satsuma equation with a \(4\times 4\) matrix Lax pairhttps://zbmath.org/1496.351702022-11-17T18:59:28.764376Z"Hu, Beibei"https://zbmath.org/authors/?q=ai:hu.beibei"Zhang, Ling"https://zbmath.org/authors/?q=ai:zhang.ling"Lin, Ji"https://zbmath.org/authors/?q=ai:lin.jiSummary: In this paper, we consider a new two-component Sasa-Satsuma equation, which can simulate the propagation and interaction of ultrashort pulses and describe the propagation of femtosecond pulses in optical fibers. The unified transformation method is used to construct a \(4\times 4\) matrix Riemann-Hilbert problem. Then, the solution of the initial-boundary value problems for the new two-component generalized Sasa-Satsuma equation well can be obtained by solving this matrix Riemann-Hilbert problem. In addition, we obtain that the spectral functions satisfy an important global relation.Bounded weak solutions to the thin film Muskat problem via an infinite family of Liapunov functionalshttps://zbmath.org/1496.352392022-11-17T18:59:28.764376Z"Laurençot, Philippe"https://zbmath.org/authors/?q=ai:laurencot.philippe"Matioc, Bogdan-Vasile"https://zbmath.org/authors/?q=ai:matioc.bogdan-vasileSummary: A countably infinite family of Liapunov functionals is constructed for the thin film Muskat problem, which is a second-order degenerate parabolic system featuring cross-diffusion. More precisely, for each \(n\geq 2\) we construct an homogeneous polynomial of degree \(n\), which is convex on \([0,\infty )^2\), with the property that its integral is a Liapunov functional for the problem. Existence of global bounded non-negative weak solutions is then shown in one space dimension.Existence of weak solutions to the two-dimensional incompressible Euler equations in the presence of sources and sinkshttps://zbmath.org/1496.352882022-11-17T18:59:28.764376Z"Bravin, Marco"https://zbmath.org/authors/?q=ai:bravin.marco"Sueur, Franck"https://zbmath.org/authors/?q=ai:sueur.franckSummary: A classical model for sources and sinks in a two-dimensional perfect incompressible fluid occupying a bounded domain dates back to \textit{V. I. Yudovich}'s paper [``A two-dimensional non-stationary problem on the flow of an ideal incompressible fluid through a given region'', Mat. Sb. 64(106), 562--588 (1964)]. In this model, on the one hand, the normal component of the fluid velocity is prescribed on the boundary and is nonzero on an open subset of the boundary, corresponding either to sources (where the flow is incoming) or to sinks (where the flow is outgoing). On the other hand the vorticity of the fluid which is entering into the domain from the sources is prescribed.
In this paper, we investigate the existence of weak solutions to this system by relying on \textit{a priori} bounds of the vorticity, which satisfies a transport equation associated with the fluid velocity vector field. Our results cover the case where the vorticity has a \(L^p\) integrability in space, with \(p\) in \([1,+\infty]\), and prove the existence of solutions obtained by compactness methods from viscous approximations. More precisely we prove the existence of solutions which satisfy the vorticity equation in the distributional sense in the case where \(p > \frac{4}{3}\), in the renormalized sense in the case where \(p > 1\), and in a symmetrized sense in the case where \(p =1\).On the non-chiral intermediate long wave equation. II: Periodic casehttps://zbmath.org/1496.353072022-11-17T18:59:28.764376Z"Berntson, Bjorn K."https://zbmath.org/authors/?q=ai:berntson.bjorn-k"Langmann, Edwin"https://zbmath.org/authors/?q=ai:langmann.edwin"Lenells, Jonatan"https://zbmath.org/authors/?q=ai:lenells.jonatanOn the non-chiral intermediate long wave equationhttps://zbmath.org/1496.353082022-11-17T18:59:28.764376Z"Berntson, Bjorn K."https://zbmath.org/authors/?q=ai:berntson.bjorn-k"Langmann, Edwin"https://zbmath.org/authors/?q=ai:langmann.edwin"Lenells, Jonatan"https://zbmath.org/authors/?q=ai:lenells.jonatanReducibility of quasi-periodic linear KdV equationhttps://zbmath.org/1496.353422022-11-17T18:59:28.764376Z"Geng, Jiansheng"https://zbmath.org/authors/?q=ai:geng.jiansheng"Ren, Xiufang"https://zbmath.org/authors/?q=ai:ren.xiufang"Yi, Yingfei"https://zbmath.org/authors/?q=ai:yi.yingfeiSummary: In this paper, we consider the following one-dimensional, quasi-periodically forced, linear KdV equations
\[
u_t+(1+ a_1(\omega t)) u_{xxx}+ a_2(\omega t,x) u_{xx}+ a_3(\omega t,x)u_x +a_4(\omega t,x)u=0
\]
under the periodic boundary condition \(u(t,x+2\pi )=u(t,x)\), where \(\omega \)'s are frequency vectors lying in a bounded closed region \(\Pi_*\subset{\mathbb{R}}^b\) for some \(b>1\), \(a_1:{\mathbb{T}}^b\rightarrow{\mathbb{R}}\), \(a_i: {\mathbb{T}}^b\times{\mathbb{T}}\rightarrow{\mathbb{R}}\), \(i=2,3,4\), are real analytic, bounded from the above by a small parameter \(\epsilon_*>0\) under a suitable norm, and \(a_1\), \(a_3\) are even, \(a_2\), \(a_4\) are odd. Under the real analyticity assumption of the coefficients, we re-visit a result of \textit{P. Baldi} et al. [Math. Ann. 359, No. 1--2, 471--536 (2014; Zbl 1350.37076)] by showing that there exists a Cantor set \(\Pi_{\epsilon_*}\subset \Pi_*\) with \(|\Pi_*\setminus \Pi_{\epsilon_*}|=O(\epsilon_*^{\frac{1}{100}})\) such that for each \(\omega \in \Pi_{\epsilon_*} \), the corresponding equation is smoothly reducible to a constant-coefficient one. Our main result removes a condition originally assumed in [loc. cit.] and thus can yield general existence and linear stability results for quasi-periodic solutions of a reversible, quasi-periodically forced, nonlinear KdV equation with much less restrictions on the nonlinearity. The proof of our reducibility result makes use of some special structures of the equations and is based on a refined Kuksin's estimate for solutions of homological equations with variable coefficients.\(M\)-lump solution, semirational solution, and self-consistent source extension of a novel \(( 2 + 1)\)-dimensional KdV equationhttps://zbmath.org/1496.353442022-11-17T18:59:28.764376Z"Hai, Rihan"https://zbmath.org/authors/?q=ai:hai.rihan"Gegen, Hasi"https://zbmath.org/authors/?q=ai:gegen.hasi(no abstract)A higher dispersion KdV equation on the half-linehttps://zbmath.org/1496.353452022-11-17T18:59:28.764376Z"Himonas, A. Alexanddrou"https://zbmath.org/authors/?q=ai:himonas.a-alexanddrou"Yan, Fangchi"https://zbmath.org/authors/?q=ai:yan.fangchiSummary: The initial-boundary value problem (ibvp) for the \(m\)-th order dispersion Korteweg-de Vries (KdV) equation on the half-line with rough data and solution in restricted Bourgain spaces is studied using the Fokas Unified Transform Method (UTM). Thus, this work advances the implementation of the Fokas method, used earlier for the KdV on the half-line with smooth data and solution in the classical Hadamard space, consisting of function that are continuous in time and Sobolev in the spatial variable, to the more general Bourgain spaces framework of dispersive equations with rough data on the half-line. The spaces needed and the estimates required arise at the linear level and in particular in the estimation of the linear pure ibvp, which has forcing and initial data zero but non-zero boundary data. Using the iteration map defined by the Fokas solution formula of the forced linear ibvp in combination with the bilinear estimates in modified Bourgain spaces introduced by this map, well-posedness of the nonlinear ibvp is established for rough initial and boundary data belonging in Sobolev spaces.On the focusing generalized Hartree equationhttps://zbmath.org/1496.353492022-11-17T18:59:28.764376Z"Arora, Anudeep Kumar"https://zbmath.org/authors/?q=ai:arora.anudeep-kumar"Roudenko, Svetlana"https://zbmath.org/authors/?q=ai:roudenko.svetlana"Yang, Kai"https://zbmath.org/authors/?q=ai:yang.kaiSummary: In this paper we give a review of the recent progress on the focusing generalized Hartree equation, which is a nonlinear Schrodinger-type equation with the nonlocal nonlinearity, expressed as a convolution with the Riesz potential. We describe the local well-posedness in \(H^1\) and \(H^s\) settings, discuss the extension to the global existence and scattering, or finite time blow-up. We point out different techniques used to obtain the above results, and then show the numerical investigations of the stable blow-up in the \(L^2\)-critical setting. We finish by showing known analytical results about the stable blow-up dynamics in the \(L^2\)-critical setting.Global well-posedness for the derivative nonlinear Schrödinger equationhttps://zbmath.org/1496.353502022-11-17T18:59:28.764376Z"Bahouri, Hajer"https://zbmath.org/authors/?q=ai:bahouri.hajer"Perelman, Galina"https://zbmath.org/authors/?q=ai:perelman.galinaSummary: This paper is dedicated to the study of the derivative nonlinear Schrödinger equation on the real line. The local well-posedness of this equation in the Sobolev spaces \(H^s(\mathbb{R})\) is well understood since a couple of decades, while the global well-posedness is not completely settled. For the latter issue, the best known results up-to-date concern either Cauchy data in \(H^{\frac{1}{2}}(\mathbb{R})\) with mass strictly less than \(4\pi\) or general initial conditions in the weighted Sobolev space \(H^{2,2}(\mathbb{R})\). In this article, we prove that the derivative nonlinear Schrödinger equation is globally well-posed for general Cauchy data in \(H^{\frac{1}{2}}(\mathbb{R})\) and that furthermore the \(H^{\frac{1}{2}}\) norm of the solutions remains globally bounded in time. The proof is achieved by combining the profile decomposition techniques with the integrability structure of the equation.Small multi solitons in a double power nonlinear Schrödinger equationhttps://zbmath.org/1496.353512022-11-17T18:59:28.764376Z"Bai, Mengxue"https://zbmath.org/authors/?q=ai:bai.mengxue"Zhang, Jian"https://zbmath.org/authors/?q=ai:zhang.jianSummary: We consider a nonlinear Schrödinger equation with double power nonlinearity
\[
i \varphi_t + \Delta \varphi + | \varphi |^{\frac{ 4}{ d}} \varphi + | \varphi |^{p - 1} \varphi = 0, \quad (t, x) \in \mathbb{R} \times \mathbb{R}^d, \, d \geq 2 \quad (N L S)
\]
for \(1 < p < 1 + \frac{ 4}{ d}\). Let \(q = q(x)\) be the unique positive solution of the nonlinear elliptic equation
\[
\Delta u + | u |^{\frac{ 4}{ d}} u - u = 0, \quad u \in H^1 (\mathbb{R}^d)
\]
and \(Q_\omega\) be the unique positive solution of the nonlinear elliptic equation
\[
\Delta u + | u |^{\frac{ 4}{ d}} u + | u |^{p - 1} u - \omega u = 0, \quad 1 < p < 1 + \frac{ 4}{ d}, \, u \in H^1 (\mathbb{R}^d).
\]
Then we prove that
\[
\omega_q = s u p \{ \omega | \| Q_\omega \|_{L^2 (\mathbb{R}^d)} < \| q \|_{L^2 (\mathbb{R}^d)} \} > 0.
\]
Furthermore for \(\omega \in(0, \omega_q)\), the soliton \(e^{i \omega t} Q_\omega(x)\) of \((N L S)\) is orbitally stable. Moreover for \(K \geq 2\) and \(k = 1, 2, \cdot \cdot \cdot, K\), taking \(\omega_k \in(0, \omega_q)\), \(\gamma_k \in \mathbb{R}\), \(x_k \in \mathbb{R}^d\), \(v_k \in \mathbb{R}^d\) with \(v_k \neq v_{k^\prime}\) to \(k \neq k^\prime\) and
\[
(R_k (t, x) = Q_{\omega_k} (x - x_k - v_k t) e^{i (\frac{ 1}{ 2} v_k x - \frac{ 1}{ 4} | v_k |^2 t + \omega_k t + \gamma_k)}
\]
with \((t, x) \in \mathbb{R} \times \mathbb{R}^d\) and \(\sum_{k = 1}^K \| R_k (t) \|_{L^2 (\mathbb{R}^d)} < \| q \|_{L^2 (\mathbb{R}^d)}\), there exists a solution \(\varphi(t, x)\) of \((N L S)\) such that
\[
\lim_{t \to + \infty} | | \varphi(t, \cdot) - \sum_{k = 1}^K R_k(t, \cdot) | |_{H^1 (\mathbb{R}^d)} = 0.
\]
This \(\varphi(t, x)\) is called the small multi soliton of \((N L S)\).Arnold-Liouville theorem for integrable PDEs: a case study of the focusing NLS equationhttps://zbmath.org/1496.353622022-11-17T18:59:28.764376Z"Kappeler, T."https://zbmath.org/authors/?q=ai:kappeler.thomas"Topalov, P."https://zbmath.org/authors/?q=ai:topalov.peter-jSymplectic non-squeezing for the cubic NLS on the linehttps://zbmath.org/1496.353632022-11-17T18:59:28.764376Z"Killip, Rowan"https://zbmath.org/authors/?q=ai:killip.rowan"Visan, Monica"https://zbmath.org/authors/?q=ai:visan.monica"Zhang, Xiaoyi"https://zbmath.org/authors/?q=ai:zhang.xiaoyiSummary: We prove symplectic non-squeezing for the cubic nonlinear Schrödinger equation on the line via finite-dimensional approximation.Attosecond soliton switching through the interactions of two and three solitons in an inhomogeneous fiberhttps://zbmath.org/1496.353692022-11-17T18:59:28.764376Z"Veni, S. Saravana"https://zbmath.org/authors/?q=ai:veni.s-saravana"Rajan, M. S. Mani"https://zbmath.org/authors/?q=ai:rajan.m-s-maniSummary: We obtain the exact two and three soliton solutions for higher order NLS equation with variable coefficients using Darboux transformation method with some algebraic manipulations based on constructed Lax Pair. For the first time, switching characteristics of two and three solitons in the attosecond regime via inelastic soliton interactions are discussed through some graphical illustrations. Additionally, effects of the inhomogeneous coefficients on propagation features of solitons are analyzed graphically. Our results have certain applications in the construction of optical switching and soliton management in optical communication systems.Defocusing NLS equation with nonzero background: large-time asymptotics in a solitonless regionhttps://zbmath.org/1496.353712022-11-17T18:59:28.764376Z"Wang, Zhaoyu"https://zbmath.org/authors/?q=ai:wang.zhaoyu"Fan, Engui"https://zbmath.org/authors/?q=ai:fan.enguiSummary: We consider the Cauchy problem for the defocusing Schrödinger (NLS) equation with a nonzero background
\[
\begin{aligned}
&i q_t + q_{x x} - 2(| q |^2 - 1) q = 0,\\
&q(x, 0) = q_0(x), \quad \lim_{x \to \pm \infty} q_0(x) = \pm 1.
\end{aligned}
\]
Recently, for the space-time region \(| x /(2 t) | < 1\) which is a solitonic region without stationary phase points on the jump contour, \textit{S. Cuccagna} and \textit{R. Jenkins} [Commun. Math. Phys. 343, No. 3, 921--969 (2016; Zbl 1342.35326)] presented the asymptotic stability of the \(N\)-soliton solutions for the NLS equation by using the \(\overline{\partial}\) generalization of the Deift-Zhou nonlinear steepest descent method. Their large-time asymptotic expansion takes the form
\[
q (x, t) = T (\infty)^{- 2} q^{s o l, N} (x, t) + \mathcal{O} (t^{- 1})
\tag{0.1}
\]
whose leading term is N-soliton and the second term \(\mathcal{O}(t^{- 1})\) is a residual error from a \(\overline{\partial}\)-equation. In this paper, we are interested in the large-time asymptotics in the space-time region \(| x /(2 t) | > 1\) which is outside the soliton region, but there will be two stationary points appearing on the jump contour \(\mathbb{R}\). We found an asymptotic expansion that is different from (0.1)
\[
q (x, t) = e^{- i \alpha (\infty)} (1 + t^{- 1 / 2} h (x, t)) + \mathcal{O} (t^{- 3 / 4}),
\tag{0.2}
\]
whose leading term is a nonzero background, the second \(t^{- 1 / 2}\) order term is from the continuous spectrum and the third term \(\mathcal{O}(t^{- 3 / 4})\) is a residual error from a \(\overline{\partial}\)-equation. The above two asymptotic results (0.1) and (0.2) imply that the region \(| x /(2 t) | < 1\) considered by Cuccagna and Jenkins [loc. cit.] is a fast decaying soliton solution region, while the region \(| x /(2 t) | > 1\) considered by us is a slow decaying nonzero background region.Auto-Bäcklund transformations, Lax pair, bilinear forms and bright solitons for an extended \((3+1)\)-dimensional nonlinear Schrödinger equation in an optical fiberhttps://zbmath.org/1496.353742022-11-17T18:59:28.764376Z"Zhou, Tian-Yu"https://zbmath.org/authors/?q=ai:zhou.tian-yu"Tian, Bo"https://zbmath.org/authors/?q=ai:tian.boSummary: In this Letter, we investigate an extended \((3+1)\)-dimensional nonlinear Schrödinger equation in an optical fiber. Via the truncated Laurent expansions, auto-Bäcklund transformations are obtained. According to the Ablowitz-Kaup-Newell-Segur procedure, we derive a Lax pair under certain optical-fiber coefficient constraints. Via the Hirota method, we obtain some bilinear forms, bright two-soliton and bright three-soliton solutions under certain optical-fiber coefficient constraints.On Chien's question to the Hu-Washizu three-field functional and variational principlehttps://zbmath.org/1496.353812022-11-17T18:59:28.764376Z"Sun, Bohua"https://zbmath.org/authors/?q=ai:sun.bohua(no abstract)Long time dynamics of a phase-field model of prostate cancer growth with chemotherapy and antiangiogenic therapy effectshttps://zbmath.org/1496.353942022-11-17T18:59:28.764376Z"Biswas, Tania"https://zbmath.org/authors/?q=ai:biswas.tania"Rocca, Elisabetta"https://zbmath.org/authors/?q=ai:rocca.elisabettaThe authors study a model of prostate cancer growth and chemotherapy, where the critical nutrient controls the advancement of the tumor. It consists of a system of three (coupled) semilinear parabolic equations: an Allen-Cahn-type equation which describes the tumor phase, a reaction-diffusion equation controlling the nutrient properties and an additional reaction-diffusion equation which governs the concentration of prostate-specific antigen in the prostatic tissue. The first equation is accompanied by the homogeneous Dirichlet boundary condition and the other two equations, by the homogeneous Neumann condition. The solution operator associated with the corresponding initial-boundary value problem defines a strongly continuous semigroup on a suitable phase space. The authors show that this semigroup admits a global attractor and then establish a long time behaviour result for the considered initial-boundary value problem.
Reviewer: Catalin Popa (Iaşi)Pattern formation in electrically coupled pacemaker cellshttps://zbmath.org/1496.353962022-11-17T18:59:28.764376Z"Fatoyinbo, Hammed Olawale"https://zbmath.org/authors/?q=ai:fatoyinbo.hammed-olawaleFrom the text: Excitable cells such as neurons, smooth and skeletal muscles, and endocrine glands can generate an electrical signal (action potential) when stimulated by external stimulus. The signal helps in the coordination of different physiological activities including transfer of information among neurons, vasomotion in muscle cells and secretion of hormones. Experimental and theoretical studies have shown that under certain conditions, the electrical signal can be produced spontaneously in excitable cells. Such behaviour is referred to as \textit{pacemaker dynamics}. This research aims to study the pacemaker electrical activity in a population of electrically coupled smooth muscles.Threshold dynamics of a reaction-diffusion equation model for cholera transmission with waning vaccine-induced immunity and seasonalityhttps://zbmath.org/1496.354022022-11-17T18:59:28.764376Z"Zhou, Mengchen"https://zbmath.org/authors/?q=ai:zhou.mengchen"Wang, Wei"https://zbmath.org/authors/?q=ai:wang.wei.21"Fan, Xiaoting"https://zbmath.org/authors/?q=ai:fan.xiaoting"Zhang, Tonghua"https://zbmath.org/authors/?q=ai:zhang.tonghuaSummary: Cholera is an acute intestinal infectious disease caused by the bacterium \textit{Vibrio cholerae}. To explore the multiple effects of spatial mobility, spatial heterogeneity and the seasonality on the transmission of cholera, we propose a time periodic reaction-diffusion equation model with latent period. Based on the basic reproduction number \(\mathscr{R}_0\), we establish a threshold-type result. And in the case where all the parameters are constants and \(\mathscr{R}_0 >1\), we show the global attractivity of the endemic steady state by constructing Lyapunov functionals. Finally, we perform some numerical simulations. Our simulations show that (i) increasing the vaccination rate of susceptible individuals and vaccine protective efficacy can reduce the transmission risk \(\mathscr{R}_0\); (ii) decreasing the transmission coefficient of contact with infected individuals, the transmission coefficient of contact with hyperinfectious vibrios and the transmission coefficient of contact with hypoinfectious vibrios can reduce the transmission risk \(\mathscr{R}_0\); (iii) it is possible to underestimate the transmission risk \(\mathscr{R}_0\) in the periodic system if the spatial averaged system is used, based on some experimental data.Propagation of stochastic travelling waves of cooperative systems with noisehttps://zbmath.org/1496.354692022-11-17T18:59:28.764376Z"Wen, Hao"https://zbmath.org/authors/?q=ai:wen.hao"Huang, Jianhua"https://zbmath.org/authors/?q=ai:huang.jianhua"Li, Yuhong"https://zbmath.org/authors/?q=ai:li.yuhongSummary: We consider the cooperative system driven by a multiplicative Itô type white noise. The existence and their approximations of the travelling wave solutions are proven. With a moderately strong noise, the travelling wave solutions are constricted by choosing a suitable marker of wavefront. Moreover, the stochastic Feynman-Kac formula, sup-solution, sub-solution and equilibrium points of the dynamical system corresponding to the stochastic cooperative system are utilized to estimate the asymptotic wave speed, which is closely related to the white noise.Loosely Bernoulli odometer-based systems whose corresponding circular systems are not loosely Bernoullihttps://zbmath.org/1496.370012022-11-17T18:59:28.764376Z"Gerber, Marlies"https://zbmath.org/authors/?q=ai:gerber.marlies"Kunde, Philipp"https://zbmath.org/authors/?q=ai:kunde.philippIn 1932 \textit{J. von Neumann} [Ann. Math. (2) 33, 587--642 (1932; Zbl 0005.12203)]
suggested a classification scheme for the statistical behavior of differentiable systems. This may be understood as categorizing compact manifold diffeomorphisms up to measure isomorphisms. \textit{M. Foreman} and \textit{B. Weiss} [J. Eur. Math. Soc. (JEMS) 24, No. 8, 2605--2690 (2022; Zbl 07523087)] showed that this is theoretically impossible and used a functor \(\mathcal{F}\) mapping odometer-based systems to establish an anti-classification result for smooth ergodic diffeomorphisms, up to measure isomorphisms. Note that the functor \(\mathcal{F}\) is a map from the set of odometer-based systems to circular systems. \textit{M. Foreman} and \textit{B. Weiss} [J. Mod. Dyn. 15, 345--423 (2019; Zbl 07350263)] proved also that any finite-entropy system with an odometer factor can be modeled as an odometer-based system.
As pointed out in [\textit{M. Foreman} and \textit{B. Weiss}, J. Eur. Math. Soc. (JEMS) 24, No. 8, 2605--2690 (2022; Zbl 07523087)] J. P. Thouvenot asked whether the functor \(\mathcal{F}\) maps loosely Bernoulli automorphisms to loosely Bernoulli automorphisms. Another question that arises is whether \(\mathcal{F}\) preserves the Kakutani equivalence.
In the present paper, the authors give examples answering negatively to both of these questions. The authors construct a uniquely readable uniform odometer-based system \(\mathbb{E}\) of positive entropy that is loosely Bernoulli. Under some given conditions, they provide an example of a positive-entropy odometer-based system \(\mathbb{E}\) that is loosely Bernoulli, but the circular system \(\mathcal{F}(\mathbb{E})\) is not loosely Bernoulli. In Section 5, the authors prove the following:
Theorem. There exist circular coefficients and a loosely Bernoulli odometer-based system \(\mathbb{K}\) of zero measure-theoretic entropy with a uniform and uniquely readable construction sequence such that \(\mathcal{F}(\mathbb{K})\) is not loosely Bernoulli.
