Recent zbMATH articles in MSC 34https://zbmath.org/atom/cc/342023-11-13T18:48:18.785376ZWerkzeugLaplacians on infinite graphshttps://zbmath.org/1521.050022023-11-13T18:48:18.785376Z"Kostenko, Aleksey"https://zbmath.org/authors/?q=ai:kostenko.aleksey-s"Nicolussi, Noema"https://zbmath.org/authors/?q=ai:nicolussi.noemaSummary: ``The main focus in this memoir is on Laplacians on both weighted graphs and weighted metric graphs. Let us emphasize that we consider infinite locally finite graphs and do not make any further geometric assumptions. Whereas the existing literature usually treats these two types of Laplacian operators separately, we approach them in a uniform manner in the present work and put particular emphasis on the relationship between them. One of our main conceptual messages is that these two settings should be regarded as complementary (rather than opposite) and exactly their interplay leads to important further insight on both sides. Our central goal is twofold. First of all, we explore the relationships between these two objects by comparing their basic spectral (self-adjointness, spectral gap, etc.), parabolic (Markovian uniqueness, recurrence, stochastic completeness, etc.), and metric (quasi isometries, intrinsic metrics, etc.) properties. In turn, we exploit these connections either to prove new results for Laplacians on metric graphs or to provide new proofs and perspective on the recent progress in weighted graph Laplacians. We also demonstrate our findings by considering several important classes of graphs (Cayley graphs, tessellations, and antitrees).''
Contents: Chapter 1. Introduction; Chapter 2. Laplacians on graphs; Chapter 3. Connections via boundary triplets; Chapter 4. Connections between parabolic properties; Chapter 5. One-dimensional Schrödinger operators with point interactions; Chapter 6. Graph Laplacians as boundary operators; Chapter 7. From continuous to discrete and back; Chapter 8. Examples.
Appendix A: Boundary triplets and Weyl functions; Appendix B: Dirichlet forms; Appendix C: Heat kernel bounds; Appendix D: Glossary of notation.
Reviewer: Shuchao Li (Wuhan)An affine Weyl group action on the basic hypergeometric series arising from the \(q\)-Garnier systemhttps://zbmath.org/1521.170882023-11-13T18:48:18.785376Z"Idomoto, Taiki"https://zbmath.org/authors/?q=ai:idomoto.taiki"Suzuki, Takao"https://zbmath.org/authors/?q=ai:suzuki.takaoSummary: Recently, we formulated the \(q\)-Garnier system in a framework of an extended affine Weyl group of type \(A^{(1)}_{2n+1}\times A^{(1)}_1\times A^{(1)}_1\). On the other hand, the \(q\)-Garnier system admits a particular solution in terms of the basic hypergeometric series \({}_{n+1}\phi_n\). In this article, we investigate an action of the extended affine Weyl group on \({}_{n+1}\phi_n\).Quadratic relations between Bessel momentshttps://zbmath.org/1521.320172023-11-13T18:48:18.785376Z"Fresán, Javier"https://zbmath.org/authors/?q=ai:fresan.javier"Sabbah, Claude"https://zbmath.org/authors/?q=ai:sabbah.claude"Yu, Jeng-Daw"https://zbmath.org/authors/?q=ai:yu.jeng-dawThis article is motivated by conjectures by Broadhust and Roberts in a series of papers. They put forward a program to understand the motivic origin of the Bessel moments \(\int_0^{\infty} I_0(t)^a K_0(t)^b t^c dt\), where \(I_0(t)\) and \(K_0(t)\) are solutions to the ordinary differential equation \(((t\partial_t)^2-t^2)u=0\). The main insight of Broadhurst and Roberts is to consider these integrals over finite fields, pursuing the analogy between the Bessel differential equations and the Kloosterman \(\ell\)-adic sheaf. That is, \(I_0(t)\) and \(K_0(t)\) are to correspond to the eigenvalues of the Frobenius, and out of them one forms the \(k\)-th symmetric power moments of Kloosterman sums. Then the \(L\)-function \(L_k(s)\) is built and the critical values agree, up to rational factors and powers of \(\pi\), with certain determinants of the Bessel moments.
The article of the authors [Duke Math. J. 171, No. 8, 1649--1747 (2022; Zbl 1498.14019)] considers, after the change of variables \(z=t^2/4\), the associated rank-two vector bundle with connection on \(\mathbb{G}_m\), called the Kloosterman connection and denoted by \(Kl_2\). In the same article, motives \(M_k\) associated with symmetric powers of Kloosterman connection were constructed over \(\mathbb{Q}\), as pure of weight \(k+1\), of rank \(k^{\prime}=\lfloor{(k-1)/2}\rfloor\) (resp. \(k^{\prime}-1\)) if \(k\) is not a multiple of \(4\) (resp. \(k\) is a multiple of \(4\)). In the article, the \(L\)-function of this motive \(M_k\) was shown to coincide with \(L_k(s)\), and the Hodge numbers of \(M_k\) were computed. \smallskip
The present article is a continuation of [loc. cit.], and investigates the period realizations of the motive \(M_k\). The de Rham realization of \(M_k\) is isomorphic to the moddle de Rham cohomology of the \(k\)-th symmetric power Sym\(^k Kl_2\). Extending the earlier work [loc. cit.], a basis of middle de Rahm cohomology is exhibited. Then an explicit formula to compute the matrix of Sym\(^k\) on this basis is presented. Also it is proved that the dual of the Betti realization of \(M_k\) is isomorphic to the middle twisted homology of Sym\(^k Kl_2\). The main result is to present formulas for the period pairings. Theorem: Assume that \(k\) is not a multiple of \(4\).
\begin{itemize}
\item[(1)] With respect to the basis \(\{\omega_i\,|\, i=1,\dots, k^{\prime}\}\), the matrix of the de Rahm intersection pairing \(S_k^{mid}\) is a lower-right triangle matrix with coefficients in \(\mathbb{Q}\) and \((i,j)\) antidiagonal entires are explicitly exhibited depending on \(k\) odd or \(k\) even.
\item[(2)] The middle cohomology classes \(\{\,\alpha_i\,|\, i=1,\dots, k^{\prime}\,\}\) form a basis and the matrix of the Betti intersection pairing \(B_k^{mid}\) on this basis is given explitly by a formula, involving the \(n\)-th Bernoulli number.
\item[(3)] With respect to the bases \(\{\,\alpha_i\,|\, i=1,\dots, k^{\prime}\,\}\) and \(\{\,\omega_i\,|\, i=1,\dots, k^{\prime}\,\}\), the matrix of the period pairing \(P_k^{mid}\) consists of the Bessel moments, up to rational factors and powers of \(\pi\), \(\int_0^{\infty} I_0(t)^i K_0(t)^{k-i} t^{2j-1} dt\).
\item[(4)] The following quadratic relations hold: \(P_k^{mid}\cdot (S_k^{mid})^{-1}\cdot ^t P_k^{mid}=(-2\pi i)^{k+1} B_k^{mid}\).
\end{itemize}
(Quadratic relations of the shape in (4) was conjectured by Broadhurst and Roberts, and presumably they exhaust all algebraic relations between the Bessel moments.)
Deligne's conjecture about the critical values of \(L_k(s)\) is made explicit by identifying (up to a rational factor) the periods that coincide with some determinants of Bessel moments.