Finally, they present an example \(\mathbb{M}\) of a loosely Bernoulli odometer-based system with zero entropy whose corresponding circular system is loosely Bernoulli.
Reviewer: Hasan Akin (Trieste)A family of non-monotonic toral mixing mapshttps://zbmath.org/1496.370022022-11-17T18:59:28.764376Z"Hill, J. Myers"https://zbmath.org/authors/?q=ai:hill.j-myers"Sturman, R."https://zbmath.org/authors/?q=ai:sturman.rob"Wilson, M. C. T."https://zbmath.org/authors/?q=ai:wilson.mark-c-tSummary: We establish the mixing property for a family of Lebesgue measure preserving toral maps composed of two piecewise linear shears, the first of which is non-monotonic. The maps serve as a basic model for the `stretching and folding' action in laminar fluid mixing, in particular flows where boundary conditions give rise to non-monotonic flow profiles. The family can be viewed as the parameter space between two well-known systems, Arnold's Cat Map and a map due to [\textit{S. Cerbelli} and \textit{M. Giona}, Chaos Solitons Fractals 35, No. 1, 13--37 (2008; Zbl 1142.37008)], both of which possess finite Markov partitions and straightforward to prove mixing properties. However, no such finite Markov partitions appear to exist for the present family, so establishing mixing properties requires a different approach. In particular, we follow a scheme of \textit{A. Katok} and \textit{J.-M. Strelcyn} [Invariant manifolds, entropy and billiards; smooth maps with singularities. With the collab. of F. Ledrappier and F. Przytycki. Lecture Notes in Mathematics, 1222. Berlin etc.: Springer-Verlag (1986; Zbl 0658.58001)], proving strong mixing properties with respect to the Lebesgue measure on two open parameter spaces. Finally, we comment on the challenges in extending these mixing windows and the potential for using the same approach to prove mixing properties in similar systems.On the cohomological equation of a linear contractionhttps://zbmath.org/1496.370032022-11-17T18:59:28.764376Z"Leclercq, Régis"https://zbmath.org/authors/?q=ai:leclercq.regis"Zeggar, Abdellatif"https://zbmath.org/authors/?q=ai:zeggar.abdellatifSummary: In this paper, we study the discrete cohomological equation of a contracting linear automorphism \(A\) of the Euclidean space \(\mathbb{R}^d\). More precisely, if \(\delta\) is the cobord operator defined on the Fréchet space \(E = C^l (\mathbb{R}^d)\) (\(0 \leq l \leq \infty \)) by: \( \delta(h) = h - h \circ A\), we show that:
\begin{itemize}
\item If \(E = C^0(\mathbb{R}^d)\), the range \(\delta(E)\) of \(\delta\) has infinite codimension and its closure is the hyperplane \(E_0\) consisting of the elements of \(E\) vanishing at 0. Consequently, \(H^1 (A, E)\) is infinite dimensional non Hausdorff topological vector space and then the automorphism \(A\) is not cohomologically \(C^0\)-stable.
\item If \(E = C^l(\mathbb{R}^d)\), with \(1 \leq l \leq \infty\), the space \(\delta(E)\) coincides with the closed hyperplane \(E_0\). Consequently, the cohomology space \(H^1 (A, E)\) is of dimension 1 and the automorphism \(A\) is cohomologically \(C^l\)-stable.
\end{itemize}Optimal time averages in non-autonomous nonlinear dynamical systemshttps://zbmath.org/1496.370042022-11-17T18:59:28.764376Z"Doering, Charles R."https://zbmath.org/authors/?q=ai:doering.charles-r"McMillan, Andrew N."https://zbmath.org/authors/?q=ai:mcmillan.andrew-nSummary: We discuss an auxiliary function method that allows the computation of extremal long-time averages of functions of dynamical variables in autonomous, nonlinear ordinary differential equations via convex optimization. For dynamical systems defined by autonomous polynomial vector fields, it is operationally realized as a semidefinite program utilizing sum of squares technology. In this contribution, we review the method and extend it for application to periodically driven, non-autonomous nonlinear vector fields involving trigonometric functions of the dynamical variables. The damped driven Duffing oscillator and periodically driven pendulum are presented as examples to illustrate the auxiliary function method's utility.Entropy as an integral operatorhttps://zbmath.org/1496.370052022-11-17T18:59:28.764376Z"Rahimi, Mehdi"https://zbmath.org/authors/?q=ai:rahimi.mehdiSummary: In this paper, we introduce the concept of entropy kernel operator for compact dynamical systems of finite Kolmogorov entropy. It is a compact positive operator on a Hilbert space. Then we state the Kolmogorov entropy in terms of the eigenvalues of the entropy kernel operator.Optimal stochastic forcings for sensitivity analysis of linear dynamical systemshttps://zbmath.org/1496.370062022-11-17T18:59:28.764376Z"Nechepurenko, Yuri M."https://zbmath.org/authors/?q=ai:nechepurenko.yuri-m"Zasko, Grigory V."https://zbmath.org/authors/?q=ai:zasko.grigory-vSummary: The paper is devoted to the construction of optimal stochastic forcings for studying the sensitivity of linear dynamical systems to external perturbations. The optimal forcings are sought to maximize the Schatten norms of the response. As an example, we consider the problem of constructing the optimal stochastic forcing for the linear dynamical system arising from the analysis of large-scale structures in a stratified turbulent Couette flow.Lifting the regionally proximal relation and characterizations of distal extensionshttps://zbmath.org/1496.370072022-11-17T18:59:28.764376Z"Cao, Kai"https://zbmath.org/authors/?q=ai:cao.kai"Dai, Xiongping"https://zbmath.org/authors/?q=ai:dai.xiongpingSummary: We consider a commutative diagram (CD) of flows with discrete phase group T and extensions as follows:
\[
\begin{tikzcd}
{(T,X)} \arrow[rr, "\pi"] \arrow[rd, "\phi"] & & {(T,Y)} \arrow[ld, "\psi" '] \arrow[rd, "\psi_K", dotted] & \\
& {(T,Z)} & & {(T,Y_K)} \arrow[ll, "\psi^{\prime}_K" ', dotted]
\end{tikzcd}
\]
where \((T,Y)\) is minimal.
By \(\operatorname{RP}_{\phi}\) and \(\operatorname{RP}_{\psi}\) we denote the relativized regionally proximal relations in \(X\) and \(Y\), respectively. We mainly prove, among other things, the following:
1. If \(X\) is topologically transitive \(\phi\)-distal, then \(X\) is minimal.
2. \((\pi \times \pi)\operatorname{RP}_{\phi} = \operatorname{RP}_{\psi}\).
3. If \(Y\) is locally \(\psi\)-Bronstein, then \(\operatorname{RP}_{\psi} \circ P\psi\) is an equivalence relation, and, \(\bar{y} \in \operatorname{RP}_{\psi}\) whenever \(\bar{y} \in \operatorname{RP}_{\psi} \circ \operatorname{RP}_{\psi}\) is almost periodic.
4. If \(Y_d\) is the maximal distal extension of \(Z\) below \(Y\) and \(Z^d\) is the universal minimal distal extension of \(Z\), then \(Y \bot_{Y_d} Z^d\).
5. (a) \(Y\) is locally \(\psi\)-Bronstein iff \(F^d < F^{\prime} A\) where \(A\), \(F\), \(F^d\) are respectively
the Ellis groups of \(Y\), \(Z\), \(Z^d\). (b) If \(Y\) is locally \(\psi\)-Bronstein and \(K\) a \(\tau\)-
closed group with \(F^{\prime} A < K < F\), then there is a unique \(Y_K\) which is \(\psi^{\prime}_K\)-equicontinuous and has the Ellis group \(K\).
We also prove the above theorems in the case \(T\) is a semigroup. Moreover, we show the following in minimal semiflows:
6. \(Y\) is \(\psi\)-distal iff \(psi\) has a DE-tower iff there is a least group-like extension \(X\) via \(\phi\) (i.e., \(\phi^{-1} \phi x = \operatorname{Aut}_\phi (T, X)x\) for all \(x \in X\)).
7. \(\psi\) is group-like iff \(\psi\) has a G-tower that consists of group extensions and inverse limits.Equicontinuity and sensitivity in mean formshttps://zbmath.org/1496.370082022-11-17T18:59:28.764376Z"Li, Jie"https://zbmath.org/authors/?q=ai:li.jie"Ye, Xiangdong"https://zbmath.org/authors/?q=ai:ye.xiangdong"Yu, Tao"https://zbmath.org/authors/?q=ai:yu.taoThis paper deals with topological dynamical systems of the form \((X,T)\), where \(X\) is compact metric and \(T\) a continuous and surjective self map of \(X\). Starting from the observation that equicontinuous systems are `easy' the authors generalize further earlier weakenings equicontinuity: mean equicontinuity [\textit{J. Li} et al., Ergodic Theory Dyn. Syst. 35, No. 8, 2587--2612 (2015; Zbl 1356.37016)] and equicontinuity in the mean [\textit{W. Huang} et al., Ergodic Theory Dyn. Syst. 41, No. 2, 494--533 (2021; Zbl 1456.37015)]; both are equivalent for minimal systems. The further weakenings are called almost mean equicontinuity and almost equicontinuity in the mean; both localize their condition to a point. Both consider the behavior of the means \[M_n(x,y)=\frac1n\sum_{i=0}^nd(T^ix,T^iy)\] of the initial segments of sequences of distances \(d(T^ix,T^iy)\); the former wants \(\limsup_nM_n(x,y)\) to be small for all \(y\) close to a singled out~\(x\); the latter (stronger) condition wants \(\sup_nM_n(x,y)\) to be small.\par The authors provide an example that shows that almost equicontinuity in the mean is indeed stronger than almost mean equicontinuity, and also of a system that is almost equicontinuous in the mean and exhibits Devaney chaos and uniform positive entropy.\par Next there is a treatment of variants of sensitivity: mean sensitivity and sensitivity in the mean. The authors give sufficient conditions for these properties to hold and show that they are equivalent for minimal systems. The paper ends with an investigation of these notions in the context of measure-preserving systems.
Reviewer: K. P. Hart (Delft)Generalized fractal dimensions of invariant measures of full-shift systems over compact and perfect spaces: generic behaviorhttps://zbmath.org/1496.370092022-11-17T18:59:28.764376Z"Carvalho, Silas L."https://zbmath.org/authors/?q=ai:carvalho.silas-l"Condori, Alexander"https://zbmath.org/authors/?q=ai:condori.alexanderSummary: In this paper, we show that, for topological dynamical systems with a dense set (in the weak topology) of periodic measures, a typical (in Baire's sense) invariant measure has, for each \(q>0\), zero lower \(q\)-generalized fractal dimension. This implies, in particular, that a typical invariant measure has zero upper Hausdorff dimension and zero lower rate of recurrence. Of special interest is the full-shift system \((X,T)\) (where \(X=M^{\mathbb{Z}}\) is endowed with a sub-exponential metric and the alphabet \(M\) is a compact and perfect metric space), for which we show that a typical invariant measure has, for each \(q>1\), infinite upper \(q\)-correlation dimension. Under the same conditions, we show that a typical invariant measure has, for each \(s\in(0,1)\) and each \(q>1\), zero lower \(s\)-generalized and infinite upper \(q\)-generalized dimensions.Towards the Heider balance with a cellular automatonhttps://zbmath.org/1496.370102022-11-17T18:59:28.764376Z"Malarz, Krzysztof"https://zbmath.org/authors/?q=ai:malarz.krzysztof"Wołoszyn, Maciej"https://zbmath.org/authors/?q=ai:woloszyn.maciej"Kułakowski, Krzysztof"https://zbmath.org/authors/?q=ai:kulakowski.krzysztofSummary: The state of structural balance (termed also `Heider balance') of a social network is often discussed in social psychology and sociophysics. In this state, actors at network nodes classify other individuals as enemies or friends. Hence, the network contains two kinds of links: positive and negative. Here a new cellular automaton is designed and investigated, which mimics the time evolution towards the structural balance. The automaton is deterministic and synchronous. The medium is the triangular lattice with some fraction \(f\) of links removed. We analyse the number of unbalanced triads (parameterized as `energy'), the frequencies of balanced triads and correlations between them. The time evolution enhances the local correlations of balanced triads. Local configurations of unbalanced triads are found which are blinking with period of two time steps.Turnpike properties for discrete-time optimal control problems with a Lyapunov functionhttps://zbmath.org/1496.370112022-11-17T18:59:28.764376Z"Zaslavski, Alexander J."https://zbmath.org/authors/?q=ai:zaslavski.alexander-jSummary: We study the turnpike phenomenon for discrete disperse dynamical systems introduced in [Sib. Mat. Zh. 21, No. 4, 136--145 (1980; Zbl 0453.90024)] by \textit{A. M. Rubinov}, which have a prototype in mathematical economics.Parametric topological entropy and differential equations with time-dependent impulseshttps://zbmath.org/1496.370122022-11-17T18:59:28.764376Z"Andres, Jan"https://zbmath.org/authors/?q=ai:andres.janThe author studies a parametric topological entropy of nonautonomous dynamical systems and its application to the theory of impulsive differential equations.
One of the main results is a lower estimation of the parametric topological entropy of sequences of self-maps in terms of the Nielsen numbers, the so-called Ivanov-like inequality.
This result is then applied to the following impulsive system on tori:
\begin{align*}
&x' = F(t,x), \\
&x(t_j+) = I_j(x(t_j^-)),\ j \in {\mathbb N},
\end{align*}
where \(F : [t_0,\infty)\times {\mathbb R} \to {\mathbb R}\) is a Carathéodory function, the impulsive functions \(I_j : {\mathbb R}^n \to {\mathbb R}^n\) are equicontinuous, \(t_{j+1} = t_j + \omega_j\), with \(\omega_j > 0\), \(j \in {\mathbb N}\), provided
\[
F(t,\ldots,x_k,\ldots) = F(t,\ldots,x_k +1,\ldots)
\]
and
\[
I_j(\ldots,x_k,\ldots) = I_j(\ldots,x_k + 1,\ldots)\ (\mathrm{mod}\ 1), \ j \in {\mathbb N}
\]
for \(k = 1,\ldots,n\). Sufficient conditions for the positivity of the parametric topological entropy for this impulsive system on the torus \({\mathbb R}^n / {\mathbb Z}^n\) are given in terms of the associated Poincaré translation operators. In the reviewer's opinion this result has no analogy in the related literature.
Reviewer: Jan Tomeček (Olomouc)An operator theoretical approach to the sequence entropy of dynamical systemshttps://zbmath.org/1496.370132022-11-17T18:59:28.764376Z"Rahimi, M."https://zbmath.org/authors/?q=ai:rahimi.mahboobeh|rahimi.mohammad-reza-ostad|rahimi.mohammad-a|rahimi.mohammad-naqib|rahimi.mohamadtaghi|rahimi.m-y|rahimi.mostafa|rahimi.maryam|rahimi.morteza|rahimi.m-ostad|rahimi.mona|rahimi.mehdi|rahimi.mansour|rahimi.mehran"Mohammadi Anjedani, M."https://zbmath.org/authors/?q=ai:mohammadi-anjedani.mSummary: In this paper, given a sequence of positive integers, we assign a linear operator on a Hilbert space, to any compact topological dynamical system of finite entropy. Then we represent the sequence entropy of the systems in terms of the eigenvalues of the linear operator. In this way, we present a spectral approach to the sequence entropy of the dynamical systems. This spectral representation to the sequence entropy of a system is given for systems with some additional condition called admissibility condition. We also prove that, there exist a large family of dynamical systems, satisfying the admissibility condition.New entropy bounds via uniformly convex functionshttps://zbmath.org/1496.370142022-11-17T18:59:28.764376Z"Sayyari, Yamin"https://zbmath.org/authors/?q=ai:sayyari.yaminSummary: In this paper we give extensions of Jensen's discrete inequality considering the class of uniformly convex functions. We also introduce lower and upper bounds for Jensen's inequality (for uniformly convex functions), and we apply this results in information theory and obtain new and strong bounds for Shannon's entropy of a probability distribution.One-dimensional dynamical systemshttps://zbmath.org/1496.370152022-11-17T18:59:28.764376Z"Efremova, Lyudmila S."https://zbmath.org/authors/?q=ai:efremova.lyudmila-s"Makhrova, Elena N."https://zbmath.org/authors/?q=ai:makhrova.elena-nRemarks on sensitivity and chaos in nonautonomous dynamical systemshttps://zbmath.org/1496.370162022-11-17T18:59:28.764376Z"Salman, Mohammad"https://zbmath.org/authors/?q=ai:salman.mohammad"Das, Ruchi"https://zbmath.org/authors/?q=ai:das.ruchiSummary: In this article, we show that (1) there exists a nonautonomous system which is sensitive with dense minimal points but not syndetically sensitive; (2) there exists a nonautonomous system which is syndetically transitive but not syndetically sensitive; (3) there exists a \(\mathscr{F}_s\)-transitive nonautonomous system with two different minimal subsets and not \(\mathscr{F}_{ts}\)-sensitive. These examples disprove Theorem 2, one part of Theorem 4 and Theorem 10 proved by \textit{N. Li} and \textit{L. Wang} [Int. J. Bifurcation Chaos Appl. Sci. Eng. 30, No. 10, Article ID 2050146, 11 p. (2020; Zbl 1458.37027)].Shadowing as a structural property of the space of dynamical systemshttps://zbmath.org/1496.370172022-11-17T18:59:28.764376Z"Meddaugh, Jonathan"https://zbmath.org/authors/?q=ai:meddaugh.jonathanSummary: We demonstrate that there is a large class of compact metric spaces for which the shadowing property can be characterized as a structural property of the space of dynamical systems. We also demonstrate that, for this class of spaces, in order to determine whether a system has shadowing, it is sufficient to check that \textit{continuously generated} pseudo-orbits can be shadowed.Erratum and addendum to: ``A forward ergodic closing lemma and the entropy conjecture for nonsingular endomorphisms away from tangencies''https://zbmath.org/1496.370182022-11-17T18:59:28.764376Z"Hayashi, Shuhei"https://zbmath.org/authors/?q=ai:hayashi.shuheiSummary: We add a lemma implicitly used in the proof of the forward Ergodic Closing Lemma in our paper [ibid. 40, No. 4, 2285--2313 (2020; Zbl 1441.37026)].On existence of an energy function for \(\Omega\)-stable surface diffeomorphismshttps://zbmath.org/1496.370192022-11-17T18:59:28.764376Z"Barinova, M. K."https://zbmath.org/authors/?q=ai:barinova.m-kSummary: If the chain recurrent set of a diffeomorphism \(f\) given on a closed \(n\)-manifold \(M^n\) is hyperbolic (equivalently, \(f\) is an \(\Omega \)-stable) then it coincides with the closure of the periodic points set \(Per_f\) and its chain recurrent components coincide with the basic sets. Due to \textit{C. Conley} [Isolated invariant sets and the Morse index. Providence, RI: American Mathematical Society (AMS) (1978; Zbl 0397.34056)] for such a diffeomorphism there is a Lyapunov function which is a continuous function \(\varphi:M^n\to\mathbb{R}\) increasing out of the chain recurrent set and a constant on the chain components. But in general a Lyapunov function has critical points out of the chain recurrent set, that is it is not an energy function. In this paper we investigate the problem of the existence of an energy function for diffeomorphisms of a surface. \textit{D. Pixton} [Topology 16, 167--172 (1977; Zbl 0355.58004)]
constructed a Morse energy function for Morse-Smale 2-diffeomorphisms (all basic sets are trivial). It was proved by \textit{V. Z. Grines} and \textit{O. V. Pochinka} [J. Math. Sci., New York 250, No. 4, 537--568 (2020; Zbl 1452.37024); translation from Sovrem. Mat., Fundam. Napravl. 63, No. 2, 191--222 (2017)]
that every \(\Omega \)-stable diffeomorphism \(f:M^2\to M^2\), whose all non-trivial basic sets are attractors or repellers, possesses a smooth energy function which is a Morse function outside non-trivial basic sets. The question about an existence of an energy function for 2-diffeomorphisms with zero-dimensional basic sets was open until now. The main result of this paper is that every \(\Omega \)-stable 2-diffeomorphism with a zero-dimensional non-trivial basic set without pairs of conjugated points does not possess an energy function.On the topological classification of structurally stable diffeomorphisms on 3-manifolds with a 2-dimensional expanding attractorhttps://zbmath.org/1496.370202022-11-17T18:59:28.764376Z"Grines, V. Z."https://zbmath.org/authors/?q=ai:grines.vyacheslav-z"Kruglov, E. V."https://zbmath.org/authors/?q=ai:kruglov.evgenii-valentinovich"Pochinka, O. V."https://zbmath.org/authors/?q=ai:pochinka.olga-vSummary: The paper is devoted to the topological classification of structurally stable diffeomorphisms on three-dimensional manifolds whose non-wandering set contains a 2-dimensional expanding attractor. The first author and \textit{E. V. Zhuzhoma} [Russ. Math. Surv. 34, No. 4, 163--164 (1979; Zbl 0464.58013)]
obtained the topological classification of similar cascades in the dimension greater than 3. They proposed that the embedding of frames of saddle separatrices is tame for dimension equals 3. The tameness of embedding of ones was proved in the paper of the first author et al. [Russ. J. Nonlinear Dyn. 16, No. 4, 595--606 (2020; Zbl 1464.37031)]. This fact allowed to obtain the topological classification of the considering class of diffeomorphisms in dimension 3.Homoclinic orbits and chaos in nonlinear dynamical systems: auxiliary systems methodhttps://zbmath.org/1496.370212022-11-17T18:59:28.764376Z"Grechko, D. A."https://zbmath.org/authors/?q=ai:grechko.d-a"Barabash, N. V."https://zbmath.org/authors/?q=ai:barabash.nikita-v"Belykh, V. N."https://zbmath.org/authors/?q=ai:belykh.vladimir-nikitichSummary: The auxiliary systems method in other words the method of two-dimensional comparison systems plays an essential role in the nonlocal bifurcational dynamical systems theory. In this paper we demonstrate this method in a particular case of 4-dimensional nonlinear dynamical system formed by a coupled Van der Pol-Duffing oscillator and a linear oscillator. For this system, using the auxiliary systems method, a rigorous proof of the existence of a homoclinic orbit of a saddle-focus is carried out for which the Shilnikov condition of chaos is satisfied. The paper is dedicated to the memory of Gennady A. Leonov, who made a significant contribution to the development of methods for the analytical study of dynamical systems.A family of periodic motions to chaos with infinite homoclinic orbits in the Lorenz systemhttps://zbmath.org/1496.370222022-11-17T18:59:28.764376Z"Guo, Siyu"https://zbmath.org/authors/?q=ai:guo.siyu"Luo, Albert C. J."https://zbmath.org/authors/?q=ai:luo.albert-c-jSummary: In this paper, the bifurcation dynamics of a family of \((n_1,1,n_2)\)-periodic motions to chaos with infinite homoclinic orbits \((\min{\{n_1,n_2\}}=1\), \(\max{\{n_1,n_2}\}=1,2,{\ldots},)\) in the Lorenz system is studied through the discrete mapping method. The bifurcation trees of \((n_1,1,n_2)\)-periodic motion to chaos are presented through discrete nodes and harmonic amplitudes. The stability and bifurcations of periodic motions are determined through eigenvalue analysis. The bifurcation scenarios of \((n_1,1,n_2)\)-period-1 motions to chaos are similar each other. The critical values for existence of \((n_1,1,n_2)\)-related periodic motions are determined for saddle-node and period-doubling bifurcations. The homoclinic orbits are associated with unstable periodic motions on the bifurcation trees of the \((n_1,1,n_2)\)-periodic motions to chaos. The homoclinic obits and periodic motions are illustrated from the bifurcation trees of the \((n_1,1,n_2)\)-periodic motions to chaos. The numerical and analytical trajectories of unstable periodic motions were presented for comparison. If the numerical simulations did not have any computational errors, the numerical and analytical solutions of unstable periodic motions in the Lorenz system should be identical. Thus, one observed so-called strange attractors in the Lorenz system through numerical simulations, which are not real strange attractors. This paper is specially dedicated to the good friend and colleague in memory of Gennady A. Leonov for his contributions on nonlinear dynamics.On discrete homoclinic attractors of three-dimensional diffeomorphismshttps://zbmath.org/1496.370232022-11-17T18:59:28.764376Z"Gonchenko, A. S."https://zbmath.org/authors/?q=ai:gonchenko.alexander-s"Gonchenko, S. V."https://zbmath.org/authors/?q=ai:gonchenko.sergey-vSummary: We give a short review on discrete homoclinic attractors. Such strange attractors contain only one saddle fixed point and, hence, entirely its unstable invariant manifold. We discuss the most important peculiarities of these attractors such as their geometric and homoclinic structures, phenomenological scenarios of their appearance, pseudohyperbolic properties etc.On examples of pseudohyperbolic attractors in flows and mapshttps://zbmath.org/1496.370242022-11-17T18:59:28.764376Z"Kainov, M."https://zbmath.org/authors/?q=ai:kainov.m"Kazakov, A."https://zbmath.org/authors/?q=ai:kazakov.a-yu|kazakov.aleksandr-leonidovich|kazakov.alexander-andreevich|kazakov.a-k|kazakov.alexey-o|kazakov.a-v|kazakov.alexander-o|kazakov.anton|kazakov.aleksei-olegovich|kazakov.a-n|kazakov.alexander-yakov|kazakov.a-yaSummary: In this paper we give some known examples of pseudohyperbolic attractors of systems of differential equations and diffeomorphisms and also describe our numerical method for the verification of strange attractors on pseudohyperbolicity. By means of this method we give numerical evidence of the pseudohyperbolicity of the Lorenz attractor in the Lyubimov-Zaks model, the wild spiral attractor of Turaev and Shilnikov type in a four-dimensional Lorenz system, various discrete attractors of Lorenz type in three-dimensional Hénon maps, and the figure-eight attractor in the nonholonomic model of Chaplygin top.Further results on eigenvalues of symmetric decomposable tensors from multilinear dynamical systemshttps://zbmath.org/1496.370252022-11-17T18:59:28.764376Z"Chen, Haibin"https://zbmath.org/authors/?q=ai:chen.haibin"Li, Mengzhen"https://zbmath.org/authors/?q=ai:li.mengzhen"Yan, Hong"https://zbmath.org/authors/?q=ai:yan.hong"Zhou, Guanglu"https://zbmath.org/authors/?q=ai:zhou.guangluThe authors consider a discrete-time version of multilinear dynamical systems associated with symmetric decomposable tensors, including orthogonal and non-orthogonal decomposable tensors as subclasses. Special emphasis is given to the study of the asymptotic stability of such systems. The results are supported by numerical analysis.
Reviewer: P. Shakila Banu (Erode)Dynamical phenomena connected with stability loss of equilibria and periodic trajectorieshttps://zbmath.org/1496.370262022-11-17T18:59:28.764376Z"Neishtadt, Anatolii I."https://zbmath.org/authors/?q=ai:neishtadt.anatolii-i"Treschev, Dmitry V."https://zbmath.org/authors/?q=ai:treshchev.dmitrij-vEvolution maps and symmetryhttps://zbmath.org/1496.370272022-11-17T18:59:28.764376Z"Barreira, Luis"https://zbmath.org/authors/?q=ai:barreira.luis-m"Valls, Claudia"https://zbmath.org/authors/?q=ai:valls.claudiaSummary: We describe in detail how the symmetries of equivariance and reversibility induce corresponding properties for the stable and unstable invariant manifolds and for the stable and unstable foliations for any sufficiently small Lipschitz perturbation of a linear hyperbolic dynamics. This requires establishing first a faithful correspondence between the reversibility and equivariance properties of the original dynamics and corresponding properties for its evolution map.Uniqueness of minimal unstable lamination for discretized Anosov flowshttps://zbmath.org/1496.370282022-11-17T18:59:28.764376Z"Guelman, Nancy"https://zbmath.org/authors/?q=ai:guelman.nancy"Martinchich, Santiago"https://zbmath.org/authors/?q=ai:martinchich.santiagoFor a partially hyperbolic system, there are invariant foliations tangent to the stable and unstable directions. This paper concerns unstable laminations, that is, closed invariant non-empty subsets of the unstable foliation, and shows in certain settings that there is a unique minimal unstable lamination. This has dynamical consequences as any quasi-attractor has its own minimal unstable lamination, and so a system with a unique minimal unstable lamination has at most one quasi-attractor.
Theorem A of the paper concerns a special type of dynamical system called discretized Anosov flow. This is a partially hyperbolic diffeomorphism of the form \(f(x) = \varphi_{\tau(x)}(x)\) where \(\varphi_t\) is a topological Anosov flow in dimension 3 and the function \(\tau\) gives the flow time. In certain families of 3-manifolds, it has been shown that every partially hyperbolic diffeomorphism isotopic to the identity is a discretized Anosov flow [\textit{T. Barthelmé} et al., ``Partially hyperbolic diffeomorphisms homotopic to the identity in dimension 3. I: The dynamically coherent case'', Preprint, \url{arXiv:1908.06227}].
Theorem A states that if \(f\) is a discretized Anosov flow and the corresponding topological Anosov flow is both transitive and not a suspension, then \(f\) has a unique minimal unstable lamination.
The other main result of the paper concerns skew-products defined over non-trivial circle bundles. Theorem B states that if \(f\) is a partially hyperbolic skew product where the center foliation is a circle bundle which is not virtually trivial and the quotient dynamics on \(M / \mathcal{W}^c\) is transitive, then \(f\) has a unique minimal unstable lamination.
Reviewer: Andy Hammerlindl (Melbourne)On the density of certain spectral points for a class of \(C^2\) quasiperiodic Schrödinger cocycleshttps://zbmath.org/1496.370292022-11-17T18:59:28.764376Z"Wu, Fan"https://zbmath.org/authors/?q=ai:wu.fan"Fu, Linlin"https://zbmath.org/authors/?q=ai:fu.linlin"Xu, Jiahao"https://zbmath.org/authors/?q=ai:xu.jiahaoSummary: For \(C^2\) cos-type potentials, large coupling constants, and fixed \(Diophantine\) frequency, we show that the density of the spectral points associated with the Schrödinger operator is larger than 0. In other words, for every fixed spectral point \(E , \liminf\limits_{\epsilon\to 0}\frac{|(E-\epsilon,E+\epsilon)\bigcap\Sigma_{\alpha,\lambda\upsilon}|}{2\epsilon} = \beta\), where \(\beta\in [\frac{1}{2},1]\). Our approach is a further improvement on the papers [\textit{J. Xu} et al., The Hölder continuity of Lyapunov exponents for a class of Cos-type quasiperiodic Schrödinger cocycles, Preprint, \url{arXiv:2006.03381}] and [\textit{Y. Wang} and \textit{Z. Zhang}, Int. Math. Res. Not. 2017, No. 8, 2300--2336 (2017; Zbl 1405.37086)].Bernoulli property for certain skew products over hyperbolic systemshttps://zbmath.org/1496.370302022-11-17T18:59:28.764376Z"Dong, Changguang"https://zbmath.org/authors/?q=ai:dong.changguang"Kanigowski, Adam"https://zbmath.org/authors/?q=ai:kanigowski.adamA measure-preserving transformation \(T\) on a probability measure space \((X, \mathcal{B},\mu)\) is Bernoulli if there exists a finite measurable partition \(\mathcal{P}=\{P_1, \dots, P_k\}\) on \(X\) such that the partitions \(\{T^n \mathcal{P}: n\in\mathbb{Z}\}\) are independent and their join span \(\mathcal{B}\). The Bernoulli property has been established for several classes of dynamical systems including Anosov flows [\textit{M. Ratner}, Isr. J. Math. 17, 380--391 (1974; Zbl 0304.28011)] and ergodic toral automorphisms [\textit{Y. Katznelson}, Isr. J. Math. 10, 186--195 (1971; Zbl 0219.28014)].
Here the authors consider a class of partially hyperbolic systems arising from skew-products over hyperbolic systems. More precisely, let \((\Sigma_A, \sigma)\) be a transitive subshift of finite type, \(\mu\) be a Gibbs measure on \(\Sigma_A\), \(\phi:\Sigma_A \to \mathbb{R}\) be an aperiodic and Hölder continuous function with \(\int_{\Sigma_A} \phi d\mu =0\). They prove that for any quasi-elliptic ergodic zero entropy flow \((K_t, N, \nu, d)\) with a regular generating partition, the skew product \(\sigma_{\phi}(x, y)=(\sigma(x), K_{\phi(x)}y)\) on \((\Sigma_{A}\times N, \mu\times \nu)\) is Bernoulli.
Reviewer: Pengfei Zhang (Norman)Chaos and integrability in \(\operatorname{SL}(2,\mathbb{R})\)-geometryhttps://zbmath.org/1496.370312022-11-17T18:59:28.764376Z"Bolsinov, Aleksei V."https://zbmath.org/authors/?q=ai:bolsinov.alexey-v"Veselov, Aleksandr P."https://zbmath.org/authors/?q=ai:veselov.alexander-p"Ye, Yiru"https://zbmath.org/authors/?q=ai:ye.yiruOn ergodicity of foliations on \({\mathbb{Z}^d}\)-covers of half-translation surfaces and some applications to periodic systems of Eaton lenseshttps://zbmath.org/1496.370322022-11-17T18:59:28.764376Z"Frączek, Krzysztof"https://zbmath.org/authors/?q=ai:fraczek.krzysztof-m"Schmoll, Martin"https://zbmath.org/authors/?q=ai:schmoll.martin-johannesSummary: We consider the geodesic flow defined by periodic Eaton lens patterns in the plane and discover ergodic ones among those. The ergodicity result on Eaton lenses is derived from a result for quadratic differentials on the plane that are pull backs of quadratic differentials on tori. Ergodicity itself is concluded for \({\mathbb{Z}^d}\)-covers of quadratic differentials on compact surfaces with vanishing Lyapunov exponents.Emergence of order from chaos: a phenomenological model of coupled oscillatorshttps://zbmath.org/1496.370332022-11-17T18:59:28.764376Z"Ghosh, Anupam"https://zbmath.org/authors/?q=ai:ghosh.anupam"Sujith, R. I."https://zbmath.org/authors/?q=ai:sujith.r-iSummary: This paper aims to study the transition to order from chaos using a mathematical model of coupled \textit{non-identical} oscillators. In the course of our analysis, we adopt a different control parameter other than the conventional parameters: coupling strength and frequency-mismatch, generally used in literature to obtain the aforementioned transition. Both the interacting oscillators become periodic, and they lead to the synchronized state as the control parameter changes monotonically. Initially, the participating oscillators, with low amplitude of oscillations, are in the chaotic desynchronized state. Besides, during the periodic oscillations, the oscillators exhibit high amplitude of oscillations. We illustrate the corresponding results using two examples of coupled oscillators. Such emergence of order from chaos with an accompanying increase in amplitude is ascertained in various numerical simulations and experimental observations.Constructing chaotic repellorshttps://zbmath.org/1496.370342022-11-17T18:59:28.764376Z"Li, Chunbiao"https://zbmath.org/authors/?q=ai:li.chunbiao"Gu, Zhenyu"https://zbmath.org/authors/?q=ai:gu.zhenyu"Liu, Zuohua"https://zbmath.org/authors/?q=ai:liu.zuohua"Jafari, Sajad"https://zbmath.org/authors/?q=ai:jafari.sajad"Kapitaniak, Tomasz"https://zbmath.org/authors/?q=ai:kapitaniak.tomaszSummary: The introduction of surfaces of equilibria in a dynamical system is a useful tool for constructing a chaotic repellor. To transform an attractor to a repellor, there are infinitely many available functions for introducing a surface of equilibria. Chaotic repellors can be constructed thereafter from a single chaotic attractor, a symmetric pair of chaotic attractors or even from those systems with attractor doubling and self-reproducing. Offset boosting of a variable driven by an embedded function or extra supplementary functions shows a flexible control on system attractors along with those coexisting repellors, which also rescales the frequency of oscillation even without destroying the amplitude of them.Stochastic deformations of coupling-induced oscillatory regimes in a system of two logistic mapshttps://zbmath.org/1496.370352022-11-17T18:59:28.764376Z"Bashkirtseva, I."https://zbmath.org/authors/?q=ai:bashkirtseva.irina-adolfovna"Ryashko, L."https://zbmath.org/authors/?q=ai:ryashko.lev-borisovichSummary: We consider a system of two coupled identical logistic maps, which in isolation demonstrate stable equilibrium modes. Under increasing coupling strength, this system exhibits transitions from initial equilibrium regime to synchronized oscillatory behavior with complex modes, both regular (periodic or quasiperiodic) and chaotic. Moreover, the coupled subsystem can demonstrate synchronization with anti-phase oscillations in tonic or burst form. A novelty of this paper is related to analysis of noise-induced deformations of these corporative oscillatory regimes. For this model, the following constructive stochastic effects are studied: (i) noise-induced temporal destruction of the anti-phase synchronization with transition from tonic to burst oscillations, (ii) noise-induced destruction of 2-torus with the transition from quasiperiodic oscillations to the bursting, (iii) stochastic transformations of burst oscillations from regular to chaotic. An important role of transients and ``riddled'' basins in the study of stochastic shifts of crisis bifurcation points is discussed. For the analytical investigation of these phenomena, a method based on the stochastic sensitivity of attractors (discrete cycles and tori) and confidence domains is applied.On \(q\)-deformed logistic mapshttps://zbmath.org/1496.370362022-11-17T18:59:28.764376Z"Cánovas, Jose S."https://zbmath.org/authors/?q=ai:canovas.jose-sThis paper is devoted to the study of \(q\)-deformations of the logistic family of maps with either a single or several \(q\)-deformations. The author first considers the stability of fixed points of these maps and characterizes when they are locally asymptotically stable and globally asymptotically stable. The complexity of \(q\)-deformations of maps in the logistic family is studied by computing their topological entropy. In the several \(q\)-deformations case, the author observes Parrondo's paradox, according to which the composition of dynamically simple maps results in complex dynamical behavior. Parrondo's paradox was also observed by \textit{J. Cánovas} and \textit{M. Muñoz-Guillermo} [Phys. Lett., A 383, No. 15, 1742--1754 (2019; Zbl 1476.62268)].
Reviewer: Steve Pederson (Atlanta)Dynamical properties of a novel one dimensional chaotic maphttps://zbmath.org/1496.370372022-11-17T18:59:28.764376Z"Kumar, Amit"https://zbmath.org/authors/?q=ai:kumar.amit.2|kumar.amit-n|kumar.amit.1|kumar.amit"Alzabut, Jehad"https://zbmath.org/authors/?q=ai:alzabut.jehad-o"Kumari, Sudesh"https://zbmath.org/authors/?q=ai:kumari.sudesh"Rani, Mamta"https://zbmath.org/authors/?q=ai:rani.mamta"Chugh, Renu"https://zbmath.org/authors/?q=ai:chugh.renuSummary: In this paper, a novel one dimensional chaotic map \(K(x) = \frac{\mu x(1\, -x)}{1+ x}\), \(x\in [0, 1]\), \(\mu > 0\) is proposed. Some dynamical properties including fixed points, attracting points, repelling points, stability and chaotic behavior of this map are analyzed. To prove the main result, various dynamical techniques like cobweb representation, bifurcation diagrams, maximal Lyapunov exponent, and time series analysis are adopted. Further, the entropy and probability distribution of this newly introduced map are computed which are compared with traditional one-dimensional chaotic logistic map. Moreover, with the help of bifurcation diagrams, we prove that the range of stability and chaos of this map is larger than that of existing one dimensional logistic map. Therefore, this map might be used to achieve better results in all the fields where logistic map has been used so far.Heat flow for harmonic maps from graphs into Riemannian manifoldshttps://zbmath.org/1496.370382022-11-17T18:59:28.764376Z"Baird, Paul"https://zbmath.org/authors/?q=ai:baird.paul"Fardoun, Ali"https://zbmath.org/authors/?q=ai:fardoun.ali"Regbaoui, Rachid"https://zbmath.org/authors/?q=ai:regbaoui.rachidSummary: We introduce the notion of harmonic map from a graph into a Riemannian manifold via a discrete version of the energy density. Existence and basic properties are established. Global existence and convergence of the associated heat flow are proved without any assumption on the curvature of the target manifold. We discuss a variant of the Steiner problem which replaces length by elastic energy.Elliptic fixed points with an invariant foliation: some facts and more questionshttps://zbmath.org/1496.370392022-11-17T18:59:28.764376Z"Chenciner, Alain"https://zbmath.org/authors/?q=ai:chenciner.alain"Sauzin, David"https://zbmath.org/authors/?q=ai:sauzin.david"Sun, Shanzhong"https://zbmath.org/authors/?q=ai:sun.shanzhong"Wei, Qiaoling"https://zbmath.org/authors/?q=ai:wei.qiaolingSummary: We address the following question: let \(F\colon (\mathbb{R}^2, 0)\to (\mathbb{R}^2, 0)\) be an analytic local diffeomorphism defined in the neighborhood of the nonresonant elliptic fixed point 0 and let \(\Phi\) be a formal conjugacy to a normal form \(N\). Supposing \(F\) leaves invariant the foliation by circles centered at \(0\), what is the analytic nature of \(\Phi\) and \(N\)?Injectivity of non-singular planar maps with one convex componenthttps://zbmath.org/1496.370402022-11-17T18:59:28.764376Z"Sabatini, Marco"https://zbmath.org/authors/?q=ai:sabatini.marcoSummary: We prove that if a non-singular planar map \(\Lambda \in C^2(\mathbb{R}^2,\mathbb{R}^2)\) has a convex component, then it is injective. We do not assume strict convexity.One-parameter set of diffeormorphisms of the plane with stable periodic pointshttps://zbmath.org/1496.370412022-11-17T18:59:28.764376Z"Vasil'eva, E. V."https://zbmath.org/authors/?q=ai:vasileva.ekaterina-vladimirovna|vasileva.ekaterina-viktorovnaSummary: In this paper we consider two-dimensional diffeomorphisms with hyperbolic fixed points and nontransverse homoclinic points. It is assumed that the tangency of a stable and unstable manifolds is not a tangency of finite order. It is shown that there exists a continuous one-parameter set of two-dimensional diffeomorphisms such that each diffeomorphism in a neighborhood of a homoclinic point has an infinite set of stable periodic points whose characteristic exponents are separated from zero.An infinite sequence of bitransitive Sierpiński carpets for \(z\mapsto\lambda(z+\frac{1}{z})\)https://zbmath.org/1496.370422022-11-17T18:59:28.764376Z"Look, Daniel M."https://zbmath.org/authors/?q=ai:look.daniel-mThe paper is concerned with the dynamics generated by functions of the form \(f_{\lambda}(z)=\lambda(z+1/z)\), where \(\lambda\) is a parameter. More precisely, the author constructs a sequence of parameters \(\lambda_n\) on the imaginary axis which tends to \(0\) such that the Julia set of \(f_{\lambda_n}\) is homeomorphic to the well-known Sierpiński carpet. However, the dynamics on these Julia sets are not conjugate.
Reviewer: Weiwei Cui (Lund)McMullen's and geometric pressures and approximating the Hausdorff dimension of Julia sets from belowhttps://zbmath.org/1496.370432022-11-17T18:59:28.764376Z"Przytycki, Feliks"https://zbmath.org/authors/?q=ai:przytycki.feliksSummary: We introduce new variants of the notion of geometric pressure for rational functions on the Riemann sphere, including non-hyperbolic functions, in the hope that some of them will turn out useful to achieve fast approximation from below of the hyperbolic Hausdorff dimension of Julia sets.Asymptotics of Andronov-Hopf dynamic bifurcationshttps://zbmath.org/1496.370442022-11-17T18:59:28.764376Z"Kalyakin, L. A."https://zbmath.org/authors/?q=ai:kalyakin.leonid-anatolevichSummary: We consider a system of two nonlinear differential equations with a slowly varying parameter \(\mu = \epsilon t \). For a frozen parameter \(\mu = \mathrm{const}\) the system has a focus type equilibrium state the stability of which changes when passing through the value \(\mu = 0\), i.e., we deal with an Andronov-Hopf bifurcation. Using the normal form method combined with the averaging method, we study asymptotics with respect to a small parameter \(\epsilon \rightarrow 0\) for solutions having a narrow transient layer near the bifurcation point in the domain \( |\epsilon t | \ll 1\). We express the leading term of asymptotics in terms of the solution to the Bernoulli equation.Complex periodic bursting structures in the Rayleigh-van der Pol-Duffing oscillatorhttps://zbmath.org/1496.370452022-11-17T18:59:28.764376Z"Ma, Xindong"https://zbmath.org/authors/?q=ai:ma.xindong"Bi, Qinsheng"https://zbmath.org/authors/?q=ai:bi.qinsheng"Wang, Lifeng"https://zbmath.org/authors/?q=ai:wang.lifengSummary: In the present paper, complex bursting patterns caused by the coupling effect of different frequency scales in the Rayleigh-van der Pol-Duffing oscillator (RVDPDO) driven by the external excitation term are presented theoretically. Seven different kinds of bursting, i.e., bursting of compound ``Homoclinic/Homoclinic'' mode via ``Homoclinic/Homoclinic'' hysteresis loop, bursting of compound ``fold/Homoclinic-Homoclinic/Hopf'' mode via ``fold/Homoclinic'' hysteresis loop, bursting of compound ``fold/Homoclinic-Hopf/Hopf'' mode via ``fold/Homoclinic'' hysteresis loop, bursting of ``fold/Homoclinic'' mode via ``fold/Homoclinic'' hysteresis loop, bursting of ``fold/Hopf'' mode via ``fold/fold'' hysteresis loop, bursting of ``Hopf/Hopf'' mode via ``fold/fold'' hysteresis loop and bursting of ``fold/fold'' mode are studied by using the phase diagram, time domain signal analysis, phase portrait superposition analysis and slow-fast analysis. With the help of the Melnikov method, the parameter properties related to the beingness of the Homoclinic and Heteroclinic bifurcations chaos of the periodic excitations are investigated. Then, by acting the external forcing term as a slowly changing state variable, the stability and bifurcation characteristics of the generalized autonomous system are given, which performs a major part in the interpretative principles of different bursting patterns. This paper aims to show the sensitivity of dynamical characteristics of RVDPDO to the variation of parameter \(\mu\) and decide how the choice of the parameters influences the manifold of RVDPDO during the repetitive spiking states. Finally, the validity of the research is tested and verified by the numerical simulations.Dynamic behaviors of a symmetrically coupled period-doubling systemhttps://zbmath.org/1496.370462022-11-17T18:59:28.764376Z"Yu, Zhiheng"https://zbmath.org/authors/?q=ai:yu.zhiheng"Li, Lin"https://zbmath.org/authors/?q=ai:li.lin.1"Zhang, Wenmeng"https://zbmath.org/authors/?q=ai:zhang.wenmengSummary: A system of two coupled mappings demonstrates a variety of nonlinear phenomena such as the inverse state, spatiotemporal intermittence, traveling wave and the synchronization. In this paper, we are concerned with a system of symmetrically coupled quadratic mappings. \textit{B. P. Bezruchko} et al. [Chaos Solitons Fractals 15, No. 4, 695--711 (2003; Zbl 1031.70012)] employed numerical method to study the bifurcation problem of such a system, but did not give a full investigation in theory because of the complicated computation. In this paper, we adopt the \textit{complete discrimination system theory} and \textit{the real root isolation algorithm} to overcome the difficulty. We will give a completed description of the bifurcations in theory for such a system, including the transcritical bifurcation, pitchfork bifurcation, flip bifurcation and the Neimark-Sacker bifurcation.Hopf bifurcations in a class of reaction-diffusion equations including two discrete time delays: an algorithm for determining Hopf bifurcation, and its applicationshttps://zbmath.org/1496.370472022-11-17T18:59:28.764376Z"Bilazeroğlu, Ş."https://zbmath.org/authors/?q=ai:bilazeroglu.seyma"Merdan, H."https://zbmath.org/authors/?q=ai:merdan.huseyinSummary: We analyze Hopf bifurcation and its properties of a class of system of reaction-diffusion equations involving two discrete time delays. First, we discuss the existence of periodic solutions of this class under Neumann boundary conditions, and determine the required conditions on parameters of the system at which Hopf bifurcation arises near equilibrium point. Bifurcation analysis is carried out by choosing one of the delay parameter as a bifurcation parameter and fixing the other in its stability interval. Second, some properties of periodic solutions such as direction of Hopf bifurcation and stability of bifurcating periodic solution are studied through the normal form theory and the center manifold reduction for functional partial differential equations. Moreover, an algorithm is developed in order to determine the existence of Hopf bifurcation (and its properties) of variety of system of reaction-diffusion equations that lie in the same class. The benefit of this algorithm is that it puts a very complex and long computations of existence of Hopf bifurcation for each equation in that class into a systematic schema. In other words, this algorithm consists of the conditions and formulae that are useful for completing the existence analysis of Hopf bifurcation by only using coefficients in the characteristic equation of the linearized system. Similarly, it is also useful for determining the direction analysis of the Hopf bifurcation merely by using the coefficients of the second degree Taylor polynomials of functions in the right hand side of the system. Finally, the existence of Hopf bifurcation for three different problems whose governing equations stay in that class is given by utilizing the algorithm derived, and thus the feasibility of the algorithm is presented.Chaos explosion and topological horseshoe in three-dimensional impacting hybrid systems with a single impact surfacehttps://zbmath.org/1496.370482022-11-17T18:59:28.764376Z"Wang, Lei"https://zbmath.org/authors/?q=ai:wang.lei.18"Yang, Xiao-Song"https://zbmath.org/authors/?q=ai:yang.xiaosongSummary: For a class of three-dimensional impacting hybrid systems comprising a linear system of ordinary differential equations and a reset map, and having a single impact surface, this paper studies the phenomena of impacting homoclinic bifurcation leading to periodic orbits, horseshoes and chaos explosions. More precisely, it is proved that the homoclinic bifurcation can result in the impacting periodic sinks or impacting periodic saddle orbits when the impact surface is a plane and the reset map satisfies some basic conditions, but it is not easy to find horseshoes in this case. Furthermore, when the single impact surface is not a plane and the reset map has a certain rotational property, it is proved that a topological horseshoe will appear suddenly when bifurcation parameter passes through some threshold, thus, a kind of chaos explosion takes place. These main results are illustrated by several examples, in which it can be seen that the newborn chaotic invariant sets might be chaotic attractors from the perspective of numerical simulation.On \(1:3\) resonance under reversible perturbations of conservative cubic Hénon mapshttps://zbmath.org/1496.370492022-11-17T18:59:28.764376Z"Gonchenko, Marina S."https://zbmath.org/authors/?q=ai:gonchenko.marina-s"Kazakov, Alexey O."https://zbmath.org/authors/?q=ai:kazakov.alexey-o"Samylina, Evgeniya A."https://zbmath.org/authors/?q=ai:samylina.evgeniya-a"Shykhmamedov, Aikan"https://zbmath.org/authors/?q=ai:shykhmamedov.aikanSummary: We consider reversible nonconservative perturbations of the conservative cubic Hénon maps \(H_3^{\pm} : \bar{x}=y,\bar{y}=-x+M_1 +M_2 y\pm y^3\) and study their influence on the 1:3 resonance, i.e., bifurcations of fixed points with eigenvalues \(e^{\pm i2\pi /3}\). It follows from [\textit{H. R. Dullin} and \textit{J. D. Meiss}, Physica D 143, No. 1--4, 262--289 (2000; Zbl 0961.37010)] that this resonance is degenerate for \(M_1 =0,M_2 =-1\) when the corresponding fixed point is elliptic. We show that bifurcations of this point under reversible perturbations give rise to four 3-periodic orbits, two of them are symmetric and conservative (saddles in the case of map \(H_3^+\) and elliptic orbits in the case of map \(H_3^-)\), the other two orbits are nonsymmetric and they compose symmetric couples of dissipative orbits (attracting and repelling orbits in the case of map \(H_3^+\) and saddles with the Jacobians less than 1 and greater than 1 in the case of map \(H_3^-)\). We show that these local symmetry-breaking bifurcations can lead to mixed dynamics due to accompanying global reversible bifurcations of symmetric nontransversal homo- and heteroclinic cycles. We also generalize the results of [loc. cit.] to the case of the \(p:q\) resonances with odd \(q\) and show that all of them are also degenerate for the maps \(H_3^{\pm}\) with \(M_1 =0\).Relating boundary and interior solutions of the cohomological equation for cocycles by isometries of negatively curved spaces. The Li\v{v}sic casehttps://zbmath.org/1496.370502022-11-17T18:59:28.764376Z"Moraga, Alexis"https://zbmath.org/authors/?q=ai:moraga.alexis"Ponce, Mario"https://zbmath.org/authors/?q=ai:ponce.marioA cocycle by isometries is a pair \((T,A)\) where \(T : \Omega \to \Omega\) is a self-homeomorphism of a compact metric space and \(A : \Omega \to \mathrm{Isom}(\mathcal{H})\) is a continuous map to the topological group of isometries of another metric space \(\mathcal{H}\). Such a cocycle is called reducible if the cohomological equation
\[
A(\omega) \equiv B(T \omega) \cdot [B(\omega)]^{-1}
\]
admits a continuous solution \(B : \Omega \to \mathrm{Isom}(\mathcal{H})\).