Reviewer: Noriko Yui (Kingston)Qualitative theory of ODEs. An introduction to dynamical systems theoryhttps://zbmath.org/1521.340012023-11-13T18:48:18.785376Z"Żołądek, Henryk"https://zbmath.org/authors/?q=ai:zoladek.henryk"Murillo, Raul"https://zbmath.org/authors/?q=ai:murillo.raulThis book represents lecture notes on the qualitative theory of ODEs and an introduction to the dynamical systems theory. It consists of the following six chapters: singular points of vector fields, phase portraits of vector fields, bifurcation theory, equations with a small parameter, irregular dynamics in differential equations, basic concepts and theorems of the theory of ODEs. The first chapter is devoted to the stability of equilibrium points in general and, in particular, of hyperbolic equilibrium points. The second chapter is devoted to the study of periodic solutions, the Poincaré-Bendixson criterion, the Dulac criterion, and special attention is paid to drawing phase portraits of vector fields on the plane. The third chapter presents the theory of bifurcations, in particular, saddle-node, Andronov-Hopf, limit cycle, separatrix connection, Bogdanov-Takens, and other bifurcations. Several issues in which a small parameter appears (in different contexts), including averaging theory, KAM theory, and the theory of relaxation oscillations, are discussed in the fourth chapter. The fifth chapter discusses the ergodicity of translations on tori, the dynamics of orientation-preserving diffeomorphisms of the circle, the Smale horseshoe and symbolic dynamics, and other examples of chaotic behavior such as Anosov systems and attractors. The sixth chapter is an appendix which collects the basic facts from the ODEs course. The book is full of various examples. Each chapter contains quite a lot of exercises (of varying degrees of complexity) designed for independent solution. The book can be useful for both students and graduates.
Reviewer: Eduard Musafirov (Grodno)Dynamic equations and almost periodic fuzzy functions on time scaleshttps://zbmath.org/1521.340022023-11-13T18:48:18.785376Z"Wang, Chao"https://zbmath.org/authors/?q=ai:wang.chao"Agarwal, Ravi P."https://zbmath.org/authors/?q=ai:agarwal.ravi-pThis book establishes an almost periodic theory of multidimensional fuzzy dynamic equations and fuzzy vector-valued functions on a time scale domain. The time scales domain considered here are complete-closed time scales under non-translational shifts. This book consists of six chapters.
The first chapter provides some necessary knowledge of interval and fuzzy arithmetic. A generalization of the Hukuhara difference and its properties are introduced. Applications of Hukuhara difference in solving interval and fuzzy linear equations and fuzzy differential equations. In Chapter 2, authors give an embedding theorem for a fuzzy multidimensional space and new types of multiplication in a fuzzy multidimensional space which are useful for further analysis. In Chapter 3, authors introduce the basic notions of generalized Hukuhara \(\Delta\)-derivatives of fuzzy vector-valued functions on time scales. Also, the \(\Delta\)-integral of fuzzy vector-valued functions is introduces. Further, some fundamental properties of generalized Hukuhara \(\Delta\)-derivatives and \(\Delta\)-integral of fuzzy vector-valued functions are established. Chapter 4 deals with the study of the notion of shift almost periodic fuzzy vector-valued functions on complete-closed time scales under non-translational shifts. Chapter 5 presents basic results of fuzzy multidimensional spaces and the calculus of fuzzy vector-valued functions on time scales. Finally, in Chapter 6, the authors develop a theory of almost periodic fuzzy multidimensional dynamic systems on time scales. Several applications of this theory are provided. A study of new type of fuzzy dynamic systems, called fuzzy \(q\)-dynamic systems is also presented and studied. An Appendix of the book introduces the notion of the almost anti-periodic discrete process and provides a new avenue to study the almost anti-periodic process on time scales.
The references in the book are not properly cited. Some reference items are wrong, particularly, in the Preface. However, the text material of the book is presented in a readable format. The book may be a good reference material for researchers working in the related field.
Reviewer: Sanket Tikare (Mumbai)Analytically integrable system orbitally equivalent to a semi-quasihomogeneous systemhttps://zbmath.org/1521.340032023-11-13T18:48:18.785376Z"Algaba, Antonio"https://zbmath.org/authors/?q=ai:algaba.antonio"García, Cristóbal"https://zbmath.org/authors/?q=ai:garcia.cristobal"Reyes, Manuel"https://zbmath.org/authors/?q=ai:reyes.manuel"Giné, Jaume"https://zbmath.org/authors/?q=ai:gine.jaumeSummary: For perturbations of integrable non-Hamiltonian quasi-homogeneous planar vector field whose origin is a non-degenerate singular point, orbital linearization and analytic integrability are equivalent. We show a class of analytically integrable vector fields whose origin is a degenerate singular point which is orbitally equivalent to a semi-quasi-homogeneous system, that is, it is not orbital equivalent to its lowest-degree quasi-homogeneous term.Arbitrary order differential equations with fuzzy parametershttps://zbmath.org/1521.340042023-11-13T18:48:18.785376Z"Allahviranloo, Tofigh"https://zbmath.org/authors/?q=ai:allahviranloo.tofigh"Salahshour, Soheil"https://zbmath.org/authors/?q=ai:salahshour.soheilSummary: In the last decades, some generalization of theory of ordinary differential equations has been considered to the arbitrary order differential equations by many researchers, the so-called theory of arbitrary order differential equations (often called as fractional order differential equations [FDEs]). Because of the ability for modeling real phenomena, arbitrary order differential equations have been applied in various fields such as control systems, biosciences, bioengineering, and references therein. In this chapter, the authors propose arbitrary order differential equations with respect to another function using fuzzy parameters (initial values and the unknown solutions). The generalized fuzzy Laplace transform is applied to obtain the Laplace transform of arbitrary order integral and derivative of fuzzy-valued functions to solve linear FDEs. To obtain the large class of solutions for FDEs, the concept of generalized Hukuhara differentiability is applied.