The main theorem of the paper is a reducibility criterion. The authors assume that \(T\) is a hyperbolic homeomorphism and that \(A\) satisfies a Hölder continuity condition. They also assume that the space \(\mathcal{H}\) is Gromov hyperbolic, and that it is uniquely visible, in the sense that every pair of boundary points can be connected by a unique geodesic. In this setting, the authors obtain a sufficient and necessary condition for reducibility, which is formulated in terms of the induced action on the boundary \(\partial \mathcal{H}\). Namely, the space \(\Omega \times \partial \mathcal{H}\) must be saturated by (regular enough) sections that are invariant under the corresponding skew-product dynamics.
The classical Li\v{v}sic theorem is an ingredient in the construction of the solution \(B\); it is applied to an \(\mathbb{R}\)-valued cocycle arising from the action on the space of horospheres.
Reviewer: Jairo Bochi (University Park)Random attractors for dissipative systems with rough noiseshttps://zbmath.org/1496.370512022-11-17T18:59:28.764376Z"Duc, Luu Hoang"https://zbmath.org/authors/?q=ai:duc.luu-hoangThe author studies the long-term behavior of the dynamics generated by rough differential equations with Hölder driving noises. The existence and upper semi-continuity of the global pullback attractor for certain dissipative systems with noises is proved. Further, it is shown that if one starts with a strictly disspative system, then the random attractor is a singleton if the perturbations possess sufficiently small noise intensity.
Reviewer: Marks Ruziboev (Wien)Destruction of cluster structures in an ensemble of chaotic maps with noise-modulated nonlocal couplinghttps://zbmath.org/1496.370522022-11-17T18:59:28.764376Z"Nikishina, Nataliya N."https://zbmath.org/authors/?q=ai:nikishina.nataliya-n"Rybalova, Elena V."https://zbmath.org/authors/?q=ai:rybalova.elena-v"Strelkova, Galina I."https://zbmath.org/authors/?q=ai:strelkova.galina-ivanova"Vadivasova, Tatiyana E."https://zbmath.org/authors/?q=ai:vadivasova.tatiyana-eSummary: We study numerically the spatio-temporal dynamics of a ring network of nonlocally coupled logistic maps when the coupling strength is modulated by colored Gaussian noise. Two cases of noise modulation are considered: 1) when the coupling coefficients characterizing the influence of neighbors on different elements are subjected to independent noise sources, and 2) when the coupling coefficients for all the network elements are modulated by the same stochastic signal. Without noise, the ring of chaotic maps exhibits a chimera state. The impact of noise-modulated coupling between the ring elements is explored when the parameter, which controls the correlation time and the spectral width of colored noise, and the noise intensity are varied. We investigate how the spatio-temporal structures observed in the ring evolve as the noise parameters change. The numerical results obtained are used to construct regime diagrams for the two cases of noise modulation. Our findings show the possibility of controlling the spatial structures in the ring in the presence of noise. Depending on the type of noise modulation, the spectral properties and intensity of colored noise, one can suppress the incoherent clusters of chimera states, and induce the regime of solitary states or synchronize chaotic oscillations of all the ring elements.Analogues of Khintchine's theorem for random attractorshttps://zbmath.org/1496.370532022-11-17T18:59:28.764376Z"Baker, Simon"https://zbmath.org/authors/?q=ai:baker.simon|baker.simon.1"Troscheit, Sascha"https://zbmath.org/authors/?q=ai:troscheit.saschaSummary: In this paper we study random iterated function systems. Our main result gives sufficient conditions for an analogue of a well known theorem due to \textit{A. Khintchine} [Math. Ann. 92, 115--125 (1924; JFM 50.0125.01)] from Diophantine approximation to hold almost surely for stochastically self-similar and self-affine random iterated function systems.Invariant densities for random continued fractionshttps://zbmath.org/1496.370542022-11-17T18:59:28.764376Z"Kalle, Charlene"https://zbmath.org/authors/?q=ai:kalle.charlene"Matache, Valentin"https://zbmath.org/authors/?q=ai:matache.valentin"Tsujii, Masato"https://zbmath.org/authors/?q=ai:tsujii.masato"Verbitskiy, Evgeny"https://zbmath.org/authors/?q=ai:verbitskiy.evgeny-aSummary: We continue the study of random continued fraction expansions, generated by random application of the Gauss and the Rényi backward continued fraction maps. We show that this random dynamical system admits a unique absolutely continuous invariant measure with smooth density.Existence of unstable stationary solutions for nonlinear stochastic differential equations with additive white noisehttps://zbmath.org/1496.370552022-11-17T18:59:28.764376Z"Lv, Xiang"https://zbmath.org/authors/?q=ai:lv.xiangSummary: This paper is concerned with the existence of unstable stationary solutions for nonlinear stochastic differential equations (SDEs) with additive white noise. Assume that the nonlinear term \(f\) is monotone (or anti-monotone) and the global Lipschitz constant of \(f\) is smaller than the positive real part of the principal eigenvalue of the competitive matrix \(A\), the random dynamical system (RDS) generated by SDEs has an unstable \(\mathscr{F}_+\)-measurable random equilibrium, which produces a stationary solution for nonlinear SDEs. Here, \(\mathscr{F}_+ = \sigma \{ \omega \mapsto W_t (\omega):t\geq 0\}\) is the future \(\sigma\)-algebra. In addition, we get that the \(\alpha\)-limit set of all pull-back trajectories starting at the initial value \(x(0) = x\in\mathbb{R}^n\) is a single point for all \(\omega\in\Omega\), i.e., the unstable \(\mathscr{F}_+\)-measurable random equilibrium. Applications to stochastic neural network models are given.On essential numerical ranges and essential spectra of Hamiltonian systems with one singular endpointhttps://zbmath.org/1496.370562022-11-17T18:59:28.764376Z"Zhu, Li"https://zbmath.org/authors/?q=ai:zhu.li"Sun, Huaqing"https://zbmath.org/authors/?q=ai:sun.huaqingSummary: This paper is concerned with essential numerical ranges and essential spectra of Hamiltonian systems with one singular endpoint. For semi-bounded systems, the characterization of each element of the essential numerical range in terms of certain singular sequences is given, the concept of form perturbation small at the singular endpoint is introduced, and the stability of the essential numerical range is obtained under this perturbation, which shows the stability of the infimum or supremum of the essential spectrum. Some sufficient conditions for the invariance of the essential numerical range are given in terms of coefficients of Hamiltonian systems.The influence of a parameter that controls the asymmetry of a potential energy surface with an entrance channel and two potential wellshttps://zbmath.org/1496.370572022-11-17T18:59:28.764376Z"Agaoglou, Makrina"https://zbmath.org/authors/?q=ai:agaoglou.makrina"Katsanikas, Matthaios"https://zbmath.org/authors/?q=ai:katsanikas.matthaios"Wiggins, Stephen"https://zbmath.org/authors/?q=ai:wiggins.stephenSummary: In this paper we study an asymmetric valley-ridge inflection point (VRI) potential, whose energy surface (PES) features two sequential index-1 saddles (the upper and the lower), with one saddle having higher energy than the other, and two potential wells separated by the lower index-1 saddle. We show how the depth and the flatness of our potential changes as we modify the parameter that controls the asymmetry as well as how the branching ratio (ratio of the trajectories that enter each well) is changing as we modify the same parameter and its correlation with the area of the lobes as they have been formed by the stable and unstable manifolds that have been extracted from the gradient of the LD scalar fields.Zeros of rational functions and solvable nonlinear evolution equationshttps://zbmath.org/1496.370582022-11-17T18:59:28.764376Z"Calogero, Francesco"https://zbmath.org/authors/?q=ai:calogero.francesco-aSummary: Recently a convenient technique to relate the time evolution of the \(N\) zeros of a time-dependent polynomial \(p_N(z; t\) of degree \(N\) in the complex variable \(z\) to the time evolution of its \(N\) coefficients has been exploited to identify large classes of dynamical systems solvable by \textit{algebraic} operations. These models also include \(N\)-body problems that evolve in the complex plane (or, equivalently, in the real Cartesian plane) according to systems of nonlinearly-coupled equations of motion of Newtonian type (``accelerations equal forces''). Many of these models feature remarkable properties: for instance, they are Hamiltonian and integrable and/or multiply periodic or even isochronous (featuring completely periodic solutions with a fixed period largely independent of the initial data), or \textit{asymptotically isochronous} (featuring isochrony only up to corrections vanishing in the remote future). In this paper, an analogous technique is introduced that focuses instead on the time evolution of the \(N\) zeros of an appropriate class of time-dependent rational functions \(R_N(z; t)\), thereby opening large vistas of new dynamical systems solvable by \textit{algebraic} operations and featuring remarkable properties. A few examples are reported.{
\copyright 2018 American Institute of Physics}On the integrability of circulatory systemshttps://zbmath.org/1496.370592022-11-17T18:59:28.764376Z"Kozlov, V. V."https://zbmath.org/authors/?q=ai:kozlov.victor-v|kozlov.vasilii-vasilevich|kozlov.vladimir-vasilievich|kozlov.valerij-vasilievichSummary: This paper discusses conditions for the existence of polynomial (in velocities) first integrals of the equations of motion of mechanical systems in a nonpotential force field (circulatory systems). These integrals are assumed to be single-valued smooth functions on the phase space of the system (on the space of the tangent bundle of a smooth configuration manifold). It is shown that, if the genus of the closed configuration manifold of such a system with two degrees of freedom is greater than unity, then the equations of motion admit no nonconstant single-valued polynomial integrals. Examples are given of circulatory systems with configuration space in the form of a sphere and a torus which have nontrivial polynomial laws of conservation. Some unsolved problems involved in these phenomena are discussed.Integrability by separation of variableshttps://zbmath.org/1496.370602022-11-17T18:59:28.764376Z"Llibre, Jaume"https://zbmath.org/authors/?q=ai:llibre.jaume"Ramírez, Rafael"https://zbmath.org/authors/?q=ai:ramirez.rafael-oSummary: We study the integrability in the Jacobi sense (integrability by separation of variables), of the Hamiltonian differential systems using the Levi-Civita Theorem. In particular we solve the Stark problem for \(N > 3\).Eigenfunctions of a discrete elliptic integrable particle model with hyperoctahedral symmetryhttps://zbmath.org/1496.370612022-11-17T18:59:28.764376Z"van Diejen, Jan Felipe"https://zbmath.org/authors/?q=ai:van-diejen.jan-felipe"Görbe, Tamás"https://zbmath.org/authors/?q=ai:gorbe.tamas-fThe main purpose of the authors is to carry out a finite-dimensional reduction of the eigenvalue problem for a second-order difference operator describing the quantum Hamiltonian of an elliptic Ruijsenaars type \(n\)-particle model on the circle with hyperoctahedral symmetry.
Reviewer: Mohammed El Aïdi (Bogotá)Tensor invariants of geodesic, potential, and dissipative systems on tangent bundles of two-dimensional manifoldshttps://zbmath.org/1496.370622022-11-17T18:59:28.764376Z"Shamolin, M. V."https://zbmath.org/authors/?q=ai:shamolin.m-vSummary: Tensor invariants (differential forms) for homogeneous dynamical systems on tangent bundles of smooth two-dimensional manifolds are presented. The connection between the presence of these invariants and the full set of first integrals necessary for the integration of geodesic, potential, and dissipative systems is shown. The force fields introduced into the considered systems make them dissipative with dissipation of different signs and generalize previously considered force fields.Birkhoff averages and rotational invariant circles for area-preserving mapshttps://zbmath.org/1496.370632022-11-17T18:59:28.764376Z"Sander, E."https://zbmath.org/authors/?q=ai:sander.eric|sander.evelyn|sander.edward-a"Meiss, J. D."https://zbmath.org/authors/?q=ai:meiss.james-dSummary: Rotational invariant circles of area-preserving maps are an important and well-studied example of KAM tori. John Greene conjectured that the locally most robust rotational circles have rotation numbers that are noble, i.e., have continued fractions with a tail of ones, and that, of these circles, the most robust has golden mean rotation number. The accurate numerical confirmation of these conjectures relies on the map having a time-reversal symmetry, and such high accuracy has not been obtained in more general maps. In this paper, we develop a method based on a weighted Birkhoff average for identifying chaotic orbits, island chains, and rotational invariant circles that does not rely on these symmetries. We use Chirikov's standard map as our test case, and also demonstrate that our methods apply to three other, well-studied cases.Multiplicity of homoclinic solutions for second order Hamiltonian systems with local conditions at the originhttps://zbmath.org/1496.370642022-11-17T18:59:28.764376Z"Liu, Peng"https://zbmath.org/authors/?q=ai:liu.peng.1|liu.peng"Guo, Fei"https://zbmath.org/authors/?q=ai:guo.feiSummary: In this paper, the multiplicity of homoclinic solutions for second order non-autonomous Hamiltonian systems \(\ddot{u}(t) - L(t)u(t) + \nabla_uW( t, u (t)) = \textbf{0}\) is obtained via a new Symmetric Mountain Pass Lemma established by \textit{R. Kajikiya} [J. Funct. Anal. 225, No. 2, 352--370 (2005; Zbl 1081.49002)], where \(L \in C( \mathbb{R},\mathbb{R}^{ N \times N } )\) is symmetric but non-periodic, \(W \in C^1(\mathbb{R} \times \mathbb{R}^N , \mathbb{R} )\) is locally even in \(u\) and only satisfies some growth conditions near \(u = \textbf{0} \), which improves some previous results.Homoclinic orbits of sub-linear Hamiltonian systems with perturbed termshttps://zbmath.org/1496.370652022-11-17T18:59:28.764376Z"Lv, Haiyan"https://zbmath.org/authors/?q=ai:lv.haiyan"Chen, Guanwei"https://zbmath.org/authors/?q=ai:chen.guanweiSummary: By using variational methods, we obtain the existence of homoclinic orbits for perturbed Hamiltonian systems with sub-linear terms. To the best of our knowledge, there is no published result focusing on the perturbed and sub-linear Hamiltonian systems.Multiple homoclinic solutions for \(p\)-Laplacian Hamiltonian systems with concave-convex nonlinearitieshttps://zbmath.org/1496.370662022-11-17T18:59:28.764376Z"Wan, Lili"https://zbmath.org/authors/?q=ai:wan.liliSummary: The multiplicity of homoclinic solutions is obtained for a class of the \(p\)-Laplacian Hamiltonian systems \(\frac{d}{dt}(|\dot{u}(t)|^{p-2}\dot{u}(t))-a(t)|u(t)|^{p-2}u(t)+ \nabla W(t,u(t))=0\) via variational methods, where \(a(t)\) is neither coercive nor bounded necessarily and \(W(t,u)\) is under new concave-convex conditions. Recent results in the literature are generalized even for \(p=2\).On locally superquadratic Hamiltonian systems with periodic potentialhttps://zbmath.org/1496.370672022-11-17T18:59:28.764376Z"Ye, Yiwei"https://zbmath.org/authors/?q=ai:ye.yiweiSummary: In this paper, we study the second-order Hamiltonian systems
\[
\ddot{u}-L(t)u+\nabla W(t,u)=0,
\] where \(t\in \mathbb{R}\), \(u\in \mathbb{R}^N\), \(L\) and \(W\) depend periodically on \(t, 0\) lies in a spectral gap of the operator \(-d^2/dt^2+L(t)\) and \(W(t,x)\) is locally superquadratic. Replacing the common superquadratic condition that \(\lim_{|x|\rightarrow \infty}\frac{W(t,x)}{|x|^2}=+\infty\) uniformly in \(t\in \mathbb{R}\) by the local condition that \(\lim_{|x|\rightarrow \infty}\frac{W(t,x)}{|x|^2}=+\infty\) a.e. \(t\in J\) for some open interval \(J\subset \mathbb{R} \), we prove the existence of one nontrivial homoclinic soluiton for the above problem.First approximation formulas in the problem of perturbation of definite and indefinite multipliers of linear Hamiltonian systemshttps://zbmath.org/1496.370682022-11-17T18:59:28.764376Z"Yumagulov, M. G."https://zbmath.org/authors/?q=ai:yumagulov.marat-gayazovich"Ibragimova, L. S."https://zbmath.org/authors/?q=ai:ibragimova.liliya-sunagatovna"Belova, A. S."https://zbmath.org/authors/?q=ai:belova.anna-sergeevnaSummary: New formulas of the first approximation are proposed in the problem of perturbing definite and indefinite multipliers of linear periodic Hamiltonian systems. The resultant formulas are based on the analysis of the spectral properties of the monodromy matrix system. The proposed formulas lead to new criteria according to the Lyapunov stability for linear periodic Hamiltonian systems in critical cases. Applications to the problem of parametric resonance in fundamental resonances are considered. The results obtained are formulated in terms of the original equations and brought to effective formulas and algorithms.A generalized Poincaré-Birkhoff theoremhttps://zbmath.org/1496.370692022-11-17T18:59:28.764376Z"Moreno, Agustin"https://zbmath.org/authors/?q=ai:moreno.agustin-s"van Koert, Otto"https://zbmath.org/authors/?q=ai:van-koert.ottoSummary: We prove a generalization of the classical Poincaré-Birkhoff theorem for Liouville domains, in arbitrary even dimensions. This is inspired by the existence of global hypersurfaces of section for the spatial case of the restricted three-body problem [\textit{A. Moreno} and \textit{O. van Koert}, Nonlinearity 35, No. 6, 2920--2970 (2022; Zbl 07539293)].Some reversing orbits for a rattleback modelhttps://zbmath.org/1496.370702022-11-17T18:59:28.764376Z"Arioli, Gianni"https://zbmath.org/authors/?q=ai:arioli.gianni"Koch, Hans"https://zbmath.org/authors/?q=ai:koch.hans-friedrichSummary: A physical rattleback is a toy that can exhibit counter-intuitive behavior when spun on a horizontal plate. Most notably, it can spontaneously reverse its direction of rotation. Using a standard mathematical model of the rattleback, we prove the existence of reversing motion, reversing motion combined with rolling, and orbits that exhibit such behavior repeatedly.Contact Hamiltonian and Lagrangian systems with nonholonomic constraintshttps://zbmath.org/1496.370712022-11-17T18:59:28.764376Z"de León, Manuel"https://zbmath.org/authors/?q=ai:de-leon.manuel"Jiménez, Víctor M."https://zbmath.org/authors/?q=ai:jimenez.victor-manuel"Lainz, Manuel"https://zbmath.org/authors/?q=ai:lainz.manuelThis paper aims at using contact and Jacobi geometry to develop the natural geometric framework for studying the dynamics of mechanical systems that are subject to both nonholonomic constraints and Rayleigh dissipation.
A \textit{nonholonomic mechanical system} is a mechanical system subject to \textit{nonholonomic constraints}, i.e., constraints (on the position and velocities) that do not derive from constraints only on the positions. Examples include mechanical systems that have rolling contact (like a ball rolling without slipping on a plane) or some kind of sliding contact (like a rigid body sliding on a plane). In the Lagrangian formalism, a mechanical system is described by a \textit{Lagrangian function} \(L\colon TQ\to\mathbb{R},\ (q,\dot q)\mapsto L(q,\dot q)\), where the smooth manifold \(Q\) denotes the \textit{configuration space} of the system. Then a nonholonomic constraint is given by a submanifold \(\mathcal{D}\subset TQ\) such that \(\tau_Q(\mathcal{D})=Q\), where \(\tau_Q\colon TQ\to Q\) denotes the bundle map. In the following, one only considers nonholonomic constraints that are linear in the velocities, i.e., \(\mathcal{D}\subset TQ\) is a vector subbundle.
If the mechanical system is conservative, i.e., \(L=K_g-V\), where \(V\in C^\infty(Q)\) and \(K_g\) is the kinetic energy of some pseudo-Riemannian metric \(g\) on \(Q\), then the Lagrangian \(L\) is regular, i.e., the associated Legendre transform \(\mathbb{F}L:TQ\to T^\ast Q\) is a local diffeomorphism. In this case, the natural geometric description of their dynamics is provided in terms of Hamiltonian systems on symplectic manifolds (see, e.g., [\textit{R. Abraham} and \textit{J. E. Marsden}, Foundations of mechanics. 2nd ed., rev., enl., and reset. With the assistance of Tudor Ratiu and Richard Cushman. Reading, Massachusetts: The Benjamin/Cummings Publishing Company, Inc (1978; Zbl 0393.70001)] and references therein). Indeed, the unconstrained dynamics is obtained as the projection on \(Q\) of the flow of the Euler-Lagrange vector field \(\Gamma_L\), i.e., the Hamiltonian vector field of the system \((TQ,\omega_L,E_L)\), where \(E_L=\Delta(L)-L\) is the energy, with \(\Delta\) the Euler vector field on \(TQ\), and \(\omega_L=(\mathbb{F}L)^\ast\omega_{\text{can}}\) is the pull-back along \(\mathbb{F}L\) of the canonical symplectic form on \(T^\ast Q\). This \(\Gamma_L\) is a SODE (second-order differential equation) on \(TQ\) and its flow is obtained integrating the standard Euler-Lagrange equations. This description of the dynamics is consistent with the one arising from D'Alembert principle.