For the entire collection see [Zbl 1439.74003].Existence and uniqueness of solutions of nonlinear fractional stochastic differential systems with nonlocal functional boundary conditionshttps://zbmath.org/1521.340052023-11-13T18:48:18.785376Z"Abdelhamid, Ouaddah"https://zbmath.org/authors/?q=ai:abdelhamid.ouaddah"Graef, John R."https://zbmath.org/authors/?q=ai:graef.john-r"Ouahab, Abdelghani"https://zbmath.org/authors/?q=ai:ouahab.abdelghaniSummary: The authors study the existence and uniqueness of solutions to nonlinear first-order fractional stochastic differential systems driven by Brownian motion and with nonlocal functional boundary conditions. The technique of proof involves Perov's fixed point theorem with matrices that converge to zero and the Leray-Schauder theorem.On inclusion problems involving Caputo and Hadamard fractional derivativeshttps://zbmath.org/1521.340062023-11-13T18:48:18.785376Z"Ahmad, Bashir"https://zbmath.org/authors/?q=ai:ahmad.bashir.2"Ntouyas, Sotiris K."https://zbmath.org/authors/?q=ai:ntouyas.sotiris-k"Tariboon, Jessada"https://zbmath.org/authors/?q=ai:tariboon.jessadaSummary: In this paper, we study the existence of solutions to new inclusion problems involving both Caputo and Hadamard fractional derivatives, and separated boundary conditions. We apply the modern tools of the fixed point theory to study the cases when the multi-valued map (the right hand-side of the inclusions) takes convex as well as non-convex values. Examples illustrating the abstract results are also presented.Analytical and numerical analysis of damped harmonic oscillator model with nonlocal operatorshttps://zbmath.org/1521.340072023-11-13T18:48:18.785376Z"Alharthi, Nadiyah Hussain"https://zbmath.org/authors/?q=ai:alharthi.nadiyah-hussain"Atangana, Abdon"https://zbmath.org/authors/?q=ai:atangana.abdon"Alkahtani, Badr S."https://zbmath.org/authors/?q=ai:alkahtani.badr-saad-tSummary: Nonlocal operators with different kernels were used here to obtain more general harmonic oscillator models. Power law, exponential decay, and the generalized Mittag-Leffler kernels with Delta-Dirac property have been utilized in this process. The aim of this study was to introduce into the damped harmonic oscillator model nonlocalities associated with these mentioned kernels and see the effect of each one of them when computing the Bode diagram obtained from the Laplace and the Sumudu transform. For each case, we applied both the Laplace and the Sumudu transform to obtain a solution in a complex space. For each case, we obtained the Bode diagram and the phase diagram for different values of fractional orders. We presented a detailed analysis of uniqueness and an exact solution and used numerical approximation to obtain a numerical solution.Asymptotic behavior of fractional-order nonlinear systems with two different derivativeshttps://zbmath.org/1521.340082023-11-13T18:48:18.785376Z"Chen, Liping"https://zbmath.org/authors/?q=ai:chen.liping"Xue, Min"https://zbmath.org/authors/?q=ai:xue.min"Lopes, António"https://zbmath.org/authors/?q=ai:lopes.antonio-m"Wu, Ranchao"https://zbmath.org/authors/?q=ai:wu.ranchao"Chen, YangQuan"https://zbmath.org/authors/?q=ai:chen.yangquanSummary: This paper addresses the asymptotic behavior of systems described by nonlinear differential equations with two fractional derivatives. Using the Mittag-Leffler function, the Laplace transform, and the generalized Gronwall inequality, a sufficient asymptotic stability condition is derived for such systems. Numerical examples illustrate the theoretical results.A computational approach to exponential-type variable-order fractional differential equationshttps://zbmath.org/1521.340092023-11-13T18:48:18.785376Z"Garrappa, Roberto"https://zbmath.org/authors/?q=ai:garrappa.roberto"Giusti, Andrea"https://zbmath.org/authors/?q=ai:giusti.andreaSummary: We investigate the properties of some recently developed variable-order differential operators involving order transition functions of exponential type. Since the characterization of such operators is performed in the Laplace domain, it is necessary to resort to accurate numerical methods to derive the corresponding behaviours in the time domain. In this regard, we develop a computational procedure to solve variable-order fractional differential equations of this novel class. Furthermore, we provide some numerical experiments to show the effectiveness of the proposed technique.Existence of the positive solutions for boundary value problems of mixed differential equations involving the Caputo and Riemann-Liouville fractional derivativeshttps://zbmath.org/1521.340102023-11-13T18:48:18.785376Z"Liu, Yujing"https://zbmath.org/authors/?q=ai:liu.yujing"Yan, Chenguang"https://zbmath.org/authors/?q=ai:yan.chenguang"Jiang, Weihua"https://zbmath.org/authors/?q=ai:jiang.weihua.1|jiang.weihuaIn recent years, due to applications in mathematics, physics, biology, neural networks, and so on, the theory of fractional calculus has become the main focus of many scholars. At the same time, the theory of fractional differential equations is becoming more and more extensive and systematic. Some authors studied the existence of solutions for a class of mixed fractional differential equations.
In this paper, a class of two point boundary value problems of mixed fractional differential equation systems was investigated, in which, the highest derivative terms are composite with the left-sided Riemann-Liouville and the right-sided Caputo fractional derivatives. The existence of the solutions for the boundary value problems was obtained by using the fixed-point theorem in cone expansion and compression of norm type. Finally, an example is given out to illustrate the main results.
Reviewer: Xiping Liu (Shanghai)Existence and H-U stability of a tripled system of sequential fractional differential equations with multipoint boundary conditionshttps://zbmath.org/1521.340112023-11-13T18:48:18.785376Z"Murugesan, Manigandan"https://zbmath.org/authors/?q=ai:murugesan.manigandan"Muthaiah, Subramanian"https://zbmath.org/authors/?q=ai:muthaiah.subramanian"Alzabut, Jehad"https://zbmath.org/authors/?q=ai:alzabut.jehad-o"Nandha Gopal, Thangaraj"https://zbmath.org/authors/?q=ai:gopal.thangaraj-nandhaSummary: In this paper, we introduce a new coupled system of sequential fractional differential equations with coupled boundary conditions. We establish existence and uniqueness results using the Leray-Schauder alternative and Banach contraction principle. We examine the stability of the solutions involved in the Hyers-Ulam type. As an application, we present a few examples to illustrate the main results.Local and global canonical forms for differential-algebraic equations with symmetrieshttps://zbmath.org/1521.340122023-11-13T18:48:18.785376Z"Kunkel, Peter"https://zbmath.org/authors/?q=ai:kunkel.peter"Mehrmann, Volker"https://zbmath.org/authors/?q=ai:mehrmann.volkerIn this paper, the authors give complementary results for specific linear time-varying DAEs, featuring symmetries. A real system of square matrices, writing
\[
E(t)\dot{x} = A(t)x+f(t),
\]
is said to be self-adjoint when
\[
E^T = -E \text{ and } A^T=A+\dot{E},
\]
skew-adjoint when
\[
E^T = E \text{ and }A^T=-A-\dot{E}.
\]
The so-called port-Hamiltonian systems (see e.g. van der Schaft 1998) lead to typical examples of such skew-adjoint symmetric DAEs. As specialists of linear time-varying DAEs, the authors use a geometrical approach to give insights on the flow of the underlying dynamical systems. This classifying is facilitated by the use of two equivalence relations between system pairs \((E, A)\): \textit{equivalence} (definition 2.2) and \emph{congruence} (definition 2.6). \((E_1, A_1)\) is congruent to \((E_2, A_2)\) when there is a differentiable pointwise nonsingular matrix function \(Q\) such that
\[
E_2 = Q^T E_1 Q \;\; , A_2 = Q^T A_1 Q - Q^T \dot{E_1} Q.
\]
In the studied cases, one may find an orthogonal \(Q\) matrix which makes congruent pairs equivalent also.
The main results consist of building canonical forms for both classes of symmetrics linear DAEs, over a finite partition of the initial definition interval of the studied system. Theorem 3.4 shows as a corollary that the flow is symplectic for the self-adjoint DAEs, while Theorem 4.4 and Corollary 4.5 show that it is orthogonal in the skew-adjoint case.
Original proofs are stated well, many examples illustrate the main results, which makes this paper lively.
Reviewer: Gabriel Thomas (Grenoble)Discrete breathers of nonlinear dimer lattices: bridging the anti-continuous and continuous limitshttps://zbmath.org/1521.340132023-11-13T18:48:18.785376Z"Hofstrand, Andrew"https://zbmath.org/authors/?q=ai:hofstrand.andrew"Li, Huaiyu"https://zbmath.org/authors/?q=ai:li.huaiyu"Weinstein, Michael I."https://zbmath.org/authors/?q=ai:weinstein.michael-iThis article studies the asymptotic theory and numerical simulations of infinite array of nonlinear dimer oscillators which are linearly coupled as in the classical model of Su, Schrieffer and Heeger (SSH), where the ratio of in-cell and out-of-cell couplings of the SSH model defines distinct phases. The authors obtain two main results.
The first one is the existence of discrete breather solutions for sufficiently small values of the out-of-cell coupling parameter, for the case of weak out-of-cell coupling with any prescribed isolated dimer frequency \(\omega_b\) satisfying non-resonance and non-degeneracy assumptions, with states being \(\frac{2\pi}{\omega_b}\)-periodic in time and exponentially localized in space.
The second one is the global continuation with respect to this coupling parameter. They first consider the case where the seeding discrete breather frequency \(\omega_b\) is in the phonon gap of the underlying linear infinite array, and where the phonon gap decreases in width and tends to a point due to the coupling is increased. In this limit, the spatial scale of the discrete breather grows and its amplitude decreases, which indicates the weakly nonlinear long-wave regime. Then they show that in this regime the discrete breather envelope is determined by a vector gap soliton of the limiting envelope equations via asymptotic analysis. Afterwards, they use the envelope theory to describe discrete breathers for SSH-coupling parameters corresponding to topologically trivial and, by exploiting an emergent symmetry, topologically nontrivial regimes, when the spectral gap is small. Their asymptotic theory indicates excellent agreement with extensive numerical simulations over a wide range of parameters. Analogous asymptotic and numerical results are obtained for the continuations from the anti-continuous regime for frequencies \(\omega_b\) below the acoustic or above the optical phonon bands.
Reviewer: Caidi Zhao (Wenzhou)A new approach for modelling the damped Helmholtz oscillator: applications to plasma physics and electronic circuitshttps://zbmath.org/1521.340142023-11-13T18:48:18.785376Z"El-Tantawy, S. A."https://zbmath.org/authors/?q=ai:el-tantawy.s-a"Salas, Alvaro H."https://zbmath.org/authors/?q=ai:salas.alvaro-h"Alharthi, M. R."https://zbmath.org/authors/?q=ai:alharthi.muteb-rSummary: In this paper, a new approach is devoted to find novel analytical and approximate solutions to the damped quadratic nonlinear Helmholtz equation (HE) in terms of the Weiersrtrass elliptic function. The exact solution for undamped HE (integrable case) and approximate/semi-analytical solution to the damped HE (non-integrable case) are given for any arbitrary initial conditions. As a special case, the necessary and sufficient condition for the integrability of the damped HE using an elementary approach is reported. In general, a new ansatz is suggested to find a semi-analytical solution to the non-integrable case in the form of Weierstrass elliptic function. In addition, the relation between the Weierstrass and Jacobian elliptic functions solutions to the integrable case will be derived in details. Also, we will make a comparison between the semi-analytical solution and the approximate numerical solutions via using Runge-Kutta fourth-order method, finite difference method, and homotopy perturbation method for the first-two approximations. Furthermore, the maximum distance errors between the approximate/semi-analytical solution and the approximate numerical solutions will be estimated. As real applications, the obtained solutions will be devoted to describe the characteristics behavior of the oscillations in \textit{RLC} series circuits and in various plasma models such as electronegative complex plasma model.On topological entropy of piecewise smooth vector fieldshttps://zbmath.org/1521.340152023-11-13T18:48:18.785376Z"Antunes, André Amaral"https://zbmath.org/authors/?q=ai:antunes.andre-amaral"Carvalho, Tiago"https://zbmath.org/authors/?q=ai:de-carvalho.tiago"Varão, Régis"https://zbmath.org/authors/?q=ai:varao.regisSummary: Non-smooth vector fields do not have necessarily the property of uniqueness of solution passing through a point and this is responsible to enrich the behavior of the system. Even on the plane, non-smooth vector fields can be chaotic, a feature impossible for the smooth or continuous case. We propose a new approach to better understand chaos for non-smooth vector fields by using the notion of entropy of a system. We construct a metric space of all possible trajectories of a non-smooth vector field, where we define a flow inherited by the vector field and then define the topological entropy in this scenario. As a consequence, we are able to obtain some general results and give some examples of planar non-smooth vector fields with positive (finite and infinite) entropy.The global dynamics of linear refracting systems of focus-node or center-node typehttps://zbmath.org/1521.340162023-11-13T18:48:18.785376Z"Shao, Yi"https://zbmath.org/authors/?q=ai:shao.yi"Guan, Huanhuan"https://zbmath.org/authors/?q=ai:guan.huanhuan"Li, Shimin"https://zbmath.org/authors/?q=ai:li.shimin"Fu, Haoliang"https://zbmath.org/authors/?q=ai:fu.haoliangIn this paper, the authors study the global dynamics of the following planar piecewise linear refracting system of focus-node type, center-node type, focus-improper node type and center-improper node type defined in two zones separated by the straight line \(\{(x, y)^T\in \mathbb{R}^2: \ x=0 \}\):
\[
\begin{pmatrix} \dot{x} \\ \dot{y} \end{pmatrix} =
\begin{cases}
\begin{pmatrix} 2\gamma_L & -1 \\ \gamma_L^2+1 & 0 \end{pmatrix} \begin{pmatrix} x\\ y \end{pmatrix}- \begin{pmatrix} 0\\ \alpha_L \end{pmatrix}, & \text{if } x< 0, \\
\begin{pmatrix} 2\gamma_R & -1 \\ \gamma_R^2-m_R^2 & 0 \end{pmatrix} \begin{pmatrix} x\\ y \end{pmatrix}- \begin{pmatrix} 0\\ \alpha_R \end{pmatrix}, & \text{if } x> 0,
\end{cases}
\]
where \(\gamma_L\), \(\gamma_R\), \(m_R\), \(\alpha_L\) and \(\alpha_R\) are real parameters with \(\gamma_R<-1\) and \(m_R\in \{0, 1\}\). If \(\alpha_R>0\), then the right subsystem has a node for \(m_R=1\) and has an improper node for \(m_R=0\). For those four types of systems, the authors obtained sufficient and necessary conditions for the existence of limit cycles for all parameter regions and proved that the systems have at most one limit cycles. They also show that those systems have 26 kinds of different global topological phase portraits in the Poincaré disc.