If the conservative mechanical system is additionally subject to nonholonomic constraints, then its dynamics can be still described in terms of Hamiltonian systems on symplectic manifolds (see, e.g., [\textit{C.-M. Marle}, Rep. Math. Phys. 42, No. 1--2, 211--229 (1998; Zbl 0931.37023)] and references therein). Indeed, its dynamics is the projection on \(Q\) of the flow of a nonholonomic Euler-Lagrange vector field \(\Gamma_L^\mathcal{D}\). The latter is still a SODE on \(TQ\) and is obtained from \(\Gamma_L\) by projection with respect to a certain decomposition of \((TTQ)|_\mathcal{D}\). This description of the nonholonomic dynamics is consistent with the one arising from Chetaev version of D'Alembert principle. Moreover, this nonholonomic dynamics is almost-Poisson but not Poisson. Indeed, there is a bracket \(\{-,-\}\) on \(C^\infty(\mathcal{D})\) that satisfies the Leibniz rule in each entry and, together with the energy \(E_L\) on \(TQ\), controls the time evolution of the observables, but in generally it fails to satisfy the Jacobi identity.
The authors start from the observation that there are other kinds of nonholonomic mechanical systems that do not fit in the previous framework. As a first example, one can consider a nonholonomic mechanical system that is also subject to Rayleigh dissipation and so non-conservative. Additional examples come from thermodynamics. These mechanical systems can be described by a Lagrangian function \(L\colon TQ\times\mathbb{R}\to\mathbb{R},\ (q,\dot q,z)\mapsto L(q,\dot q,z):=L_z(q,\dot q),\) where the smooth manifold \(Q\) is the configuration space and the parameter \(z\) on \(\mathbb{R}\) denotes friction (or a thermal variable in thermodynamics).
If the Lagrangian \(L\) is regular, in the sense that, for any \(z\in\mathbb{R}\), the associated Legendre transform \(\mathbb{F}L_z:TQ\to T^\ast Q\) is a local diffeomorphism, the natural geometric framework of their dynamics is provided by the theory of Hamiltonian systems on contact manifolds (cf., e.g., [\textit{M. de León} and \textit{M. Lainz Valcázar}, J. Math. Phys. 60, No. 10, 102902, 18 p. (2019; Zbl 1427.70039)] and references therein). Indeed, the unconstrained dynamics is obtained as the projection on \(Q\) of the flow of the Euler-Lagrange vector field \(\Gamma_L\), i.e., the Hamiltonian vector field of the system \((TQ\times\mathbb{R},\eta_L,E_L)\), where \(E_L=\Delta(L)-L\) is the energy, with \(\Delta\) the Euler vector field on \(TQ\), and \(\eta_L=(\mathbb{F}L\times\operatorname{id}_\mathbb{R})^\ast\eta_{\text{can}}\) is the pull-back along \(\mathbb{F}L\times\operatorname{id}_\mathbb{R}\) of the canonical contact form on \(T^\ast Q\times\mathbb{R}\). In the current setting, this \(\Gamma_L\) is still an SODE on \(TQ\times\mathbb{R}\) (in the sense recalled in Definition~5) and its flow is obtained integrating the so-called Herglotz equations. Indeed, this description of the dynamics is consistent with the one arising from the Herglotz variational principle (as recalled in Section~4).
In this paper the authors show that the theory of Hamiltonian systems on contact manifolds can be adapted to provide a geometric interpretation of the dynamics of mechanical systems that are subject to both dissipation and nonholonomic constraints. Section~5 defines a version of the Herglotz principle in presence of nonholonomic constraints: essentially, one restricts the variations so that they satisfy the constraints. Then the dynamics is described by the extremals of this Herglotz principle with constraints and they are given by the solutions of the so-called constrained Herglotz equations (see Theorem~5). Actually, this description admits a geometric description similar to the one obtained when there are no constraints. Indeed, Theorem 6 shows that, if the Lagrangian \(L\colon TQ\times\mathbb{R}\to\mathbb{R}\) is regular (i.e., \(F_z\colon TQ\to\mathbb{R}\) is regular, for any \(z\in\mathbb{R}\), as recalled above), the solutions of the constrained Herglotz equations are the projections on \(Q\) of the integral curves of the nonholonomic Euler-Lagrange vector field \(\Gamma_L^\mathcal{D}\). The latter is still a SODE on \(TQ\times\mathbb{R}\) (see Definition~5) and is obtained from \(\Gamma_L\) by projection with respect to a certain decomposition of \(T(TQ\times\mathbb{R})\) along \(\mathcal{D}\).
The authors also prove that the time evolution of these mechanical systems subject to both dissipation and nonholonomic constraints is governed by an almost-Jacobi bracket (see Definition 7). Indeed, in Section 6, they first construct a nonholonomic bracket from functions on \(TQ\times\mathbb{R}\) to functions on \(\mathcal{D}\times\mathbb{R}\) (see Equation 100), then they prove that this nonholonomic bracket (together with the Energy \(E_L\)) controls the time evolution of the observables (see Theorem 12) and it is an almost Jacobi bracket (see Proposition 6). Further, it turns out that this nonholonomic bracket is actually a Jacobi structure (i.e., it satisfies the Jacobi identity) if and only if the constraint \(\mathcal{D}\subset TQ\) is an involutive vector subbundle (see Theorem 13).
Finally, in Example 2, the authors illustrate their results applying them to a particular example given by a model of the Chaplygin's sleight subject to Rayleigh dissipation.
Reviewer: Alfonso Giuseppe Tortorella (Porto)Miura and Darboux transformations in the SUSY KP hierarchieshttps://zbmath.org/1496.370722022-11-17T18:59:28.764376Z"Chen, Huizhan"https://zbmath.org/authors/?q=ai:chen.huizhan"Cheng, Jipeng"https://zbmath.org/authors/?q=ai:cheng.jipeng"Wu, Zhiwei"https://zbmath.org/authors/?q=ai:wu.zhiweiSummary: The Miura links between the KP and modified KP hierarchies are extended to the SUSY KP (SKP) and SUSY modified KP hierarchies (SmKP) of Manin Radul and Jacobian types. The corresponding Darboux transformations in the SUSY case are constructed by using Miura links. In this sense, one can better understand why one step Darboux transformation of the SKP hierarchy can not keep the odd flows. In addition, the Darboux transformations in the SmKP case can be obtained naturally by the Miura links. Then the squared eigenfunction symmetry of the SmKP hierarchy is obtained by using Miura link from the SKP hierarchy. At last, starting from the results in the SKP hierarchy, the constrained SmKP hierarchy of Manin Radul type can also be constructed by Miura link, that is, by adding appropriate squared eigenfunction symmetry in odd flow parts. The Darboux transformations of the constrained SKP and SmKP hierarchies are also discussed by using the Miura links.Hexagonal geodesic 3-webshttps://zbmath.org/1496.370732022-11-17T18:59:28.764376Z"Agafonov, Sergey I."https://zbmath.org/authors/?q=ai:agafonov.sergey-iSummary: We prove that a surface carries a hexagonal 3-web of geodesics if and only if the geodesic flow on the surface admits a cubic 1st integral and show that the system of partial differential equations, governing metrics on such surfaces, is integrable by generalized hodograph transform method. We present some new local examples of such metrics, discuss known ones, and establish an analogue of the celebrated \textit{H. Graf} and \textit{R. Sauer} theorem for Darboux superintegrable metrics [Münch. Ber. 1924, 119--156 (1924; JFM 50.0396.02)].Lumps and interaction solutions to the \((4+1)\)-dimensional variable-coefficient Kadomtsev-Petviashvili equation in fluid mechanicshttps://zbmath.org/1496.370742022-11-17T18:59:28.764376Z"Fan, Lulu"https://zbmath.org/authors/?q=ai:fan.lulu"Bao, Taogetusang"https://zbmath.org/authors/?q=ai:bao.taogetusangStable solitons in a nearly \(\mathcal{PT}\)-symmetric ferromagnet with spin-transfer torquehttps://zbmath.org/1496.370752022-11-17T18:59:28.764376Z"Barashenkov, I. V."https://zbmath.org/authors/?q=ai:barashenkov.igor-v"Chernyavsky, Alexander"https://zbmath.org/authors/?q=ai:chernyavskii.aleksandr-grigorevichSummary: We consider the Landau-Lifshitz equation for the spin torque oscillator -- a uniaxial ferromagnet in an external magnetic field with polarised spin current driven through it. In the absence of the Gilbert damping, the equation turns out to be \(\mathcal{PT}\)-symmetric. We interpret the \(\mathcal{PT}\)-symmetry as a balance between gain and loss -- and identify the gaining and losing modes. In the vicinity of the bifurcation point of a uniform static state of magnetisation, the \(\mathcal{PT}\)-symmetric Landau-Lifshitz equation with a small dissipative perturbation reduces to a nonlinear Schrödinger equation with a quadratic nonlinearity. The analysis of the Schrödinger dynamics demonstrates that the spin torque oscillator supports stable magnetic solitons. The \(\mathcal{PT}\) near-symmetry is crucial for the soliton stability: the addition of a finite dissipative term to the Landau-Lifshitz equation destabilises all solitons that we have found.Persistence of invariant tori in infinite-dimensional Hamiltonian systemshttps://zbmath.org/1496.370762022-11-17T18:59:28.764376Z"Huang, Peng"https://zbmath.org/authors/?q=ai:huang.pengSummary: In this paper, we consider the persistence of invariant tori in infinite-dimensional Hamiltonian systems
\[
H=\langle\omega, I\rangle +P(\theta, I, \omega),
\]
where \(\theta \in \mathbb{T}^\Lambda\), \(I\in \mathbb{R}^\Lambda\), the frequency \(\omega=(\cdots, \omega_\lambda, \cdots)_{\lambda\in\Lambda}\) is regarded as parameters varying freely over some subset \(\ell^\infty(\Lambda, \mathbb{R})\) of the parameter space \(\mathbb{R}^\Lambda\), \(\omega = (\cdots, \omega_\lambda, \cdots)_{\lambda\in\Lambda}\) is a bilateral infinite sequence of rationally independent frequency, in other words, any finite segments of \(\omega = (\cdots, \omega_\lambda, \cdots)_{\lambda\in\Lambda}\) are rationally independent.Invariant tori of full dimension for higher-dimensional beam equations with almost-periodic forcinghttps://zbmath.org/1496.370772022-11-17T18:59:28.764376Z"Rui, Jie"https://zbmath.org/authors/?q=ai:rui.jie"Wang, Yi"https://zbmath.org/authors/?q=ai:wang.yi.9|wang.yi.8|wang.yi.4|wang.yi.5|wang.yi.6|wang.yi.1|wang.yi.7|wang.yi.2|wang.yi.3Summary: In this paper, we focus on the class of almost-periodically forced higher-dimensional beam equations
\[
u_{tt}+(-\Delta +\mu)^2u+\psi (\omega t)u=0,\quad \mu >0, t \in \mathbb{R}, x\in \mathbb{R}^d,
\] subject to periodic boundary conditions, where \(\psi (\omega t)\) is real analytic and almost-periodic in \(t\). We show the existence of almost-periodic solutions for this equation under some suitable hypotheses. In the proof, we improve the KAM iteration to deal with the infinite-dimensional frequency \(\omega =(\omega_1,\omega_2,\ldots)\).Ground states for infinite lattices with nearest neighbor interactionhttps://zbmath.org/1496.370782022-11-17T18:59:28.764376Z"Chen, Peng"https://zbmath.org/authors/?q=ai:chen.peng"Hu, Die"https://zbmath.org/authors/?q=ai:hu.die"Zhang, Yuanyuan"https://zbmath.org/authors/?q=ai:zhang.yuanyuan.1Summary: \textit{J. Sun} and \textit{S. Ma} [J. Differ. Equations 255, No. 8, 2534--2563 (2013; Zbl 1318.37026)]
proved the existence of a nonzero \(T\)-periodic solution for a class of one-dimensional lattice dynamical systems,
\[
\ddot{q_i}=\varPhi_{i-1}^\prime(q_{i-1}-q_i)- \varPhi_i^\prime(q_i-q_{i+1}),\quad i\in \mathbb{Z},
\] where \(q_i\) denotes the co-ordinate of the \(i\)th particle and \(\varPhi_i\) denotes the potential of the interaction between the \(i\)th and the \((i+1)\)th particle. We extend their results to the case of the least energy of nonzero \(T\)-periodic solution under general conditions. Of particular interest is a new and quite general approach. To the best of our knowledge, there is no result for the ground states for one-dimensional lattice dynamical systems.Langevin approach for intrinsic fluctuations of chemical reactions with Hopf bifurcationhttps://zbmath.org/1496.370792022-11-17T18:59:28.764376Z"Xu, Hong-Yuan"https://zbmath.org/authors/?q=ai:xu.hongyuan"Luo, Yu-Pin"https://zbmath.org/authors/?q=ai:luo.yupin"Wu, Jinn-Wen"https://zbmath.org/authors/?q=ai:wu.jinnwen"Huang, Ming-Chang"https://zbmath.org/authors/?q=ai:huang.mingchangSummary: The characteristics of Langevin equations for the intrinsic fluctuations of chemical reactions are investigated via the analyses on the Brusselator model in the parameter domain of spirally stable focus. Two forms of Langevin equations are shown to be equivalent for the results of two statistical measures. The comparisons in the results of two measures between Langevin equations and chemical master equation are given: The difference in stationary probability densities is significant for systems close to the bifurcation point, even the system size is large; the power spectra of Langevin equations with white noise are qualitatively the same as that of chemical master equation, but the discrepancy is found between Langevin equations with colored noises and chemical master equation, if the length of correlation-time is comparable with the correlation-time of a system. As the linearized Langevin equations possess singularity at the supercritical Hopf bifurcation point for the statistical measures, the Langevin equations displace the bifurcation points, and the amount of displacement is a decreasing function of the system size and the length of correlation-time of noises.Upper semicontinuity of pullback attractors for a nonautonomous damped wave equationhttps://zbmath.org/1496.370802022-11-17T18:59:28.764376Z"Wang, Yonghai"https://zbmath.org/authors/?q=ai:wang.yonghai"Hu, Minhui"https://zbmath.org/authors/?q=ai:hu.minhui"Qin, Yuming"https://zbmath.org/authors/?q=ai:qin.yumingSummary: In this paper, we study the local uniformly upper semicontinuity of pullback attractors for a strongly damped wave equation. In particular, under some proper assumptions, we prove that the pullback attractor \(\{A_{\varepsilon}(t)\}_{t\in \mathbb{R}}\) of Eq. (1.1) with \(\varepsilon \in [0,1]\) satisfies \(\lim_{\varepsilon \to \varepsilon_0}\sup_{t\in [a,b]} \operatorname{dist}_{H_0^1\times L^2}(A_{\varepsilon}(t),A_{ \varepsilon_0}(t))=0\) for any \([a,b]\subset \mathbb{R}\) and \(\varepsilon_0\in [0,1]\).Weak pullback mean random attractors for the stochastic convective Brinkman-Forchheimer equations and locally monotone stochastic partial differential equationshttps://zbmath.org/1496.370812022-11-17T18:59:28.764376Z"Kinra, K."https://zbmath.org/authors/?q=ai:kinra.kush"Mohan, M. T."https://zbmath.org/authors/?q=ai:mohan.manil-tExit versus escape for stochastic dynamical systems and application to the computation of the bursting time duration in neuronal networkshttps://zbmath.org/1496.370822022-11-17T18:59:28.764376Z"Zonca, Lou"https://zbmath.org/authors/?q=ai:zonca.lou"Holcman, David"https://zbmath.org/authors/?q=ai:holcman.davidSummary: We study the exit time of two-dimensional dynamical systems perturbed by a small noise that exhibits two peculiar behaviors: (1) The maximum of the probability density function of trajectories is not located at the point attractor. The distance between the maximum and the attractor increases with the noise amplitude \(\sigma\), as shown by using WKB approximation and numerical simulations. (2) For such systems, exiting from the basin of attraction is not sufficient to guarantee a full escape, due to trajectories that can return several times inside the basin of attraction after crossing the boundary, before eventually escaping far away. We apply these results to study neuronal networks that can generate bursting events. To analyze interburst durations and their statistics, we study the phase space of a mean-field model, based on synaptic short-term changes, that exhibit burst and interburst dynamics. We find that the interburst corresponds to an escape with multiple reentries inside the basin of attraction. To conclude, escaping far away from a basin of attraction is not equivalent to reaching the boundary, thus providing an explanation for non-Poissonian long interburst durations present in neuronal dynamics.Periodic measures of reaction-diffusion lattice systems driven by superlinear noisehttps://zbmath.org/1496.370832022-11-17T18:59:28.764376Z"Lin, Yusen"https://zbmath.org/authors/?q=ai:lin.yusenSummary: The periodic measures are investigated for a class of reaction-diffusion lattice systems driven by superlinear noise defined on \(\mathbb{Z}^k\). The existence of periodic measures for the lattice systems is established in \(l^2\) by Krylov-Bogolyubov's method. The idea of uniform estimates on the tails of solutions is employed to establish the tightness of a family of distribution laws of the solutions.Random attractor for second-order stochastic delay lattice sine-Gordon equationhttps://zbmath.org/1496.370842022-11-17T18:59:28.764376Z"Li, Xintao"https://zbmath.org/authors/?q=ai:li.xintao"She, Lianbing"https://zbmath.org/authors/?q=ai:she.lianbing"Shan, Zhenpei"https://zbmath.org/authors/?q=ai:shan.zhenpeiSummary: In this paper, we prove the existence of random \(\mathcal{D}\)-attractor for the second-order stochastic delay sine-Gordon equation on infinite lattice with certain dissipative conditions, and then establish the upper bound of Kolmogorov \(\varepsilon\)-entropy for the random \(\mathcal{D}\)-attractor.Operator-theoretic framework for forecasting nonlinear time series with kernel analog techniqueshttps://zbmath.org/1496.370852022-11-17T18:59:28.764376Z"Alexander, Romeo"https://zbmath.org/authors/?q=ai:alexander.romeo"Giannakis, Dimitrios"https://zbmath.org/authors/?q=ai:giannakis.dimitriosSummary: Kernel analog forecasting (KAF), alternatively known as kernel principal component regression, is a kernel method used for nonparametric statistical forecasting of dynamically generated time series data. This paper synthesizes descriptions of kernel methods and Koopman operator theory in order to provide a single consistent account of KAF. The framework presented here illuminates the property of the KAF method that, under measure-preserving and ergodic dynamics, it consistently approximates the conditional expectation of observables that are acted upon by the Koopman operator of the dynamical system and are conditioned on the observed data at forecast initialization. More precisely, KAF yields optimal predictions, in the sense of minimal root mean square error with respect to the invariant measure, in the asymptotic limit of large data. The presented framework facilitates, moreover, the analysis of generalization error and quantification of uncertainty. Extensions of KAF to the construction of conditional variance and conditional probability functions, as well as to non-symmetric kernels, are also shown. Illustrations of various aspects of KAF are provided with applications to simple examples, namely a periodic flow on the circle and the chaotic Lorenz 63 system.Spatiotemporal pattern in a neural network with non-smooth memristorhttps://zbmath.org/1496.370862022-11-17T18:59:28.764376Z"Shi, Xuerong"https://zbmath.org/authors/?q=ai:shi.xuerong"Wang, Zuolei"https://zbmath.org/authors/?q=ai:wang.zuolei"Zhuang, Lizhou"https://zbmath.org/authors/?q=ai:zhuang.lizhouSummary: Considering complicated dynamics of non-smooth memductance function, an improved Hindmarsh-Rose neuron model is introduced by coupling with non-smooth memristor and dynamics of the improved model are discussed. Simulation results suggest that dynamics of the proposed neuron model depends on the external stimuli but not on the initial value for the magnetic flux. Furthermore, a network composed of the improved Hindmarsh-Rose neuron is addressed via single channel coupling method and spatiotemporal patterns of the network are investigated via numerical simulations with no-flux boundary condition. Firstly, development of spiral wave are discussed for different coupling strengths, different external stimuli and various initial value for the magnetic flux. Results suggest that spiral wave can be developed for coupling strength \(0 < D < 1\) when the nodes are provided with period-1 dynamics, especially, double-arm spiral wave appear for \(D = 0.4\). External stimuli changing can make spiral wave collapse and the network demonstrates chaotic state. Alternation of initial value for the magnetic flux hardly has effect on the developed spiral wave. Secondly, formation of target wave are studied for different coupling strengths, different sizes of center area with parameter diversity and various initial value for the magnetic flux. It can be obtained that, for certain size of center area with parameter diversity, target wave can be formed for coupling strength \(0 < D < 1\), while for too small size of center area with parameter diversity, target wave can hardly be formed. Change of initial value for the magnetic flux has no effect on the formation of target wave. Research results reveal the spatiotemporal patterns of neuron network to some extent and may provide some suggestions for exploring some disease of neural system.Analysis of transmission dynamics of cholera: an optimal control strategyhttps://zbmath.org/1496.370872022-11-17T18:59:28.764376Z"Adewole, Matthew O."https://zbmath.org/authors/?q=ai:adewole.matthew-olayiwola"Onifade, Akindele"https://zbmath.org/authors/?q=ai:onifade.akindele-a"Ismail, Ahmad Izani Md"https://zbmath.org/authors/?q=ai:ismail.ahmad-izani-md"Faniran, Taye"https://zbmath.org/authors/?q=ai:faniran.taye"Abdullah, Farah A."https://zbmath.org/authors/?q=ai:abdullah.farah-ainiSummary: Cholera affects populations living with poor sanitary conditions and has caused enormous morbidity and mortality. A mathematical model is presented for the spread of cholera with focus on three human populations; susceptible human, infected human and recovered human. The infected human population was subdivided into two groups-symptomatic individuals and asymptomatic individuals. We obtain the reproductive number and a sensitivity analysis of model parameters is conducted. The sensitivity analysis reveals key parameters which can be used to propose intervention strategies. Our analysis indicates that a single intervention strategy is insufficient for the eradication of the disease. Optimal control strategy is incorporated to find effective solutions for time-dependent controls for eradicating cholera epidemics. We use numerical simulations to explore various optimal control solutions involving single and multiple controls. Our results show that, as in related previous studies, the costs of controls have a direct effect on the duration and strength of each control in an optimal strategy. It is also established that a combination of multiple intervention strategies attains better results than a single-pronged approach since the strength of each control strategy is limited by available resources and social factors.Conservative, dissipative and super-diffusive behavior of a particle propelled in a regular flowhttps://zbmath.org/1496.370882022-11-17T18:59:28.764376Z"Ariel, Gil"https://zbmath.org/authors/?q=ai:ariel.gil"Schiff, Jeremy"https://zbmath.org/authors/?q=ai:schiff.jeremySummary: A recent model of \textit{G. Ariel} et al. [Phys. Rev. Lett., 118, Article 228102, (2017)] for explaining the observation of Lévy walks in swarming bacteria suggests that self-propelled, elongated particles in a periodic array of regular vortices perform a super-diffusion that is consistent with Lévy walks. The equations of motion, which are reversible in time but not volume preserving, demonstrate a new route to Lévy walking in chaotic systems. Here, the dynamics of the model is studied both analytically and numerically. It is shown that the apparent super-diffusion is due to ``sticking'' of trajectories to elliptic islands, regions of quasi-periodic orbits reminiscent of those seen in conservative systems. However, for certain parameter values, these islands coexist with asymptotically stable periodic trajectories, causing dissipative behavior on very long time scales.Co-existence thresholds in the dynamics of the plant-herbivore interaction with Allee effect and harvesthttps://zbmath.org/1496.370892022-11-17T18:59:28.764376Z"Asfaw, Manalebish Debalike"https://zbmath.org/authors/?q=ai:asfaw.manalebish-debalike"Kassa, Semu Mitiku"https://zbmath.org/authors/?q=ai:kassa.semu-mitiku"Lungu, Edward M."https://zbmath.org/authors/?q=ai:lungu.edward-mGlobal analysis of the dynamics of a piecewise linear vector field model for prostate cancer treatmenthttps://zbmath.org/1496.370902022-11-17T18:59:28.764376Z"Carvalho, Tiago"https://zbmath.org/authors/?q=ai:de-carvalho.tiago"Cristiano, Rony"https://zbmath.org/authors/?q=ai:cristiano.rony"Rodrigues, Diego S."https://zbmath.org/authors/?q=ai:rodrigues.diego-samuel"Tonon, Durval J."https://zbmath.org/authors/?q=ai:tonon.durval-joseSummary: Oncological therapies usually are applied intermittently, i.e., not continuously over time. In the periods in between, though, the cancer cells are usually left free to grow. This intermittency is a key issue in prostate cancer hormonal therapies based on androgen suppression. Here, we address this treatment modality by analyzing a piecewise smooth vector field approach. In fact, using the PSA (prostate-specific antigen) serum level as a control variable to switch between treatment and no-treatment