Reviewer: Zhengdong Du (Chengdu)Spectrally determined singularities in a potential with an inverse square initial termhttps://zbmath.org/1521.340172023-11-13T18:48:18.785376Z"Pliakis, Demetrios A."https://zbmath.org/authors/?q=ai:pliakis.demetrios-aSummary: We study the inverse spectral problem for Bessel type operators with potential \(v(x): H_\kappa = -\partial^2_x + \frac{k}{x^2} + v(x)\). The potential is assumed smooth in \((0, R)\) and with an asymptotic expansion in powers and logarithms as \(x \to 0^+\), \(v(x) = O(x^\alpha)\), \(\alpha > -2\). Specifically we show that the coefficients of the asymptotic expansion of the potential are spectrally determined. This is achieved by computing the expansion of the trace of the resolvent of this operator which is spectrally determined and elaborating the relation of the expansion of the resolvent with that of the potential, through the singular asymptotics lemma.Some qualitative properties of solutions of higher-order lower semicontinus differential inclusionshttps://zbmath.org/1521.340182023-11-13T18:48:18.785376Z"Cubiotti, Paolo"https://zbmath.org/authors/?q=ai:cubiotti.paolo"Yao, Jen-Chih"https://zbmath.org/authors/?q=ai:yao.jen-chihSummary: Let \(n, k\in \mathbf{N}\), \(T>0\), and \(F:[0, T]\times(\mathbf{R}^n)^k\to 2^{\mathbf{R}^n}\) be a lower semicontinuos and bounded multifunction with nonempty closed values. We prove that there exists a bounded and upper semicontinuous multifunction \(G:\mathbf{R}\times(\mathbf{R}^n)^k\to2^{\mathbf{R}^n}\) with nonempty compact convex values such that every generalized solution \(u:[0, T]\to\mathbf{R}^n\) of the differential inclusion \(u^{(k)}\in G(t, u, u^\prime, \dots, u^{(k-1)})\) is a generalized solution to the differential inclusion \(u^{(k)}\in F(t, u, u^\prime, \dots, u^{(k-1)})\). As an application, we prove an existence and qualitative result for the generalized solutions of the Cauchy problem associated to the inclusion \(u^{(k)}\in F(t, u, u^\prime, \dots, u^{(k-1)})\). In particular, we prove that, if \(F\) is lower semicontinuous and bounded with nonempty closed values, then the solution multifunction admits an upper semicontinuous multivalued selection with nonempty compact connected values. Finally, by applying the latter result, we prove an analogous existence and qualitative result for the generalized solutions of the Cauchy problem associated to the differential equation \(g(u^{(k)})= f(t, u, u^\prime, \dots, u^{(k-1)})\), where \(f\) is continuous. We only assume that \(g\) is continuous and locally nonconstant.Rich dynamics in planar systems with heterogeneous nonnegative weightshttps://zbmath.org/1521.340192023-11-13T18:48:18.785376Z"López-Gómez, Julián"https://zbmath.org/authors/?q=ai:lopez-gomez.julian"Muñoz-Hernández, Eduardo"https://zbmath.org/authors/?q=ai:munoz-hernandez.eduardo"Zanolin, Fabio"https://zbmath.org/authors/?q=ai:zanolin.fabioThe authors study the global structure of the set of nodal solutions of a generalized Sturm-Liouville boundary value problem associated to the quasilinear equation \[-(\phi(u'))'=\lambda u+a(t)g(u), \quad \lambda \in \mathbb R, \] where \(a(t)\) is nonnegative with some positive humps separated away by intervals of degeneracy where \(a\equiv 0\). They obtain the multiplicity of the nodal solutions.
Reviewer: Bertin Zinsou (Johannesburg)The spectral properties of a two-term fourth-order operator with a spectral parameter in the boundary conditionhttps://zbmath.org/1521.340202023-11-13T18:48:18.785376Z"Polyakov, D. M."https://zbmath.org/authors/?q=ai:polyakov.dmitrii-mikhailovichConsider the eigenvalue problem
\[
Sy:={{y}^{\left( 4 \right)}}+qy,\,\,\,y\left( 0 \right)={y}''\left( 0 \right)={y}''\left( 1 \right)={y}'''\left( 1 \right)+\lambda y\left( 1 \right)=0
\]
in the space \({{L}_{2}}(0,1)\), where \(\lambda \) is the spectral parameter and \(q\) is a real potential with \(q\in {{L}_{1}}\left( 0,1 \right)\). The operator \(S\) is defined on
\[
\begin{aligned}
\mathrm{Dom}\left( S \right)=\{ & y\in {{L}_{2}}\left( 0,1 \right):{y}'',{y}''',{{y}^{\left( 4 \right)}}\in {{L}_{1}}\left( 0,1 \right),{{y}^{\left( 4 \right)}}+qy\in {{L}_{2}}\left( 0,1 \right),\\
& y\left( 0 \right)={y}''\left( 0 \right)={y}''\left( 1 \right)={y}'''\left( 1 \right)+\lambda y\left( 1 \right)=0\}.
\end{aligned}
\]
The main object is a fourth-order differential operator with a nonsmooth potential. The author determins the eigenvalue asymptotics at high energy and a regularized trace formula for this operator.
Reviewer: Rakib Efendiev (Baku)Variational techniques for a system of Sturm-Liouville equationshttps://zbmath.org/1521.340212023-11-13T18:48:18.785376Z"Shokooh, Saeid"https://zbmath.org/authors/?q=ai:shokooh.saeidThe paper is concerned with the sixth order Sturm-Liouville problem
\[
\begin{cases}
-\left(p_i(x)u_i'''(x)\right)'''+\left(q_i(x)u_i''(x)\right)''-\left(r_i(x)u_i'(x)\right)'+s_i(x)u_i(x) =\lambda F_{u_i}(x,u_1,\dots,u_n)\\
\text{ for } 0<x<T,\\
u_i(0)=u_i(T)=u_i''(0)=u_i''(T)=u_i^{(iv)}(0)=u_i^{(iv)}(T)=0
\end{cases} \tag{1}
\]
for \(i=1,\dots,n\), where \(n\in \mathbb{N}\), \(T>0\), \(\lambda\) is a positive parameter, the functions \(p_i,q_i,r_i,s_i\in L^{\infty}([0,T])\) with \(p_i^-:=\mathrm{ess} \inf_{x\in [0,T]}p_i(x)>0\) and
\[
\max\left\{-\frac{q_i^- T^2}{\pi^2},-\frac{q_i^- T^2}{\pi^2}-\frac{r_i^- T^4}{\pi^4},-\frac{q_i^- T^2}{\pi^2}-\frac{r_i^- T^4}{\pi^4}-\frac{s_i^- T^6}{\pi^6},\right\}<p_i^-,
\]
for any \(i=1,\dots,n\), the function \(F:[0,T]\times \mathbb{R}^n\to\mathbb{R}\) satisfies some assumptions, and \(F_{u_i}\) denotes the partial derivative of \(F\) with respect to \(u_i\) for \(i=1,\dots,n\). By using the critical point theory and variational methods, the author gives an interval for the parameter \(\lambda\) such that problem (1) has at least one nontrivial weak solution.
Reviewer: Rodica Luca (Iaşi)The bounds of eigenvalue for complex singular boundary value problemshttps://zbmath.org/1521.340222023-11-13T18:48:18.785376Z"Sun, Fu"https://zbmath.org/authors/?q=ai:sun.fu"Han, Xiaoxue"https://zbmath.org/authors/?q=ai:han.xiaoxueConsider the boundary value problem associated to the complex singular differential expression
\[
-[\left( 1-{{x}^{2}} \right){y}'{]}'+qy=\lambda \omega y \quad \text{in } L^2 [0,1) \tag{1.1}
\]
with the boundary conditions
\[
\begin{cases} \cos \alpha y\left( 0 \right)-\sin \alpha py\left( 0 \right)=0, & 0<\alpha <\frac{\pi }{2}, \\
\zeta \cos \beta [y,\nu ]\left( 1 \right)-\sin \beta [y,\nu ]\left( 1 \right)=0, & 0<\beta \leq \frac{\pi }{2}, \operatorname{Im}\zeta >0 \\
\end{cases}\tag{1.2}
\]
where \(q\) is a complex valued function and \(\omega \) is a real value function called the weight function, 0 is a regular endpoint and 1 is a limit-circle type non-oscillation endpoint. This problem is a perturbation of the Legendre eigenvalue problem with limit-circle type non-oscillation endpoints associated to the boundary condition. The eigenvalues are studied and bounds for the eigenvalues are obtained.
Reviewer: Rakib Efendiev (Baku)On the spectrum of a quasi-differential boundary value problem of the second-orderhttps://zbmath.org/1521.340232023-11-13T18:48:18.785376Z"Vatolkin, M. Yu."https://zbmath.org/authors/?q=ai:vatolkin.m-yuSummary: This paper studies the structure of the spectrum of a second-order quasi-differential boundary value problem \(\left( {{}_P^2x} \right)(t) = - \lambda \left( {{}_P^0x} \right)(t)\) \((t \in [a,b],\lambda \in \mathbb{R})\) (the coefficients in the equation are real-valued functions) with given homogeneous boundary conditions at the ends of the interval, \({}_P^0x(a) = {}_P^0x(b) = 0\). First, an auxiliary Cauchy problem with a real parameter \(\beta\) in the coefficient \({{p}_{{20}}}(t)\) of the equation, namely, \({{p}_{{22}}}(t)\left( {{{p}_{{11}}}(t){v}{\kern 1pt} '(t)} \right)' + ({{p}_{{20}}}(t) + \beta ){v}(t) = 0, {v}(a) = 0, {{p}_{{11}}}(a){v}{\kern 1pt} '(a) = 1\), is considered. A fundamental theorem on either the continuity or discreteness of the real spectrum of the original boundary value problem is formulated in terms of the solution of this problem. Examples illustrating the cases of both continuous and discrete spectra of the original boundary value problem are given.Asymptotic behavior of solution branches of nonlocal boundary value problemshttps://zbmath.org/1521.340242023-11-13T18:48:18.785376Z"Xu, Xian"https://zbmath.org/authors/?q=ai:xu.xian"Qin, Baoxia"https://zbmath.org/authors/?q=ai:qin.baoxia"Wang, Zhen"https://zbmath.org/authors/?q=au:Wang, ZhenThe authors study the following parametrized differential equation with the nonlocal boundary value condition of integral type:
\[
\begin{cases}
- u^{\prime\prime} = \lambda r(t) u + f(t,u), \quad t \in (0,1), \\
u(0) = 0, \,u(1) = \int_0^1 u(t)dg_1(t), \end{cases}
\]
where \(\lambda > 0\) is a parameter; \(r: [0,1] \to (0,+\infty)\) is a smooth function; \(f : [0,1] \times \mathbb{R} \to \mathbb{R}\) is continuous; \(g_1\) is a function of bounded variation on \([0,1]\).
It is shown that there exists a connected component of the solution set which bifurcates from infinity and oscillates infinitely often over an interval of \(\lambda\)-values.
Reviewer: Valerii V. Obukhovskij (Voronezh)Eigenvalues of one-dimensional Hamiltonian operators with an eigenparameter in the boundary conditionhttps://zbmath.org/1521.340252023-11-13T18:48:18.785376Z"Li, Kun"https://zbmath.org/authors/?q=ai:li.kun.2"Zheng, Jiajia"https://zbmath.org/authors/?q=ai:zheng.jiajia"Cai, Jinming"https://zbmath.org/authors/?q=ai:cai.jinming"Zheng, Zhaowen"https://zbmath.org/authors/?q=ai:zheng.zhaowenSummary: In this paper, one-dimensional Hamiltonian operators with spectral parameter-dependent boundary conditions are investigated. First, the eigenvalues of the problem under consideration are transformed into the eigenvalues of an operator in an appropriate Hilbert space. Then, some properties of the eigenvalues are given. Moreover, the continuity and differentiability of the eigenvalues of the problem are obtained, and the differential expressions of the eigenvalues concerning each parameter are also given. Finally, Green's function is also involved.
{\copyright 2023 American Institute of Physics}On the polynomial integrability of the critical systems for optimal eigenvalue gapshttps://zbmath.org/1521.340262023-11-13T18:48:18.785376Z"Tian, Yuzhou"https://zbmath.org/authors/?q=ai:tian.yuzhou"Wei, Qiaoling"https://zbmath.org/authors/?q=ai:wei.qiaoling"Zhang, Meirong"https://zbmath.org/authors/?q=ai:zhang.meirongSummary: This exploration consists of two parts. First, we will deduce a family of critical systems consisting of nonlinear ordinary differential equations, indexed by the exponent \(p \in (1, \infty)\) of the Lebesgue spaces concerned. These systems can be used to obtain the optimal lower or upper bounds for eigenvalue gaps of Sturm-Liouville operators and are equivalent to non-convex Hamiltonian systems of two degrees of freedom. Second, with appropriate choices of exponents \(p\), the critical systems are polynomial systems in four dimensions. These systems will be investigated from two aspects. The first one is that by applying the canonical transformation and the Darboux polynomial, we obtain the necessary and sufficient conditions for polynomial integrability of these polynomial critical systems. As a special example, we conclude that the system with \(p = 2\) is polynomial completely integrable in the sense of Liouville. The second is that the linear stability of isolated singular points is characterized. By performing the Poincaré cross section technique, we observe that the systems have very rich dynamical behaviors, including periodic trajectories, quasi-periodic trajectories, and chaos.
{\copyright 2023 American Institute of Physics}Existence of solutions for two-point integral boundary value problems with impulseshttps://zbmath.org/1521.340272023-11-13T18:48:18.785376Z"Georgiev, Svetlin G."https://zbmath.org/authors/?q=ai:georgiev.svetlin-georgiev"Tikare, Sanket"https://zbmath.org/authors/?q=ai:tikare.sanket-a"Kumar, Vipin"https://zbmath.org/authors/?q=ai:kumar.vipin.1|kumar.vipinThe authors consider a boundary value problem for a system of ODEs with impulses at fixed times in the form
\[
x'(t) = f(t,x(t)) \quad t \in [0, T], \ t \ne t_k, \ k =1,\ldots,p,
\]
\[
\Delta x(t_k) = I_k(x(t_k)), \quad k = 1,\ldots,p,
\]
\[
Ax(0) + \int_0^T h(s)x(s)\ {\mathrm d}s + Bx(T) = \int_0^T g(s,x(s))\ {\mathrm d}s,
\]
where \(0 < t_1 < t_2 < \dots < t_p < T\), \(p \in {\mathbb N}\), \(f,g : [0,T]\times {\mathbb R}^n \to {\mathbb R}^n\), \(h : [0,T] \to {\mathbb R}^n\), \(I_k : {\mathbb R}^n \to {\mathbb R}^n\), \(A,B \in {\mathbb R}^{n\times n}\).
They obtain two results: existence of at least one solution and existence of at least two nonnegative solutions to the BVP. They use fixed point theory. The feasibility of the results is illustrated by an example.
Reviewer: Jan Tomeček (Olomouc)Existence of ground state solution for a class of one-dimensional Kirchhoff-type equations with asymptotically cubic nonlinearitieshttps://zbmath.org/1521.340282023-11-13T18:48:18.785376Z"Khoutir, Sofiane"https://zbmath.org/authors/?q=ai:khoutir.sofianeIn this paper, the author considers the following Kirchoff-type equation \[-\Big(1+\int_{\mathbb{R}} |u'|^2dx\Big)u''+p(x)u=l(x)u^3+f(x,u),~~ x\in \mathbb{R},\] where \(p,l\in C(\mathbb{R})\) and \(f\in C(\mathbb{R}\times\mathbb{R},\mathbb{R}).\) By using the Non-Nehari manifold method in combination with the Mountain Pass Theorem and concentration-compactness argument, the existence of a ground state solution to the above equation is established, in the case when the nonlinearity is asymptotically cubic with respect to the unknown function.