periods of hormone therapy (androgen withdrawal), by means of typical parameter values, our theoretical analysis supports the idea that intermittent androgen suppression may prevent a prostate cancer relapse for a specific class of patients.Motility switching and front-back synchronisation in polarised cellshttps://zbmath.org/1496.370912022-11-17T18:59:28.764376Z"Estrada-Rodriguez, Gissell"https://zbmath.org/authors/?q=ai:estrada-rodriguez.gissell"Perthame, Benoit"https://zbmath.org/authors/?q=ai:perthame.benoitSummary: The combination of protrusions and retractions in the movement of polarised cells leads to understand the effect of possible synchronisation between the two ends of the cells. This synchronisation, in turn, could lead to different dynamics such as normal and fractional diffusion. Departing from a stochastic single cell trajectory, where a ``memory effect'' induces persistent movement, we derive a kinetic-renewal system at the mesoscopic scale. We investigate various scenarios with different levels of complexity, where the two ends of the cell move either independently or with partial or full synchronisation. We study the relevant macroscopic limits where we obtain diffusion, drift-diffusion or fractional diffusion, depending on the initial system. This article clarifies the form of relevant macroscopic equations that describe the possible effects of synchronised movement in cells, and sheds light on the switching between normal and fractional diffusion.Modelling the effects of ozone concentration and pulse vaccination on seasonal influenza outbreaks in Gansu Province, Chinahttps://zbmath.org/1496.370922022-11-17T18:59:28.764376Z"Jing, Shuang-Lin"https://zbmath.org/authors/?q=ai:jing.shuanglin"Huo, Hai-Feng"https://zbmath.org/authors/?q=ai:huo.hai-feng"Xiang, Hong"https://zbmath.org/authors/?q=ai:xiang.hongSummary: Common air pollutants, such as ozone \((\mathrm{O}_3)\), sulfur dioxide \((\mathrm{SO}_2)\) and nitrogen dioxide \((\mathrm{NO}_2)\), can affect the spread of influenza. We propose a new non-autonomous impulsive differential equation model with the effects of ozone and vaccination in this paper. First, the basic reproduction number of the impulsive system is obtained, and the global asymptotic stability of the disease-free periodic solution is proved. Furthermore, the uniform persistence of the system is demonstrated. Second, the unknown parameters of the ozone dynamics model are obtained by fitting the ozone concentration data by the least square method and Bootstrap. The MCMC algorithm is used to fit influenza data in Gansu Province to identify the most suitable parameter values of the system. The basic reproduction number \(R_0\) is estimated to be 1.2486 (95\%CI : (1.2470, 1.2501)). Then, a sensitivity analysis is performed on the system parameters. We find that the average annual incidence of seasonal influenza in Gansu Province is 31.3374 per 100,000 people. Influenza cases started to surge in 2016, rising by a factor of one and a half between 2014 and 2016, further increasing in 2019 (54.6909 per 100,000 population). The average incidence rate during the post-upsurge period (2017--2019) is one and a half times more than in the pre-upsurge period (2014--2016). In particular, we find that the peak ozone concentration appears 5--8 months in Gansu Province. A moderate negative correlation is seen between influenza cases and monthly ozone concentration (Pearson correlation coefficient: \( r = -0.4427\)). Finally, our results show that increasing the vaccination rate and appropriately increasing the ozone concentration can effectively prevent and control the spread of influenza.Bistability in a one-dimensional model of a two-predators-one-prey population dynamics systemhttps://zbmath.org/1496.370932022-11-17T18:59:28.764376Z"Kryzhevich, S."https://zbmath.org/authors/?q=ai:kryzhevich.sergei-gennadevich"Avrutin, V."https://zbmath.org/authors/?q=ai:avrutin.viktor"Söderbacka, G."https://zbmath.org/authors/?q=ai:soderbacka.g-j|soderbacka.gunnarSummary: In this paper, we study a classical two-predators-one-prey model. The classical model described by a system of three ordinary differential equations can be reduced to a one-dimensional bimodal map. We prove that this map has at most two stable periodic orbits. Besides, we describe the bifurcation structure of the map. Finally, we describe a mechanism that leads to bistable regimes. Taking this mechanism into account, one can easily detect parameter regions where cycles with arbitrary high periods or chaotic attractors with arbitrary high numbers of bands coexist pairwise.Homeostasis in networks with multiple input nodes and robustness in bacterial chemotaxishttps://zbmath.org/1496.370942022-11-17T18:59:28.764376Z"Madeira, João Luiz de Oliveira"https://zbmath.org/authors/?q=ai:madeira.joao-luiz-de-oliveira"Antoneli, Fernando"https://zbmath.org/authors/?q=ai:antoneli.fernando-junSummary: A biological system achieves \textit{homeostasis} when there is a regulated quantity that is maintained within a narrow range of values. Here, we consider homeostasis as a phenomenon of network dynamics. In this context, we improve a general theory for the analysis of homeostasis in network dynamical systems with distinguished input and output nodes, called `input-output networks.' The theory allows one to define `homeostasis types' of a given network in a `model independent' fashion, in the sense that the classification depends on the network topology rather than on the specific model equations. Each `homeostasis type' represents a possible mechanism for generating homeostasis and is associated with a suitable `subnetwork motif' of the original network . Our contribution is an extension of the theory to the case of networks with multiple input nodes. To showcase our theory, we apply it to bacterial chemotaxis, a paradigm for homeostasis in biochemical systems. By considering a representative model of \textit{Escherichia coli} chemotaxis, we verify that the corresponding abstract network has multiple input nodes. Thus showing that our extension of the theory allows for the inclusion of an important class of models that were previously out of reach. Moreover, from our abstract point of view, the occurrence of homeostasis in the studied model is caused by a new mechanism, called \textit{input counterweight homeostasis}. This new homeostasis mechanism was discovered in the course of our investigation and is generated by a balancing between the several input nodes of the network -- therefore, it requires the existence of at least two input nodes to occur. Finally, the framework developed here allows one to formalize a notion of `robustness' of homeostasis based on the concept of `genericity' from the theory dynamical systems. We discuss how this kind of robustness of homeostasis appears in the chemotaxis model.Dynamical analysis and chaos control of the fractional chaotic ecological modelhttps://zbmath.org/1496.370952022-11-17T18:59:28.764376Z"Mahmoud, Emad E."https://zbmath.org/authors/?q=ai:mahmoud.emad-e"Trikha, Pushali"https://zbmath.org/authors/?q=ai:trikha.pushali"Jahanzaib, Lone Seth"https://zbmath.org/authors/?q=ai:jahanzaib.lone-seth"Almaghrabi, Omar A."https://zbmath.org/authors/?q=ai:almaghrabi.omar-aSummary: In this paper the fractional version of the proposed integer order chaotic ecological system is studied. Here chaos has been observed in the competitive ecological model due to linear and nonlinear interactions among various species considering shortage of food resources. The system being important constituent of the food supply chain is analyzed using tools of dynamics viz. Lyapunov dynamics, bifurcation diagrams, existence and uniqueness of solution, the fixed point analysis and effect of fractional order on the dynamics of the system. In the presence of uncertainties and disturbances the chaos in the F.O. ecological model is controlled using adaptive SMC theory about its two fixed points. Numerical illustrations have been provided using MATLAB.Adaptive dynamics of a stoichiometric phosphorus-algae-zooplankton model with environmental fluctuationshttps://zbmath.org/1496.370962022-11-17T18:59:28.764376Z"Zhao, Shengnan"https://zbmath.org/authors/?q=ai:zhao.shengnan"Yuan, Sanling"https://zbmath.org/authors/?q=ai:yuan.sanling"Wang, Hao"https://zbmath.org/authors/?q=ai:wang.hao.2|wang.hao.13|wang.hao.3|wang.hao.12|wang.hao.7|wang.hao.9|wang.hao.11|wang.hao.1|wang.hao.4|wang.hao.6|wang.hao.10|wang.hao.5Summary: We present a stochastic evolutionary phosphorus-algae-zooplankton model with phosphorus recycling and originally investigate the patterns and outcomes of adaptive changes in algal cell size, under the influence of environmental fluctuations. The threshold that determines whether the stochastic model will ecologically persist or not is first obtained. We then introduce fitness functions with stochastic fluctuations and obtain the evolutionary conditions for continuously stable strategy (CSS) and evolutionary branching, confirmed by numerical simulation. Our results predict that environmental fluctuation could drive algal evolution toward smaller cell size. Algal cell size varies significantly with phosphorus input in the presence of zooplankton, but has no response to the changing phosphorus inflow without zooplankton, and evolutionary branching will never occur without zooplankton. With the existence of zooplankton that has a fixed trait, evolutionary branching occurs with small environmental fluctuation and moderate phosphorus inflow, and the existence of environmental fluctuation could narrow the cell size difference between the newly emerging algal species, while large fluctuation or extreme phosphorus inflow will result in CSS. Moreover, environmental fluctuation potentially benefits algal biodiversity in eutrophic environments, and oligotrophication inhibits algal diversity. For the coevolution of algae and zooplankton, evolutionary cycling could appear, i.e., algal cell size and zooplankton body size can coevolve to a stable limit cycle (the Red Queen dynamics) in an eutrophic environment. In oligotrophic or moderate phosphorus environments, the influence of environmental fluctuation on algal evolution in the coevolution process is similar to the scenario that algae evolves only.Stationary distribution, extinction and probability density function of a stochastic vegetation-water model in arid ecosystemshttps://zbmath.org/1496.370972022-11-17T18:59:28.764376Z"Zhou, Baoquan"https://zbmath.org/authors/?q=ai:zhou.baoquan"Han, Bingtao"https://zbmath.org/authors/?q=ai:han.bingtao"Jiang, Daqing"https://zbmath.org/authors/?q=ai:jiang.daqing"Hayat, Tasawar"https://zbmath.org/authors/?q=ai:hayat.tasawar"Alsaedi, Ahmed"https://zbmath.org/authors/?q=ai:alsaedi.ahmedSummary: In this paper, we study a three-dimensional stochastic vegetation-water model in arid ecosystems, where the soil water and the surface water are considered. First, for the deterministic model, the possible equilibria and the related local asymptotic stability are studied. Then, for the stochastic model, by constructing some suitable stochastic Lyapunov functions, we establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution \(\varpi (\cdot)\). In a biological interpretation, the existence of the distribution \(\varpi (\cdot)\) implies the long-term persistence of vegetation under certain conditions. Taking the stochasticity into account, a quasi-positive equilibrium \(\overline{D}^{\ast}\) related to the vegetation-positive equilibrium of the deterministic model is defined. By solving the relevant Fokker-Planck equation, we obtain the approximate expression of the distribution \(\varpi (\cdot)\) around the equilibrium \(\overline{D}^{\ast}\). In addition, we obtain sufficient condition \(\mathscr{R}_0^E <1\) for vegetation extinction. For practical application, we further estimate the probability of vegetation extinction at a given time. Finally, based on some actual vegetation data from Wuwei in China and Sahel, some numerical simulations are provided to verify our theoretical results and study the impact of stochastic noise on vegetation dynamics.On the construction of resolving control in the problem of getting close at a fixed time momenthttps://zbmath.org/1496.370982022-11-17T18:59:28.764376Z"Ushakov, Vladimir Nikolaevich"https://zbmath.org/authors/?q=ai:ushakov.vladimir-nikolaevich"Ushakov, Andreĭ Vladimirovich"https://zbmath.org/authors/?q=ai:ushakov.andrej-v"Kuvshinov, Oleg Aleksandrovich"https://zbmath.org/authors/?q=ai:kuvshinov.oleg-aleksandrovichSummary: The problem of getting close of a controlled system with a compact space in a finite-dimensional Euclidean space at a fixed time is studied. A method of constructing a solution to the problem is proposed which is based on the ideology of the maximum shift of the motion of the controlled system by the solvability set of the getting close problem.A new fractional one dimensional chaotic map and its application in high-speed image encryptionhttps://zbmath.org/1496.370992022-11-17T18:59:28.764376Z"Talhaoui, Mohamed Zakariya"https://zbmath.org/authors/?q=ai:talhaoui.mohamed-zakariya"Wang, Xingyuan"https://zbmath.org/authors/?q=ai:wang.xingyuanSummary: Chaos theory has been widely used in the design of image encryption schemes. Some low-dimensional chaotic maps have been proved to be easily predictable because of their small chaotic space. On the other hand, high-dimensional chaotic maps have a larger chaotic space. However, their structures are too complicated, and consequently, they are not suitable for real-time image encryption. Motivated by this, we propose a new fractional one-dimensional chaotic map with a large chaotic space. The proposed map has a simple structure and a high chaotic behavior in an extensive range of its control parameters values. Several chaos theoretical tools and tests have been carried out to analyze and prove the proposed map's high chaotic behavior. Moreover, we use the proposed map in the design of a novel real-time image encryption scheme. In this new scheme, we combine the substitution and permutation stages to simultaneously modify both of the pixels' positions and values. The merge of these two stages and the use of the new simple one-dimensional chaotic map significantly increase the proposed scheme's security and speed. Besides, the simulation and experimental analysis prove that the proposed scheme has high performances.Infinite sums related to the generalized Fibonacci numbershttps://zbmath.org/1496.400072022-11-17T18:59:28.764376Z"Uslu, Kemal"https://zbmath.org/authors/?q=ai:uslu.kemal"Teke, Mustafa"https://zbmath.org/authors/?q=ai:teke.mustafaSummary: Fibonacci numbers and applications related to these numbers are frequently encountered both in daily life and in various fields of science and engineering. There are many studies to sum expressions on these numbers [\textit{T. Koshy}, Fibonacci and Lucas numbers with applications. Volume I. New York, NY: Wiley (2001; Zbl 0984.11010)]. However, in later periods, generalized Fibonacci numbers, which are the more general version of Fibonacci and Lucas numbers, and also new number sequences such as \(k\)-Fibonacci numbers by Sergio Falcon have entered into the literature [\textit{S. Falcón} and \textit{Á. Plaza}, Chaos Solitons Fractals 32, No. 5, 1615--1624 (2007; Zbl 1158.11306)]. In this study, some sums of generalized Fibonacci numbers have been investigated and compared with previously obtained sums of Fibonacci and Lucas numbers, which are the special cases of these sums.Unitary conjugacy for type III subfactors and \(W^\ast\)-superrigidityhttps://zbmath.org/1496.460562022-11-17T18:59:28.764376Z"Isono, Yusuke"https://zbmath.org/authors/?q=ai:isono.yusukeThe present article establishes a new criterion for intertwining by bimodules à la Popa in the setting of type III factors. Its main result in this direction, which is contained in Theorem A and Corollary B, states that for arbitrary inclusions of \(\sigma\)-finite von Neumann algebras \(A, B \subset M\) with conditional expectations \(\mathrm{E}_A \colon 1_A M 1_A \to A\) and \(\mathrm{E}_B \colon 1_B M 1_B \to B\), respectively, intertwining \(A \preceq_M B\) is equivalent to intertwining of continuous cores in the following sense. If \((N, \omega)\) is any fixed type III\(_1\) factor with a faithful normal state, both independent of \(A\), \(B\) and \(M\), and \(\varphi, \psi \in M_*\) are faithful normal states preserved by \(\mathrm{E}_B\) and \(\mathrm{E}_A\), respectively, then intertwining of \(A\) into \(B\) inside \(M\) is equivalent to the condition
\begin{gather*}
\Pi(\mathrm{C}_{\psi \otimes \omega}(A \overline{\otimes} N)) \preceq_{\mathrm{C}_{\varphi \otimes \omega}(M \overline{\otimes} N)} \mathrm{C}_{\varphi \otimes \omega}(B \overline{\otimes} N) \text{.}
\end{gather*}
This result allows the author to employ known characterisations of intertwining in the setup of type II von Neumann algebras. In this way, he achieves a \(W^\ast\)-superrigidity result for crossed products associated with Bernoulli shifts whose base is an amenable III\(_1\) factor among all crossed product von Neumann algebras arising from state-preserving, outer actions of discrete groups on amenable factors. Further, a stable strong solidity result for arbitrary free product von Neumann algebras is obtained.
Reviewer: Sven Raum (Stockholm)Strongly outer actions of amenable groups on \(\mathcal{Z}\)-stable nuclear \(C^\ast\)-algebrashttps://zbmath.org/1496.460682022-11-17T18:59:28.764376Z"Gardella, Eusebio"https://zbmath.org/authors/?q=ai:gardella.eusebio"Hirshberg, Ilan"https://zbmath.org/authors/?q=ai:hirshberg.ilan"Vaccaro, Andrea"https://zbmath.org/authors/?q=ai:vaccaro.andreaThe paper under review concerns the action \(\alpha\) of a discrete, countable, amenable group \(G\) on a separable, simple, unital, nuclear, \(\mathcal{Z}\)-stable \(C^*\)-algebra \(A\). It is motivated by researching general conditions for the preservation of \(\mathcal{Z}\)-stability by the induced crossed product \(A \rtimes G\). Towards this end, the authors study the relation of strong outerness of \(\alpha\) to two weaker versions of the tracial Rohklin property for the stabilization \(\alpha \otimes\mathrm{id}_{\mathcal{Z}}\). The first one is the weak tracial Rohklin property that replaces projections with positive elements and has been under considerable study in the literature. The second concerns the Rohklin dimenion introduced by \textit{I. Hirshberg} et al. [Commun. Math. Phys. 335, No. 2, 637--670 (2015; Zbl 1333.46055)]. Furthermore, they provide conditions for \(\alpha\) to be cocycle conjugate to \(\alpha \otimes\mathrm{id}_{\mathcal{Z}}\), even when \(\alpha\) is not strongly outer. The paper under review contains two main results in this endeavour (under the hypotheses oulined on \(G\) and \(A\)).
First, if the orbits of the action induced by \(\alpha\) on \(T(A)\) are finite and their cardinality is uniformly bounded, they show that the strong outerness of \(\alpha\) is equivalent to the weak tracial Rohklin property for \(\alpha \otimes\mathrm{id}_{\mathcal{Z}}\); if, in particular, \(G\) is residually finite then the above are also equivalent to \(\alpha \otimes \mathrm{id}_{\mathcal{Z}}\) having finite Rohklin dimension (in fact at most \(2\)). The first equivalence extends previous results of \textit{S. Echterhoff} et al. [J. Reine Angew. Math. 639, 173--221 (2010; Zbl 1202.46081)], \textit{H. Matui} and \textit{Y. Sato} [Am. J. Math. 136, No. 6, 1441--1496 (2014; Zbl 1317.46042)], and \textit{Q.-Y. Wang} [Rocky Mt. J. Math. 48, No. 4, 1307--1344 (2018; Zbl 1408.46066)], by removing all smallness assumptions on the size of the orbits by the induced action on \(T(A)\) (modulo the uniform boundedness assumption). The second equivalence generalizes results of Liao for \(\mathbb{Z}^m\)-actions, see [\textit{H.-C. Liao}, J. Funct. Anal. 270, No. 10, 3675--3708 (2016; Zbl 1355.46055); Int. J. Math. 28, No. 7, Article ID 1750050, 22 p. (2017; Zbl 1383.46053)].
Secondly, if \(\partial_e T(A)\) is compact with \(\dim(\partial_e T(A)) < \infty\), the orbits of the induced action of \(\alpha\) on \(\partial_e T(A)\) are finite with uniformly bounded cardinality, and the orbit space \(\partial_e T(A)/ G\) is Hausdorff, then \(\alpha\) is cocycle conjugate to \(\alpha \otimes\mathrm{id}_{\mathcal{Z}}\). The conditions cover for example the case when the action induced on \(\partial_e T(A)\) factors through a finite group action. This generalizes results of \textit{Y. Sato} [Adv. Stud. Pure Math. 80, 189--210 (2019; Zbl 1434.46039)], and in turn of Matui and Sato [loc.~cit..], by weakening the assumption of a trivial action on the trace space.
In the process, the authors develop equivariant versions of complemented partitions of unity and uniform property \(\Gamma\), which are of independent interest. They show that if \(A\) is a separable, unital, nuclear \(C^*\)-algebra with non-empty trace space and with no finite-dimensional quotients, and the induced action of \(\alpha\) on \(T(A)\) has finite orbits uniformly bounded in size by \(M >0\), then: \((A, \alpha)\) has uniform property \(\Gamma\) if and only if \((A, \alpha)\) has complemented partitions of unity with constant \(M\), if and only if for every \(n \in \mathbb{N}\), there is a unital embedding of the matrix algebra \(M_n \to (A^{\mathcal{U}} \cap A') ^{\alpha^{\mathcal{U}}}\). If \(A\) is also \(\mathcal{Z}\)-stable and simple, then the above are equivalent to \((A, \alpha)\) being cocycle conjugate to \((A \otimes \mathcal{Z}, \alpha \otimes \mathrm{id}_{\mathcal{Z}})\).
Reviewer: Evgenios Kakariadis (Newcastle upon Tyne)Semigroup generations of unbounded block operator matrices based on the space decompositionhttps://zbmath.org/1496.470052022-11-17T18:59:28.764376Z"Liu, Jie"https://zbmath.org/authors/?q=ai:liu.jie|liu.jie.1|liu.jie.3|liu.jie.4|liu.jie.2|liu.jie.7|liu.jie.5"Huang, Junjie"https://zbmath.org/authors/?q=ai:huang.junjie"Chen, Alatancang"https://zbmath.org/authors/?q=ai:chen.alatancangThe authors find necessary and sufficient conditions under which an unbounded block operator matrix
\[
M = \begin{pmatrix} A & B \\
C & D \end{pmatrix}
\]
with natural domain \(\mathscr{D}(M)=(\mathscr{D}(A)\cap \mathscr{D}(C))\oplus (\mathscr{D}(B)\cap \mathscr{D}(D))\) generates a \(C_0\)-semigroup. Usually, in the literature, most of the results are presented using the diagonal domain \(\mathscr{D}(M) = \mathscr{D}(A)\oplus \mathscr{D}(D)\), using standard perturbation theorems. To prove the results in the natural domain, the authors characterize the right boundedness of \(M\) with the quadratic numerical range of \(M\), and consider the residual spectrum based on the space decomposition and quadratic complements.
Reviewer: Matheus Cheque Bortolan (Florianópolis)Hamiltonian and Lagrangian systems in contact geometryhttps://zbmath.org/1496.530032022-11-17T18:59:28.764376Z"Souto Pérez, Silvia"https://zbmath.org/authors/?q=ai:souto-perez.silvia(no abstract)Generalised Bianchi permutability for isothermic surfaceshttps://zbmath.org/1496.530072022-11-17T18:59:28.764376Z"Cho, Joseph"https://zbmath.org/authors/?q=ai:cho.joseph"Leschke, Katrin"https://zbmath.org/authors/?q=ai:leschke.katrin"Ogata, Yuta"https://zbmath.org/authors/?q=ai:ogata.yutaAuthors' abstract: Isothermic surfaces are surfaces which allow a conformal curvature line parametrisation. They form an integrable system, and Darboux transforms of isothermic surfaces obey Bianchi permutability: for two distinct spectral parameters, the corresponding Darboux transforms have a common Darboux transform which can be computed algebraically. In this paper, we discuss two-step Darboux transforms with the same spectral parameter, and show that these are obtained by a Sym-type construction: All two-step Darboux transforms of an isothermic surface are given, without further integration, by parallel sections of the associated family of the isothermic surface, either algebraically or by differentiation against the spectral parameter.
Reviewer: Ivan C. Sterling (St. Mary's City)Cartan connections and path structures with large automorphism groupshttps://zbmath.org/1496.530372022-11-17T18:59:28.764376Z"Falbel, E."https://zbmath.org/authors/?q=ai:falbel.elisha"Mion-Mouton, M."https://zbmath.org/authors/?q=ai:mion-mouton.martin"Veloso, J. M."https://zbmath.org/authors/?q=ai:veloso.jose-miguel-martinsIn this paper the authors consider compact 3-dimensional manifolds equipped with a Lagrangian contact structure or \textit{path structure}. This consists of a couple of 1-dimensional distributions \((E_1,E_2)\) such that \(E_1\oplus E_2\) is a contact distribution. If a contact form \(\theta\) is fixed, whose kernel is the contact distribution \(E_1\oplus E_2\), then the triplet \(\mathcal{T}=(E_1,E_2,\theta)\) is referred to as a \textit{strict path structure}. A (local) automorphism of \((M,\mathcal{T})\) is a (local) diffeomorphism \(f\) of \(M\) that preserves \(E_1\), \(E_2\) and \(\theta\). The group of such (local) automorphisms is denoted by \(\mathrm{Aut}^{\mathrm{loc}}(M,\mathcal{T})\). The \textit{path structure} is a geometric structure related to the geometry of second-order ordinary differential equations.
In this work, the authors use a description of the strict path geometric structure as a Cartan geometry. The fact that one has a Cartan's connection which is invariant under an automorphism group with a dense orbit implies that some components of its curvature vanish. This allows the classification of all such spaces.
The flat model for strict path geometries is the Heisenberg space \(\mathrm{Heis}(3)\) with two left-invariant directions and a fixed left-invariant contact form. Its automorphism group is \(\mathrm{Heis}(3)\times P\), where \(P\) is a group isomorphic to \(\mathbb{R}^*\). A (non-flat) constant curvature model is given by a left-invariant structure on \(\widetilde{\mathrm{SL}(2,\mathbb{R})}\) (the universal cover of \(\mathrm{SL}(2,\mathbb{R})\)), whose automorphism group is \(\widetilde{\mathrm{SL}(2,\mathbb{R})}\times \widetilde A\), where \(\widetilde A\) is a group isomorphic to \(\mathbb{R}^*\).
The main result of this paper is the following classification theorem.
\textbf{Theorem.} Let \(\mathcal{T}\) be a strict path structure on a compact three-dimensional manifold. If \(\mathcal{T}\) has a non-compact automorphism group (for the compact open topology) and a dense \(\mathrm{Aut}^{\mathrm{loc}}(M,\mathcal{T})\)-orbit, then
(1) either \((M,{\mathcal{T}})\) is, up to a constant multiplication of its contact form, isomorphic to \(\Gamma \setminus \mathrm{SL}(2,\mathbb{R})\) for some discrete subgroup \(\Gamma\) of \(\mathrm{SL}(2,\mathbb{R})\times A\) acting freely, properly and cocompactly on \(\mathrm{SL}(2,\mathbb{R})\);
(2) or \((M,{\mathcal{T}})\) is, up to finite covering, isomorphic to \(\Gamma \setminus\mathrm{Heis}(3),\) for some cocompact lattice of \(\mathrm{Heis}(3)\).
The above theorem is a generalization of \textit{É. Ghys}' theorem classifying contact-Anosov flows with smooth invariant distributions on compact three-manifolds [Ann. Sci. Éc. Norm. Supér. (4) 20, No. 2, 251--270 (1987; Zbl 0663.58025)].
Reviewer: Laura Geatti (Roma)Projectively equivalent Finsler metrics on surfaces of negative Euler characteristichttps://zbmath.org/1496.530792022-11-17T18:59:28.764376Z"Lang, Julius"https://zbmath.org/authors/?q=ai:lang.juliusThe paper deals with projectively equivalent Finsler metrics on surfaces of negative Euler characteristic. The main result lies in Theorem 1 where the author proves that on a connected closed surface of negative Euler characteristic, two real-analytic Finsler metrics have the same unparametrized oriented geodesics, if and only if they differ by a scaling constant and addition of a closed 1-form. The author proves some small results also in order to prove the main result.
Reviewer: Gauree Shanker (Bathinda)On cosymplectic dynamics. I.https://zbmath.org/1496.530832022-11-17T18:59:28.764376Z"Tchuiaga, Stephane"https://zbmath.org/authors/?q=ai:tchuiaga.stephane"Houenou, Franck"https://zbmath.org/authors/?q=ai:houenou.franck-djideme"Bikorimana, Pierre"https://zbmath.org/authors/?q=ai:bikorimana.pierreThe paper under review is an introduction to cosymplectic topology. By adapting methods from symplectic topology, the authors characterize and study several subgroups of diffeomophisms of a cosymplectic manifold.
Recall that a cosymplectic structure on a smooth manifold is given by a closed \(2\)-form \(\omega\) and a closed \(1\)-form \(\eta\) such that \(\eta \wedge \omega\) is a nowhere vanishing top-form.
In particular, among many other results, the authors show that the Reeb vector field determines the almost cosymplectic nature of a uniform limit of a sequence of almost cosymplectic diffeomorphisms. They also define and study the cosymplectic setting of Hofer and Hofer-like geometries with respect to the group of all cosymplectic diffeomorphisms isotopic to the identity map.
Reviewer: Daniele Angella (Firenze)On symplectic transformationshttps://zbmath.org/1496.530902022-11-17T18:59:28.764376Z"Springer, T. A."https://zbmath.org/authors/?q=ai:springer.tonny-albert.1Summary: This is an English translation of the Ph.D. thesis `Over symplectische transformaties' that Tonny Albert Springer, `born in's-Gravenhage in 1926', submitted as thesis for -- as is stated on the original frontispiece - \textit{the degree of doctor in mathematics and physics at Leiden University on the authority of the rector magnificus Dr. J.H. Boeke, professor in the faculty of law, to be defended against the objections of the Faculty of Mathematics and Physics on Wednesday October 17 1951 at 4 p.m.}, with promotor Prof. dr. H. D. Kloosterman.A universal coregular countable second-countable spacehttps://zbmath.org/1496.540192022-11-17T18:59:28.764376Z"Banakh, Taras"https://zbmath.org/authors/?q=ai:banakh.taras-o"Stelmakh, Yaryna"https://zbmath.org/authors/?q=ai:stelmakh.yarynaIn this interesting paper, the authors present a topological characterization of the infinite rational projective space \({\mathbb Q}P^\infty\). It is topologically, the unique countable, second countable space that possesses a superskeleton. Among its properties are the Hausdorff property, it is coregular and has very strong homogeneity properties. Moreover, it is a universal object for the class of all countable, second countable coregular spaces. The proof of the characterization theorem is quite involved and long. As the paper demonstrates, there are many spaces homeomorphic to \({\mathbb Q}P^\infty\) that surface in several seemingly unrelated situations. It is unknown whether the famous Golomb (or Kirch) space contains a subspace homeomorphic to \({\mathbb Q}P^\infty\). This paper is an absolute must for anybody interested in countable connected Hausdorff spaces.
Reviewer: Jan van Mill (Amsterdam)Surfaces of Section for Seifert fibrationshttps://zbmath.org/1496.570292022-11-17T18:59:28.764376Z"Albach, Bernhard"https://zbmath.org/authors/?q=ai:albach.bernhard"Geiges, Hansjörg"https://zbmath.org/authors/?q=ai:geiges.hansjorgConsider a Seifert manifold given by a closed oriented 3-manifold \(M\) endowed with a Seifert fibration \(M\to B\) over a closed oriented surface \(B\). With these orientability assumptions, by a classical result of \textit{D. B. A. Epstein} [Ann. Math. (2) 95, 66--82 (1972; Zbl 0231.58009)], the fibration is given by an action of the circle. This structure is determined by its Seifert invariants, \(M=M(g;(\alpha_1,\beta_1),\dots,(\alpha_n,\beta_n))\), where \(g\) is the genus of \(B\), the \(\alpha_i\in\mathbb Z^+\) are the multiplicities of the singular fibres, and every \(\beta_i\in\mathbb Z\) is coprime with \(\alpha_i\) and describes the local behaviour around the corresponding singular fibre; actually, regular fibres may be also used in this description, and they correspond to the case \(\alpha_i=1\). The Euler number of the Seifert fibration is defined as \(e=-\sum_i\beta_i/\alpha_i\). Two Seifert invariants determine the same Seifert manifold if and only if they correspond by a sequence of the following operations: permuting the pairs \((\alpha_i,\beta_i)\); adding a pair \((1,0)\); and replacing every \((\alpha_i,\beta_i)\) with \((\alpha_i,\beta_i+k_i\alpha_i)\), where the \(k_i\in\mathbb Z\) satisfy \(\sum_ik_i=0\). A global surface of section is an embedded compact surface \(\Sigma\subset M\) whose boundary \(\partial\Sigma\) is a union of fibres and whose interior intersects all other fibres transversely. The regular fibres intersecting the interior of \(\Sigma\) have the same number \(d\) of intersection points; the term a \(d\)-section is also used. Orient \(\Sigma\) so that its interior has positive intersection with the fibres. Then \(\Sigma\) is said to be positive if the induced orientation of its boundary agrees with the orientation of the fibres.
The main theorem of the paper characterizes the existence of a positive \(d\)-section \(\Sigma\) in terms of the Seifert invariants; in that case, \(\Sigma\) is unique up to isotopy. Moreover the result relates the Seifert invariants, the Euler number \(e\), the genus of \(\Sigma\), and the number of connected components of \(\Sigma\) and \(\partial\Sigma\). As a corollary, every Seifert manifold (with \(e\le0\)) admits a (positive) \(d\)-section for some \(d\in\mathbb Z^+\). The authors also describe the positive \(d\)-sections for all Seifert fibrations of \(S^3\), where the bases are weighted projective lines, and the \(d\)-sections are algebraic surfaces in weighted projective planes. These results generalize previous results for the case of Reeb dynamics.
Reviewer: Jesus A. Álvarez López (Santiago de Compostela)Positive entropy using Hecke operators at a single placehttps://zbmath.org/1496.580122022-11-17T18:59:28.764376Z"Shem-Tov, Zvi"https://zbmath.org/authors/?q=ai:shemtov.zviSummary: We prove the following statement: let \(X=\mathrm{SL}_n(\mathbb{Z})\backslash\mathrm{SL}_n(\mathbb{R})\) and consider the standard action of the diagonal group \(A<\mathrm{SL}_n(\mathbb{R})\) on it. Let \(\mu\) be an \(A\)-invariant probability measure on \(X\), which is a limit
\[
\mu=\lambda\lim\limits_i|\phi_i|^2dx,
\]
where \(\phi_i\) are normalized eigenfunctions of the Hecke algebra at some fixed place \(p\) and \(\lambda>0\) is some positive constant. Then any regular element \(a\in A\) acts on \(\mu\) with positive entropy on almost every ergodic component. We also prove a similar result for lattices coming from division algebras over \(\mathbb{Q}\) and derive a quantum unique ergodicity result for the associated locally symmetric spaces. This generalizes a result of \textit{S. Brooks} and \textit{E. Lindenstrauss} [Invent. Math. 198, No. 1, 219--259 (2014; Zbl 1343.58016)].High-efficiency chaotic time series prediction based on time convolution neural networkhttps://zbmath.org/1496.621502022-11-17T18:59:28.764376Z"Cheng, Wei"https://zbmath.org/authors/?q=ai:cheng.wei"Wang, Yan"https://zbmath.org/authors/?q=ai:wang.yan.6|wang.yan.5|wang.yan.3"Peng, Zheng"https://zbmath.org/authors/?q=ai:peng.zheng"Ren, Xiaodong"https://zbmath.org/authors/?q=ai:ren.xiaodong"Shuai, Yubei"https://zbmath.org/authors/?q=ai:shuai.yubei"Zang, Shengyin"https://zbmath.org/authors/?q=ai:zang.shengyin"Liu, Hao"https://zbmath.org/authors/?q=ai:liu.hao|liu.hao.2|liu.hao.1"Cheng, Hao"https://zbmath.org/authors/?q=ai:cheng.hao"Wu, Jiagui"https://zbmath.org/authors/?q=ai:wu.jiaguiSummary: The prediction of chaotic time series is important for both science and technology. In recent years, this type of prediction has improved significantly with the development of deep learning. Here, we propose a temporal convolutional network (TCN) model for the prediction of chaotic time series. Our TCN model offers highly stable training, high parallelism, and flexible perception field. Comparative experiments with the classic long short-term memory (LSTM) network and hybrid (CNN-LSTM) neural network show that the TCN model can reduce the training time by a factor of more than two. Furthermore, the network can focus on more important information because of the attention mechanism. By embedding the convolutional block attention module (CBAM), which combines the spatial and channel attention mechanisms, we obtain a new model, TCN-CBAM. This model is comprehensively better than the LSTM, CNN-LSTM, and TCN models in the prediction of classical systems (Chen system, Lorenz system, and sunspots). In terms of prediction accuracy, the TCN-CBAM model obtains better results for the four main evaluation indicators: root mean square error, mean absolute error, coefficient of determination, and Spearman's correlation coefficient, with a maximum increase of 41.4\%. The TCN-CBAM has also the shortest training times among the previous classic four models.Novel chaotic systems with fractional differential operators: numerical approacheshttps://zbmath.org/1496.650942022-11-17T18:59:28.764376Z"Sweilam, N. H."https://zbmath.org/authors/?q=ai:sweilam.nasser-hassan"AL-Mekhlafi, S. M."https://zbmath.org/authors/?q=ai:al-mekhlafi.seham-mahyoub"Mohamed, D. G."https://zbmath.org/authors/?q=ai:mohamed.d-gSummary: The purpose of this paper is to study numerically the behavior of two novel different classes of fractional order chaotic systems. These systems are; the fractal-fractional hyperchaotic finance system and the fractal-fractional Bloch system with time delay. The fractal-fractional derivatives are defined in the Caputo and Riemann-Liouville senses. Two Grünwald-Letnikov nonstandard finite difference schemes are presented to study the proposed chaotic systems. Moreover the stability analysis of the used methods are proved. In order to show the simplicity and effectively of the proposed methods, numerical simulations and comparative studies are given.A discrete analogue of the Lyapunov function for hyperbolic systemshttps://zbmath.org/1496.651032022-11-17T18:59:28.764376Z"Aloev, R. D."https://zbmath.org/authors/?q=ai:aloev.r-d"Hudayberganov, M. U."https://zbmath.org/authors/?q=ai:hudayberganov.m-uSummary: We study the difference splitting scheme for the numerical calculation of stable solutions of a two-dimensional linear hyperbolic system with dissipative boundary conditions in the case of constant coefficients with lower terms. A discrete analogue of the Lyapunov function is constructed and an a priori estimate is obtained for it. The obtained a priori estimate allows us to assert the exponential stability of the numerical solution.Hamiltonian preserving nonlinear opticshttps://zbmath.org/1496.651062022-11-17T18:59:28.764376Z"Baturin, S. S."https://zbmath.org/authors/?q=ai:baturin.s-sSummary: In this paper we present a method of constructing a discrete nonlinear accelerator lattice that has an approximate integral of motion that is given upfront. The integral under consideration is a Hamiltonian in normalized (canonical) coordinates that is preserved by a lattice with a given accuracy. We establish a connection between the integrator of a Hamiltonian in normalized coordinates and a real lens arrangement. We consider accelerator as an analog computer and apply known algorithms of high-order symplectic integrators, to produce several nonlinear lattices. Following this concept we introduce new lattice design based on Ruth and Yoshida integrator that could be experimentally tested at existing accelerator facilities and Paul traps.Sharp error estimates on a stochastic structure-preserving scheme in computing effective diffusivity of 3D chaotic flowshttps://zbmath.org/1496.651922022-11-17T18:59:28.764376Z"Wang, Zhongjian"https://zbmath.org/authors/?q=ai:wang.zhongjian"Xin, Jack"https://zbmath.org/authors/?q=ai:xin.jack-x"Zhang, Zhiwen"https://zbmath.org/authors/?q=ai:zhang.zhiwenImpact of leakage delay on bifurcation in fractional-order complex-valued neural networkshttps://zbmath.org/1496.682902022-11-17T18:59:28.764376Z"Xu, Changjin"https://zbmath.org/authors/?q=ai:xu.changjin"Liao, Maoxin"https://zbmath.org/authors/?q=ai:liao.maoxin"Li, Peiluan"https://zbmath.org/authors/?q=ai:li.peiluan"Yuan, Shuai"https://zbmath.org/authors/?q=ai:yuan.shuaiSummary: During the past decades, integer-order complex-valued neural networks have attracted great attention since they have been widely applied in in many fields of engineering technology. However, the investigation on fractional-order complex-valued neural networks, which are more appropriate to characterize the dynamical nature of neural networks, is rare. In this manuscript, we are to consider the stability and the existence of Hopf bifurcation of fractional-order complex-valued neural networks. By separating the coefficients and the activation functions into their real and imaginary parts and choosing the time delay as bifurcation parameter, we establish a set of sufficient conditions to ensure the stability of the equilibrium point and the existence of Hopf bifurcation for the involved network. The study shows that both the fractional order and the leakage delay have an important impact on the stability and the existence of Hopf bifurcation of the considered model. Some suitable numerical simulations are implemented to illustrate the pivotal theoretical predictions. At last, we ends this article with a simple conclusion.Discontinuous dynamical behaviors in a 2-DOF friction collision system with asymmetric dampinghttps://zbmath.org/1496.700102022-11-17T18:59:28.764376Z"Cao, Jing"https://zbmath.org/authors/?q=ai:cao.jing"Fan, Jinjun"https://zbmath.org/authors/?q=ai:fan.jinjunSummary: By using the flow switchability theory in discontinuous dynamical systems, this paper deals with the discontinuous dynamical behaviors of a two degrees of freedom system with asymmetric damping, where considering that friction and impact coexist and the static and dynamic friction coefficients are different. Because of the particularity of friction force, the flow barriers on the velocity boundary that affect the leaving flow are considered in this paper. Based on discontinuity that is caused by the sudden change of friction force or the collision between two objects, the phase space of motion for the object is divided into several different domains and boundaries; and with the help of the analysis of vector fields and G-functions on the corresponding discontinuous boundaries or in domains, the analytical conditions for all possible motions are obtained, which is used to determine the switching of motion state in this system. Finally, numerical simulations are presented to better understand the analytical conditions of the stick, grazing, impact, stuck and periodic motions.Discrete dynamic equilibrium model for a complex problem of flutter interactionshttps://zbmath.org/1496.740732022-11-17T18:59:28.764376Z"Ario, Ichiro"https://zbmath.org/authors/?q=ai:ario.ichiroSummary: The dynamic bifurcation analysis of the nonlinear oscillation of a simple fluidelastic structure is presented. This structure is a cantilever beam in a flow, and it behaves as a nonlinear system without potential energy. The structure demonstrates complex flutter behaviour that varies with the controlled flow velocity. We observe the flutter behaviour in a flow experiment, and the motion is characterised with this present model based on chaos theory of discrete dynamics. We can readily find the solution of the simple system, with which it is possible to create a map of the complex flutter behaviour.Thermal instability and chaos in a hybrid nanofluid flowhttps://zbmath.org/1496.760522022-11-17T18:59:28.764376Z"Dèdèwanou, S. J."https://zbmath.org/authors/?q=ai:dedewanou.s-j"Monwanou, A. V."https://zbmath.org/authors/?q=ai:monwanou.a-v"Koukpémèdji, A. A."https://zbmath.org/authors/?q=ai:koukpemedji.a-a"Hinvi, L. A."https://zbmath.org/authors/?q=ai:hinvi.laurent-amoussou"Miwadinou, C. H."https://zbmath.org/authors/?q=ai:miwadinou.clement-hodevewan"Chabi Orou, J. B."https://zbmath.org/authors/?q=ai:chabi-orou.jean-bDispersive wave propagation of the nonlinear Sasa-Satsuma dynamical system with computational and analytical soliton solutionshttps://zbmath.org/1496.780172022-11-17T18:59:28.764376Z"Simbawa, Eman"https://zbmath.org/authors/?q=ai:simbawa.eman-a"Seadawy, Aly R."https://zbmath.org/authors/?q=ai:seadawy.aly-r"Sugati, Taghreed G."https://zbmath.org/authors/?q=ai:sugati.taghreed-gSummary: The Sasa-Satsuma equation on a continuous background describes a nonlinear fiber system with higher-order effects including the third-order dispersion and Kerr dispersion. The Sasa-Satsuma equations describe the simultaneous propagation of two ultrashort pulses in the birefringent or two-mode fiber with the third-order dispersion, self-steepening, and stimulated Raman in scattering effects, and govern the propagation of ultra-fast pulses in optical fiber transmission systems. We consider the Sasa-Satsuma equation, which is one of the integrable extensions of the nonlinear Schrödinger equations. We find the functional integral and the Lagrangian of this model. We derived the computational and analytical soliton solutions of the nonlinear Sasa-Satsuma dynamical system. We discuss the stability analysis for our solutions.Modulated wave and modulation instability gain brought by the cross-phase modulation in birefringent fibers having anti-cubic nonlinearityhttps://zbmath.org/1496.810472022-11-17T18:59:28.764376Z"Abbagari, Souleymanou"https://zbmath.org/authors/?q=ai:abbagari.souleymanou"Saliou, Youssoufa"https://zbmath.org/authors/?q=ai:saliou.youssoufa"Houwe, Alphonse"https://zbmath.org/authors/?q=ai:houwe.alphonse"Akinyemi, Lanre"https://zbmath.org/authors/?q=ai:akinyemi.lanre"Inc, Mustafa"https://zbmath.org/authors/?q=ai:inc.mustafa"Bouetou, Thomas B."https://zbmath.org/authors/?q=ai:bouetou-bouetou.thomasSummary: In this paper, we investigate the modulated wave and W-shaped profile in birefringent fibers having the anti-cubic nonlinearity terms. We use the traveling wave hypothesis to show out the velocity of the soliton and the constraint relation on the anti-cubic nonlinear terms. We use the Jacobi elliptic function solutions to point out two types of combined solutions. After some assumption on the modulus of the Jacobi elliptic function, we have shown out the combined bright-bright soliton and dark-dark soliton-like solutions. We use the linearizing algorithm to analyze the modulation instability (MI) growth rate. We have shown that the anti-cubic nonlinear terms and cross-phase modulation (XPM) can increase MI bands and the amplitude of the MI growth rate. To corroborate the prediction made on analytical results, we use the numerical investigation to show the propagation of the modulated wave and W-shaped profile in terms of cell index. We exhibited through the numerical results that the modulated wave can conserve high energy during its propagation in birefringent fibers. The obtained results will certainly open new perspectives in optical fibers during the transmission of huge data.On the exact revival of Morse oscillator wave packetshttps://zbmath.org/1496.810502022-11-17T18:59:28.764376Z"Chatterjee, Supriya"https://zbmath.org/authors/?q=ai:chatterjee.supriyaSummary: The exact analytic expressions of the auto function and Husimi distribution function for a Morse oscillator wave packet have been derived and we use them to see the evolution of the wave packet. The dynamics of Morse oscillator wave packets for the dimers \(\mathrm{ArXe}\), \(\mathrm{Be}_2\) and \(\mathrm{Li}_2\) have been discussed. Special emphasis has been given on the revival phenomenon of such wave packets. It is obtained that the exact revivals of wave packets for \(\mathrm{ArXe}\), \(\mathrm{Be}_2\) and \(\mathrm{Li}_2\) do not occur at the revival times \((t_{rev})\) but at the instances 3.5, 8.5 and 33.5 times and their simple multiple of \(t_{rev}\) respectively.Super Hirota bilinear equations for the super modified BKP hierarchyhttps://zbmath.org/1496.811122022-11-17T18:59:28.764376Z"Chen, Huizhan"https://zbmath.org/authors/?q=ai:chen.huizhanSummary: In this paper, the super modified BKP (SmBKP) hierarchy is constructed from the perspective of the neutral free superfermions by using highest weight representations of the infinite-dimensional Lie superalgebra \(\mathfrak{b}_{\infty|\infty}(\mathfrak{g})\). Based upon this, the corresponding super Hirota bilinear identity of the SmBKP hierarchy is obtained by using the super Boson-Fermion correspondence of type B, and some specific examples of super Hirota bilinear equations are given. The super bilinear identity with respect to super wave and adjoint wave functions is also constructed. At last, we also give a class of solutions other than group orbit by the neutral free superfermions.Transition pathways in cylinder-gyroid interfacehttps://zbmath.org/1496.820272022-11-17T18:59:28.764376Z"Yao, Xiaomei"https://zbmath.org/authors/?q=ai:yao.xiaomei"Xu, Jie"https://zbmath.org/authors/?q=ai:xu.jie"Zhang, Lei"https://zbmath.org/authors/?q=ai:zhang.lei.4Summary: When two distinct ordered phases contact, the interface may exhibit rich and fascinating structures. Focusing on the Cylinder-Gyroid interface system, transition pathways connecting various interface morphologies are studied armed with the Landau-Brazovskii model. Specifically, minimum energy paths are obtained by computing transition states with the saddle dynamics. We present four primary transition pathways connecting different local minima, representing four different mechanisms of the formation of the Cylinder-Gyroid interface. The connection of Cylinder and Gyroid can be either direct or indirect via Fddd with three different orientations. Under different displacements, each of the four pathways may have the lowest energy.Very special linear gravity: a gauge-invariant graviton masshttps://zbmath.org/1496.830012022-11-17T18:59:28.764376Z"Alfaro, Jorge"https://zbmath.org/authors/?q=ai:alfaro.jorge"Santoni, Alessandro"https://zbmath.org/authors/?q=ai:santoni.alessandroSummary: Linearized gravity in the Very Special Relativity (VSR) framework is considered. We prove that this theory allows for a non-zero graviton mass \(m_g\) without breaking gauge invariance nor modifying the relativistic dispersion relation. We find the analytic solution for the new equations of motion in our gauge choice, verifying as expected the existence of only two physical degrees of freedom. Finally, through the geodesic deviation equation, we confront some results for classic gravitational waves (GW) with the VSR ones: we see that the ratios between VSR effects and classical ones are proportional to \((m_g/E)^2\), \(E\) being the energy of a graviton in the GW. For GW detectable by the interferometers LIGO and VIRGO this ratio is at most \(10^{-20}\). However, for GW in the lower frequency range of future detectors, like LISA, the ratio increases significantly to \(10^{-10}\), that combined with the anisotropic nature of VSR phenomena may lead to observable effects.Quantum corrections enhance chaos: study of particle motion near a generalized Schwarzschild black holehttps://zbmath.org/1496.830242022-11-17T18:59:28.764376Z"Bera, Avijit"https://zbmath.org/authors/?q=ai:bera.avijit"Dalui, Surojit"https://zbmath.org/authors/?q=ai:dalui.surojit"Ghosh, Subir"https://zbmath.org/authors/?q=ai:ghosh.subir.1|ghosh.subir|ghosh.subir-kumar"Vagenas, Elias C."https://zbmath.org/authors/?q=ai:vagenas.elias-cSummary: The paper is devoted to a detailed study of the effects of quantum corrections on the chaotic behaviour in the dynamics of a (massless) probe particle near the horizon of a generalized Schwarzschild black hole. Two possible origins inducing the modification of black hole metric are considered separately; the noncommutative geometry inspired metric (suggested by Nicolini, Smailagic and Spallucci) and the metric with quantum field theoretic corrections (derived by Donoghue). Our results clearly show that in both cases, the metric extensions favour chaotic behaviour, namely chaos is attained for relatively lower particle energy. This is demonstrated numerically by exhibiting the breaking of the KAM tori in Poincaré sections of particle trajectories and also via explicit computation of the (positive) Lyapunov exponents of the trajectories.Aspects of three-dimensional higher curvatures gravitieshttps://zbmath.org/1496.830252022-11-17T18:59:28.764376Z"Bueno, Pablo"https://zbmath.org/authors/?q=ai:bueno.pablo"Cano, Pablo A."https://zbmath.org/authors/?q=ai:cano.pablo-a"Llorens, Quim"https://zbmath.org/authors/?q=ai:llorens.quim"Moreno, Javier"https://zbmath.org/authors/?q=ai:moreno.javier"van der Velde, Guido"https://zbmath.org/authors/?q=ai:van-der-velde.guidoFast inertial dynamic algorithm with smoothing method for nonsmooth convex optimizationhttps://zbmath.org/1496.900602022-11-17T18:59:28.764376Z"Qu, Xin"https://zbmath.org/authors/?q=ai:qu.xin"Bian, Wei"https://zbmath.org/authors/?q=ai:bian.weiSummary: In order to solve the minimization of a nonsmooth convex function, we design an inertial second-order dynamic algorithm, which is obtained by approximating the nonsmooth function by a class of smooth functions. By studying the asymptotic behavior of the dynamic algorithm, we prove that each trajectory of it weakly converges to an optimal solution under some appropriate conditions on the smoothing parameters, and the convergence rate of the objective function values is \(o\left( t^{-2}\right)\). We also show that the algorithm is stable, that is, this dynamic algorithm with a perturbation term owns the same convergence properties when the perturbation term satisfies certain conditions. Finally, we verify the theoretical results by some numerical experiments.Nonlinearities in economics. An interdisciplinary approach to economic dynamics, growth and cycleshttps://zbmath.org/1496.910102022-11-17T18:59:28.764376ZPublisher's description: This interdisciplinary book argues that the economy has an underlying non-linear structure and that business cycles are endogenous, which allows a greater explanatory power with respect to the traditional assumption that dynamics are stochastic and shocks are exogenous.