Reviewer: Sotiris K. Ntouyas (Ioannina)On quantum star graphs with eigenparameter dependent vertex conditionshttps://zbmath.org/1521.340292023-11-13T18:48:18.785376Z"Mutlu, Gökhan"https://zbmath.org/authors/?q=ai:mutlu.gokhan"Uğurlu, Ekin"https://zbmath.org/authors/?q=ai:ugurlu.ekinSummary: We investigate the spectral properties of two different boundary value problems on a compact star graph in which the vertex conditions are dependent on the spectral parameter. We treat these boundary value problems as eigenvalue problems in some extended Hilbert spaces by associating them with vector-valued operators. We prove that the corresponding operators are self-adjoint. We construct the characteristic functions of these eigenvalue problems and prove that the corresponding operators have discrete spectrum. Moreover, we present some examples where we construct fundamental solutions and derive the resolvent operators.Phase portraits of two classes of quadratic differential systems exhibiting as solutions two cubic algebraic curveshttps://zbmath.org/1521.340302023-11-13T18:48:18.785376Z"Benterki, Rebiha"https://zbmath.org/authors/?q=ai:benterki.rebiha"Belfar, Ahlam"https://zbmath.org/authors/?q=ai:belfar.ahlamThe authors want to study the phase portraits of two families of quadratic systems which have a concrete invariant conic.
There is not much problem in their first system (2) since it is just a one parameter family with an invariant stright line which produces only two different phase portraits.
But the second family (3) has 4 parameters, and even if it is true that it always has the desired invariant curve, the results given are partially wrong and clearly incomplete. I fear that the authors are not aware of the difficulties of studying such a large family.
Even though this is not a referee report, I have found several mistakes in phase portraits from 3 to 7, and two missing phase portraits among the conditions which should produce those first pictures.
A family with so many parameters cannot be easily studied with the classical tools of locating singular points and studying their Jacobians. In these cases, the use of invariants has proved to be much more powerful.
And I also doubt that this study can be done in a completely algebraic way. From some inconsistencies I detect between phase portraits 6 and 7, I fear that there may appear also nonalgebraic bifurcations.
My recommendation to the authors is that they should start to use these much more powerful tools and make a new work on this family which promises to be an interesting one.
Reviewer: Joan C. Artés (Barcelona)On a class of global centers of linear systems with quintic homogeneous nonlinearitieshttps://zbmath.org/1521.340312023-11-13T18:48:18.785376Z"García-Saldaña, Johanna D."https://zbmath.org/authors/?q=ai:garcia-saldana.johanna-d"Llibre, Jaume"https://zbmath.org/authors/?q=ai:llibre.jaume"Valls, Claudia"https://zbmath.org/authors/?q=ai:valls.claudiaThe paper deals with the problem of characterizing the existence of a global center for a planar polynomial differential system. In the qualitative theory of ordinary differential equations, the problem of characterizing the existence of centers dates from the end of the XIX century and we can find many papers in the literature presenting results on it. However the problem of the existence of a global center started to appear at the end of the last century and there are very few results about it in the literature.
The paper provides results on the characterization of a global center for families of planar polynomial differential systems with quintic homogeneous nonlinearities. More precisely, the authors investigate a family of quintic systems possessing symmetry \((x, y, t) \rightarrow (-x, y,-t)\), with a linear or nilpotent center at the origin and without singular points at infinity.
The main results are Theorem 3 which characterizes the existence of a global center (of the linear type), and Theorem 5 which characterizes the existence of a global center (of the nilpotent type) for systems without infinity singular points. To prove these theorems the authors analyse the conditions on the parameters of the systems for the nonexistence of finity singular points different from the origin. Moreover they use the Routh-Hurwitz criterion and the Poincaré compactification to find the conditions on the parameters of the systems for nonexistence of infinity singular points.
Reviewer: Wilker Fernandes (São João del-Rei)Oscillation and nonoscillation for two-dimensional nonlinear systems of ordinary differential equationshttps://zbmath.org/1521.340322023-11-13T18:48:18.785376Z"Naito, Manabu"https://zbmath.org/authors/?q=ai:naito.manabuThe author considers two-dimensional non-linear systems of ordinary differential equations in the form
\begin{align*}
u^\prime &= a(t) \left|v\right|^{\frac{1}{\alpha}} \text{sgn}\,v, \\
v^\prime &= -b(t) \left|u\right|^{{\alpha}} \text{sgn}\,u,
\end{align*}
where \(\alpha \in (0, \infty)\) is a constant, \(a, b\) are real-valued continuous functions, and \(a\) is non-negative and not identically zero in a neighbourhood of infinity. The paper contains three main results -- two oscillation criteria (in fact, one of them consists of two results) and one non-oscillation criterion. In these criteria, the both cases \[ \int^\infty a(\tau) \, \mathrm{d}\tau = \infty, \qquad \int^\infty a(\tau) \, \mathrm{d}\tau < \infty \] are included. Since the treated systems are generalizations of half-linear equations, there exists a clear motivation for the study of these systems. In addition, for half-linear equations, many useful consequences of the main results are collected in Section 3. Then, interesting illustrative examples are mentioned in Section 4. Note that results from [\textit{H. J. Li} and \textit{C. C. Yeh}, Hiroshima Math. J. 25, No. 3, 585--594 (1995; Zbl 0872.34019)] are improved in the paper under review.
Reviewer: Michal Veselý (Brno)On the boundedness of solutions of a quasilinear system of ordinary differential equationshttps://zbmath.org/1521.340332023-11-13T18:48:18.785376Z"Mukhamadiev, Èrgashboĭ"https://zbmath.org/authors/?q=ai:mukhamadiev.ergashboi-mirzoevich"Naimov, Alizhon Nabidzhanovich"https://zbmath.org/authors/?q=ai:naimov.alizhon-nabidzhanovichSummary: In the paper, the question of the boundedness of an arbitrary solution of a quasilinear system of ordinary differential equations is investigated under boundedness of the observed values of the solution. The observed values of the solution are a finite set of scalar products of the solution with given vectors. In terms of the properties of the matrix of coefficients of the system of equations and the matrix of coefficients of observed values, theorems on the boundedness of an arbitrary solution with boundedness of the observed values are formulated and proved. The novelty of this paper is that using the method of limit equations, estimates are derived from which the boundedness or stability follows an arbitrary solution of a quasilinear system in terms of boundedness or stability of the observed values of the solution.Rareness of escaping orbits of the quasi-periodic Duffing equations with polynomial potentialshttps://zbmath.org/1521.340342023-11-13T18:48:18.785376Z"Peng, Yaqun"https://zbmath.org/authors/?q=ai:peng.yaqun"Zhuang, Yan"https://zbmath.org/authors/?q=ai:zhuang.yanSummary: We consider the equation \(\ddot{x} + x^{2n + 1} + q(t)x^l = p(t)\), where \(p\) and \(q\) are quasi-periodic functions, and prove that the set of initial values leading to escaping orbits has typically zero Lebesgue measure.