The first part of this work is formal-methodological and provides the mathematical background needed for the remainder, while the second part presents the view that signal processing involves construction and deconstruction of information and that the efficacy of this process can be measured. The third part focuses on economics and provides the related background and literature on economic dynamics and the fourth part is devoted to new perspectives in understanding nonlinearities in economic dynamics: growth and cycles.
By pursuing this approach, the book seeks to (1) determine whether, and if so where, common features exist, (2) discover some hidden features of economic dynamics, and (3) highlight specific indicators of structural changes in time series. Accordingly, it is a must read for everyone interested in a better understanding of economic dynamics, business cycles, econometrics and complex systems, as well as non-linear dynamics and chaos theory.
The articles of this volume will be reviewed individually.
Indexed articles:
\textit{Orlando, Giuseppe; Pisarchik, Alexander N.; Stoop, Ruedi}, Introduction, 1-9 [Zbl 07616822]
\textit{Orlando, Giuseppe; Taglialatela, Giovanni}, Dynamical systems, 13-37 [Zbl 07616823]
\textit{Orlando, Giuseppe; Taglialatela, Giovanni}, An example of nonlinear dynamical system: the logistic map, 39-50 [Zbl 07616824]
\textit{Orlando, Giuseppe; Stoop, Ruedi; Taglialatela, Giovanni}, Bifurcations, 51-72 [Zbl 07616825]
\textit{Yoshida, Hiroyuki}, From local bifurcations to global dynamics: Hopf systems from the applied perspective, 73-86 [Zbl 07616826]
\textit{Orlando, Giuseppe; Stoop, Ruedi; Taglialatela, Giovanni}, Chaos, 87-103 [Zbl 07616827]
\textit{Orlando, Giuseppe; Stoop, Ruedi; Taglialatela, Giovanni}, Embedding dimension and mutual information, 105-108 [Zbl 07616828]
\textit{Stoop, Ruedi}, Signal processing, 111-121 [Zbl 07616829]
\textit{Rossa, Fabio Della; Guerrero, Julio; Orlando, Giuseppe; Taglialatela, Giovanni}, Applied spectral analysis, 123-139 [Zbl 07616830]
\textit{Orlando, Giuseppe; Zimatore, Giovanna; Giuliani, Alessandro}, Recurrence quantification analysis: theory and applications, 141-150 [Zbl 07616831]
\textit{Orlando, Giuseppe; Sportelli, Mario}, On business cycles and growth, 153-168 [Zbl 07616832]
\textit{Orlando, Giuseppe}, Trade-cycle oscillations: the Kaldor model and the Keynesian Hansen-Samuelson principle of acceleration and multiplier, 169-176 [Zbl 07616833]
\textit{Orlando, Giuseppe; Sportelli, Mario; Rossa, Fabio Della}, The Harrod model, 177-189 [Zbl 07616834]
\textit{Orlando, Giuseppe; Sportelli, Mario}, Growth and cycles as a struggle: Lotka-Volterra, Goodwin and Phillips, 191-208 [Zbl 07616835]
\textit{Stoop, Ruedi}, Stable periodic economic cycles from controlling, 209-244 [Zbl 07616836]
\textit{Orlando, Giuseppe}, Kaldor-Kalecki new model on business cycles, 247-268 [Zbl 07616837]
\textit{Orlando, Giuseppe; Zimatore, Giovanna}, Recurrence quantification analysis of business cycles, 269-282 [Zbl 07616838]
\textit{Orlando, Giuseppe; Rossa, Fabio Della}, An empirical test of Harrod's model, 283-294 [Zbl 07616839]
\textit{Araujo, Ricardo Azevedo; Moreira, Helmar Nunes}, Testing a Goodwin's model with capacity utilization to the US economy, 295-313 [Zbl 07616840]
\textit{Chen, Pu; Semmler, Willi}, Financial stress, regime switching and macrodynamics, 315-335 [Zbl 07616841]Interaction patterns and coordination in two population groups: a dynamic perspectivehttps://zbmath.org/1496.910212022-11-17T18:59:28.764376Z"Xu, Bo"https://zbmath.org/authors/?q=ai:xu.bo"Wang, Ying"https://zbmath.org/authors/?q=ai:wang.ying.8|wang.ying.4|wang.ying|wang.ying.1|wang.ying.5|wang.ying.3|wang.ying.2|wang.ying.6"Han, Yu"https://zbmath.org/authors/?q=ai:han.yu"He, Yuchang"https://zbmath.org/authors/?q=ai:he.yuchang"Wang, Ziwei"https://zbmath.org/authors/?q=ai:wang.ziweiSummary: We examine the influence of interaction patterns between different groups on the replicator dynamics of evolutionary coordination game. We show that: (1) Global coordination is always a possible solution in two-group coordination games; (2) There is no stable rest point within or on the axes of the phase plane \(F\), i.e. within each population group, internal coordination is the only stable outcome at equilibrium; (3) If the interaction frequency exceeds a threshold value \((p>1/2\) in our model), a moderate wealth distribution ratio will always drive the system to converge at global coordination.Hidden and self-excited attractors in a heterogeneous Cournot oligopoly modelhttps://zbmath.org/1496.910572022-11-17T18:59:28.764376Z"Danca, Marius-F."https://zbmath.org/authors/?q=ai:danca.marius-florin"Lampart, Marek"https://zbmath.org/authors/?q=ai:lampart.marekSummary: In this paper it is numerically shown that the dynamics of a heterogeneous Cournot oligopoly model depending on two bifurcation parameters can exhibit hidden and self-excited attractors. The system has a single equilibrium and a line of equilibria. The bifurcation diagrams show that the system admits several attractor coexistence windows, where the hidden attractors can be found. Depending on the parameters ranges, the coexistence windows present combinations of periodic, quasiperiodic and chaotic attractors.Revisiting the business cycle model with cubic nonlinear investment functionhttps://zbmath.org/1496.910602022-11-17T18:59:28.764376Z"Muñoz-Guillermo, María"https://zbmath.org/authors/?q=ai:munoz-guillermo.mariaSummary: \textit{T. Puu} [Nonlinear economic dynamics. Berlin etc.: Springer-Verlag (1989; Zbl 0695.90002)] introduced a symmetric business cycle model with a cubic investment function depending on a parameter, nevertheless the fact that symmetry is not a needed condition for the investment function was highlighted in [\textit{T. Puu} and \textit{I. Sushko}, Chaos Solitons Fractals 19, No. 3, 597--612 (2004; Zbl 1068.91054)]. In this paper, the model for business cycles proposed by Puu is revisited using a topological entropy approach, in particular, we prove the existence of topological chaos for a wide range of parameter values. Moreover, a non-symmetric cubic investment map generalizing the model introduced by Puu is considered, giving answer to the posed question.The kinetic space of multistationarity in dual phosphorylationhttps://zbmath.org/1496.920282022-11-17T18:59:28.764376Z"Feliu, Elisenda"https://zbmath.org/authors/?q=ai:feliu.elisenda"Kaihnsa, Nidhi"https://zbmath.org/authors/?q=ai:kaihnsa.nidhi"de Wolff, Timo"https://zbmath.org/authors/?q=ai:de-wolff.timo"Yürük, Oğuzhan"https://zbmath.org/authors/?q=ai:yuruk.oguzhanIn biology and biochemistry, phosphorylation (as well as its inverse, dephosphorylation) is a fundamental modification process consisting in the attachment (or removal, resp.) of a phosphate group. It is important in cell signaling and is a special case of post-translational modification (PTM) in that it, e.g., changes the behavior of a compound w.r.t. a membrane. It is managed by an enzyme reaction open to quantitative description by the ODE system of a mass action reaction network. The standard machinery is applied to the enzyme reaction of a dual phosphorylation, involving a substrate with two phosphorylation sites and two enzymes, resulting in a polynomial ODE system with nine species concentrations and 12 rate constants (parameters). The objective is to study the parameter space w.r.t. points of multistationarity. In the situation investigated here, real algebraic geometry allows rather precise statements on the parameter areas of monostationarity and multistationarity, their boundaries and their connectedness. Suitable polynomials, their signs, Newton polytopes and cylindrical algebraic decompositions play a decisive role, as well as application of symbolic algorithms. The approach explored here is relevant not only for the system itself, but lends itself also to test the examination of similar models.
Reviewer: Dieter Erle (Dortmund)Mathematical models of leukaemia and its treatment: a reviewhttps://zbmath.org/1496.920322022-11-17T18:59:28.764376Z"Chulián, S."https://zbmath.org/authors/?q=ai:chulian.salvador"Martínez-Rubio, Á."https://zbmath.org/authors/?q=ai:martinez-rubio.alvaro"Rosa, M."https://zbmath.org/authors/?q=ai:rosa.mario-s|rosa.milton|rosa.maria-carlota|rosa.m-emilia|rosa.m-v|rosa.muhammad-ridho|rosa.marcio-a-f|rosa.marco|rosa.maria-j"Pérez-García, V. M."https://zbmath.org/authors/?q=ai:perez-garcia.victor-mSummary: Leukaemia accounts for around 3\% of all cancer types diagnosed in adults, and is the most common type of cancer in children of paediatric age (typically ranging from 0 to 14 years). There is increasing interest in the use of mathematical models in oncology to draw inferences and make predictions, providing a complementary picture to experimental biomedical models. In this paper we recapitulate the state of the art of mathematical modelling of leukaemia growth dynamics, in time and response to treatment. We intend to describe the mathematical methodologies, the biological aspects taken into account in the modelling, and the conclusions of each study. This review is intended to provide researchers in the field with solid background material, in order to achieve further breakthroughs in the promising field of mathematical biology.Is the Allee effect relevant to stochastic cancer model?https://zbmath.org/1496.920372022-11-17T18:59:28.764376Z"Sardar, Mrinmoy"https://zbmath.org/authors/?q=ai:sardar.mrinmoy"Khajanchi, Subhas"https://zbmath.org/authors/?q=ai:khajanchi.subhas(no abstract)Global Hopf branches in a delayed model for immune response to HTLV-1 infections: coexistence of multiple limit cycleshttps://zbmath.org/1496.920432022-11-17T18:59:28.764376Z"Li, Michael Y."https://zbmath.org/authors/?q=ai:li.michael-yi"Lin, Xihui"https://zbmath.org/authors/?q=ai:lin.xihui"Wang, Hao"https://zbmath.org/authors/?q=ai:wang.hao.4(no abstract)Nilpotent singularities and chaos: tritrophic food chainshttps://zbmath.org/1496.920892022-11-17T18:59:28.764376Z"Drubi, Fátima"https://zbmath.org/authors/?q=ai:drubi.fatima"Ibáñez, Santiago"https://zbmath.org/authors/?q=ai:ibanez.santiago"Pilarczyk, Paweł"https://zbmath.org/authors/?q=ai:pilarczyk.pawelSummary: Local bifurcation theory is used to prove the existence of chaotic dynamics in two well-known models of tritrophic food chains. To the best of our knowledge, the simplest technique to guarantee the emergence of strange attractors in a given family of vector fields consists of finding a 3-dimensional nilpotent singularity of codimension 3 and verifying some generic algebraic conditions. We provide the essential background regarding this method and describe the main steps to illustrate numerically the chaotic dynamics emerging near these nilpotent singularities. This is a general-purpose method and we hope it can be applied to a huge range of models.An eco-epidemiological model with fear effect and hunting cooperationhttps://zbmath.org/1496.920932022-11-17T18:59:28.764376Z"Liu, Junli"https://zbmath.org/authors/?q=ai:liu.junli"Liu, Bairu"https://zbmath.org/authors/?q=ai:liu.bairu"Lv, Pan"https://zbmath.org/authors/?q=ai:lv.pan"Zhang, Tailei"https://zbmath.org/authors/?q=ai:zhang.taileiSummary: In this paper, we propose an eco-epidemiological model with disease in the prey population, the model incorporates fear effect of predators on prey and hunting cooperation among predators. We assume that fear can reduce the reproduction rate of the prey population and lower the activity of the prey population, which consequently lowers the disease transmission rate. Mathematical analysis of the model with regard to the non-negativity, boundedness of solutions, stability of equilibria, permanence of the model system are analyzed. The model undergoes backward bifurcation and bistability. We conduct extensive numerical simulations to explore the roles of fear effect, hunting cooperation and other biologically related parameters (e.g. disease transmission rate of prey, death rate of predators), it is found that low levels of fear and hunting cooperation can stabilize the eco-epidemiological system, however, relatively high levels of fear and hunting cooperation can induce limit cycles. Numerical simulations show the occurrence of multiple limit cycles. It is also observed that the system shows limit cycle oscillations for small disease transmission rate/death rate of predators, and the system becomes stable when the disease transmission rate/death rate of predators is high.Qualitative analysis of a fractional model for HBV infection with capsids and adaptive immunityhttps://zbmath.org/1496.921022022-11-17T18:59:28.764376Z"Bachraoui, Moussa"https://zbmath.org/authors/?q=ai:bachraoui.moussa"Hattaf, Khalid"https://zbmath.org/authors/?q=ai:hattaf.khalid"Yousfi, Noura"https://zbmath.org/authors/?q=ai:yousfi.nouraSummary: This paper presents a mathematical model governed by fractional differential equations (FDEs) that describes the dynamics of hepatitis B virus (HBV) infection in within human body. The FDE model takes into account the HBV DNA-containing capsids, and the adaptive immunity mediated by cytotoxic T lymphocytes (CTL) cells and antibodies. Also, the incidence of infection is presented by Hattaf-Yousfi functional response that includes various forms existing in the literature. Moreover, the qualitative properties of the FDE model is rigorously established. Finally, numerical simulations are presented to support the theoretical results.Global stability of a discrete SIR epidemic model with vaccination and treatmenthttps://zbmath.org/1496.921072022-11-17T18:59:28.764376Z"Cui, Qianqian"https://zbmath.org/authors/?q=ai:cui.qianqian"Zhang, Qiang"https://zbmath.org/authors/?q=ai:zhang.qiang.7|zhang.qiang.8|zhang.qiang.3|zhang.qiang.4|zhang.qiang.6|zhang.qiang.5Summary: We study a discrete susceptible-infected-removed epidemic model with vaccination and treatment and it is shown that the global dynamics is determined by the basic reproduction number \(\mathcal R_0\). If \(\mathcal R_0 \leq 1\), then the disease-free equilibrium is globally asymptotically stable and if \(\mathcal R_0 > 1\), then the endemic equilibrium is globally asymptotically stable.Parameter estimation in epidemic models: simplified formulashttps://zbmath.org/1496.921132022-11-17T18:59:28.764376Z"Hadeler, K. P."https://zbmath.org/authors/?q=ai:hadeler.karl-peterSummary: We consider the problem of identifying the time-dependent transmission rate from incidence data and from prevalence data in epidemic SIR, SIRS, and SEIRS models. We show closed representation formulas avoiding the computation of higher derivatives of the data or solving differential equations. We exhibit the connections between the formulas given in several recent papers. In particular we explain the difficulties to estimate the initial number of susceptible or, equivalently, the initial transmission rate.Global dynamics of a time-delayed dengue transmission modelhttps://zbmath.org/1496.921252022-11-17T18:59:28.764376Z"Wang, Zhen"https://zbmath.org/authors/?q=ai:wang.zhen|wang.zhen.2|wang.zhen.6|wang.zhen.1|wang.zhen.7|wang.zhen.3|wang.zhen.5"Zhao, Xiao-Qiang"https://zbmath.org/authors/?q=ai:zhao.xiao-qiang|zhao.xiaoqiangSummary: We present a time-delayed dengue transmission model. We first introduce the basic reproduction number for this model and then show that the disease persists when \(\mathcal R_0>1\). It is also shown that the disease will die out if \(\mathcal R_0<1\), provided that the invasion intensity is not strong. We further establish a set of sufficient conditions for the global attractivity of the endemic equilibrium by the method of fluctuations. Numerical simulations are performed to illustrate our analytic results.Simplified synchronizability scheme for a class of nonlinear systems connected in chain configuration using contractionhttps://zbmath.org/1496.930582022-11-17T18:59:28.764376Z"Anand, Pallov"https://zbmath.org/authors/?q=ai:anand.pallov"Sharma, Bharat Bhushan"https://zbmath.org/authors/?q=ai:sharma.bharat-bhushanSummary: This paper derives results for the stabilizing and synchronizing controller for a generalized class of nonlinear systems connected in chain configuration. The proposed procedure utilizes contraction based backstepping approach blended with Gershgorin theorem instead of Lyapunov stability based backstepping technique for designing controllers for such systems. A systematic step by step strategy is adopted to obtain a single controller to achieve stabilization of states of systems. Further, results are extended to synchronize the systems belonging to the given generalized class of nonlinear systems. The proposed procedure leads to quite a simple controller for targeted synchronization task in comparison to existing controllers in literature for such class of systems. The systems among which the synchronization has to be done are assumed to be connected in chain formation through one-way coupling. To verify the efficacy of the proposed approach, chaotic systems such as Lorenz-Stenflo, Chen, Lü and Lorenz systems have been considered and detailed numerical validations are presented appropriately.Quasi-FM waveform using chaotic oscillator for joint radar and communication systemshttps://zbmath.org/1496.940182022-11-17T18:59:28.764376Z"Pappu, Chandra S."https://zbmath.org/authors/?q=ai:pappu.chandra-s"Carroll, Thomas L."https://zbmath.org/authors/?q=ai:carroll.thomas-lSummary: The authors propose a novel signal design for generating wideband quasi-Frequency Modulated (FM) waveforms using chaotic systems. The receiver is based on a self synchronizing chaotic system, making for fast synchronization that is robust to timing errors or Doppler shifts. The chaotic oscillator has fast and slow time scales, and the slow oscillating part of the chaotic system is used to sweep the fast oscillating part thereby generating a modulated waveform that changes its frequency as a function of time. The potentials of these waveforms are demonstrated for joint radar-communication (RadComm) systems. Using the same nonlinear system a chaos frequency shift keying (CFSK) approach is utilized to encode the digital information. To decode the information, a drive-response synchronization scheme is utilized. Results indicate that our proposed signal design closely matches the bit-error rate (BER) of theoretical noncoherent frequency shift keying (FSK) while having good radar imaging capabilities.Characterizing complexity of non-invertible chaotic maps in the Shannon-Fisher information plane with ordinal patternshttps://zbmath.org/1496.940202022-11-17T18:59:28.764376Z"Spichak, David"https://zbmath.org/authors/?q=ai:spichak.david"Kupetsky, Audrey"https://zbmath.org/authors/?q=ai:kupetsky.audrey"Aragoneses, Andrés"https://zbmath.org/authors/?q=ai:aragoneses.andresSummary: Being able to distinguish the different types of dynamics present in a given nonlinear system is of great importance in complex dynamics. It allows to characterize the system, find similarities and differences with other nonlinear systems, and classify those dynamical regimes to understand them better. For systems that develop chaos it is not always easy to distinguish determinism from stochasticity. We analyze several non-invertible maps by projecting them on the two-dimensional Fisher-Shannon plane using ordinal patterns. We find that this technique unfolds the complex structure of chaotic systems, showing more details than other methods. It also reveals signatures common to most of the non-invertible maps, and demonstrates its capability to distinguish determinism from stochasticity.Construction of one-way hash functions with increased key space using adaptive chaotic mapshttps://zbmath.org/1496.940662022-11-17T18:59:28.764376Z"Tutueva, Aleksandra V."https://zbmath.org/authors/?q=ai:tutueva.aleksandra-v"Karimov, Artur I."https://zbmath.org/authors/?q=ai:karimov.artur-i"Moysis, Lazaros"https://zbmath.org/authors/?q=ai:moysis.lazaros"Volos, Christos"https://zbmath.org/authors/?q=ai:volos.christos-k"Butusov, Denis N."https://zbmath.org/authors/?q=ai:butusov.denis-nSummary: Chaotic hash functions are a prospective branch of modern cryptography. Being compared with traditional hashing algorithms, an approach based on deterministic chaos allows achieving diffusion and confusion with less computational costs. Most of the recently proposed chaotic hash functions use piecewise maps. Cryptosystems based on such maps are not vulnerable to attack by the reconstruction of phase space but their key spaces depend on maps parameters and therefore can be insufficient. In this paper, we propose an approach for the construction of piecewise hash functions from adaptive chaotic maps. The idea is to match different values of the adaptive coefficient to several sub-domains of the chaotic map. Thus, the adaptive coefficient values are part of the hash function key. Therefore, an increase in the sub-functions number potentially enhances the cryptographic strength of the algorithm. Thus, hash functions based on novel adaptive maps have larger key space compared to conventional piecewise maps. We explicitly show that the proposed hash generation technique allows obtaining digests with the required statistical properties. Moreover, we run a collision test to prove that the collision probability is small. The obtained results can be useful in chaos-based cryptography as well for the various simulations of real processes and phenomena with chaotic behavior, in computer graphics and multimedia.