{\copyright 2023 American Institute of Physics}Structural static and vibration problemshttps://zbmath.org/1521.340352023-11-13T18:48:18.785376Z"Changizi, M. Amin"https://zbmath.org/authors/?q=ai:changizi.m-amin"Stiharu, Ion"https://zbmath.org/authors/?q=ai:stiharu.ionSummary: This chapter provides a comprehensive presentation of the application of Lie symmetry groups, which is based on defining certain canonical coordinates that satisfy conditions imposed by the application. It describes the Lie point algorithm for the reduction of the order of the ordinary differential equation. The chapter illustrates the transformation of a second-order differential equation into a first-order differential equation under the provision that a linear transformation to the specific form of differential equation is found. It presents the solution of the fundamental equation describing one degree of freedom vibrating system. The chapter also provides a number of solutions to the nonlinear problem of electrostatic attraction at small-scale devices to witness the capability of the Lie Symmetry Group approach.
For the entire collection see [Zbl 1439.74003].Continuum limits of coupled oscillator networks depending on multiple sparse graphshttps://zbmath.org/1521.340362023-11-13T18:48:18.785376Z"Ihara, Ryosuke"https://zbmath.org/authors/?q=ai:ihara.ryosuke"Yagasaki, Kazuyuki"https://zbmath.org/authors/?q=ai:yagasaki.kazuyukiSummary: The continuum limit provides a useful tool for analyzing coupled oscillator networks. Recently, \textit{G. S. Medvedev} [Commun. Math. Sci. 17, No. 4, 883--898 (2019; Zbl 1432.34045)] gave a mathematical foundation for such an approach when the networks are defined on single graphs which may be dense or sparse, directed or undirected, and deterministic or random. In this paper, we consider coupled oscillator networks depending on multiple graphs, and extend his results to show that the continuum limit is also valid in this situation. Specifically, we prove that the initial value problem (IVP) of the corresponding continuum limit has a unique solution under general conditions and that the solution becomes the limit of those to the IVP of the networks in some adequate meaning. Moreover, we show that if solutions to the networks are stable or asymptotically stable when the node number is sufficiently large, then so are the corresponding solutions to the continuum limit, and that if solutions to the continuum limit are asymptotically stable, then so are the corresponding solutions to the networks in some weak meaning as the node number tends to infinity. These results can also be applied to coupled oscillator networks with multiple frequencies by regarding the frequencies as a weight matrix of another graph. We illustrate the theory for three variants of the Kuramoto model along with numerical simulations.Flocking of a Kuramoto type model with local interaction functionshttps://zbmath.org/1521.340372023-11-13T18:48:18.785376Z"Jin, Chunyin"https://zbmath.org/authors/?q=ai:jin.chunyin"Li, Shuangzhi"https://zbmath.org/authors/?q=ai:li.shuangzhiThe authors consider the model
\[
\frac{d\mathbf{x}_i}{dt} = (\cos{\theta_i},\sin{\theta_i})
\]
\[
\frac{d\theta_i}{dt} =\frac{\lambda}{N_i(t)}\sum_{k=1}^N\psi(|\mathbf{x}_k-\mathbf{x}_i|)\sin{(\theta_k-\theta_i)}
\]
where \(\mathbf{x}_i\) and \(\theta_i\) are the position and phase, respectively, of the \(i\)th agent in the plane. \(\lambda\) is the coupling strength between agents, \(N_i(t)\) is the number of other agents within a distance \(r\) of agent \(i\) at time \(t\), and \(\psi\) is positive, non-increasing, and equal to zero for an argument greater than \(r\), i.e.~agents more than a distance \(r\) apart do not interact. The connectivity of the system can be described by a time-dependent symmetric matrix whose \((i,j)\)th entry is 1 if agents \(i\) and \(j\) are less than \(r\) apart, and zero otherwise. This matrix then describes a graph, with agents corresponding to vertices.
The main result is: assuming that the initial graph is connected, if certain inequalities on the initial conditions hold then the system will achieve flocking. The flocking state is defined by having all phases approach constant values, and the distances between all pairs of agents being less than \(r\). The phases approach their final values exponentially.
Reviewer: Carlo Laing (Auckland)A remark on renormalization group theoretical perturbation in a class of ordinary differential equationshttps://zbmath.org/1521.340382023-11-13T18:48:18.785376Z"Kuniba, Atsuo"https://zbmath.org/authors/?q=ai:kuniba.atsuoSummary: We revisit the renormalization group (RG) theoretical perturbation theory on oscillator-type second-order ordinary differential equations. For a class of potentials, we show a simple functional relation among secular coefficients of the harmonics in the naive perturbation series. It leads to an inversion formula between bare and renormalized amplitudes and an elementary proof of the absence of secular terms in all orders of the RG series. The result covers nonautonomous as well as autonomous cases and refines earlier studies, including the classic examples of Van der Pol, Mathieu, Duffing, and Rayleigh equations.A solution for reducing the degree of polynomial composition functions using Faà di Bruno's formula and Fourier transformhttps://zbmath.org/1521.340392023-11-13T18:48:18.785376Z"Ahani, Elshan"https://zbmath.org/authors/?q=ai:ahani.elshan"Ahani, Ali"https://zbmath.org/authors/?q=ai:ahani.aliSummary: Various studies concerning reducing the degree of order for nonlinear differential equations have been carried out. Yet, reducing the specified degree and a solution for them is a matter of discussion. In this study, based on Faà di Bruno's formula and using Fourier transform, a general method for reducing the degree of the equation to the desired degree has proposed. To appraise the efficacy of the proffered method, the function of the \(n\)th degree of polynomial composition function has been reduced by one and to one at the ending. The provided solution could have some uses in solving the high order differential equations of high degrees, which is practical for applied mathematics and engineering problems. The concept may implement to reduce the complexity of differential equations of higher order to more compatible forms of mathematical equations with acquirable solutions. The outcomes could be used for solving nonlinear differential equations of any order, especially those related to nonlinear engineering problems computational mechanics.Sundman transformation and alternative tangent structureshttps://zbmath.org/1521.340402023-11-13T18:48:18.785376Z"Cariñena, J. F."https://zbmath.org/authors/?q=ai:carinena.jose-f"Martínez, Eduardo"https://zbmath.org/authors/?q=ai:martinez-fernandez.eduardo"Muñoz-Lecanda, Miguel C."https://zbmath.org/authors/?q=ai:munoz-lecanda.miguel-cSummary: A geometric approach to Sundman transformation defined by basic functions for systems of second-order differential equations is developed and the necessity of a change of the tangent structure by means of the function defining the Sundman transformation is shown. Among other applications of such theory we study the linearisability of a system of second-order differential equations and in particular the simplest case of a second-order differential equation. The theory is illustrated with several examples.Steady-state solutions for the Muskat problemhttps://zbmath.org/1521.340412023-11-13T18:48:18.785376Z"Sánchez, Omar"https://zbmath.org/authors/?q=ai:sanchez.omarIn this paper, the author analyzes the existence of stationary solutions \(z=(z_1,z_2)\) for the Muskat problem with a large surface tension coefficient. Namely, it is considered the unstable case in which the heavier fluid is above the other one and the following initial conditions are imposed:
\[
z(0)=(0,0)\text{ and } z'(0)=(-\alpha,1),\quad\alpha>0.
\]
Assuming the conditions, the solution curve \(z\) can be written as the graph of a function, \(z(y)=(h(y),y))\) where \(h\) solves
\[
-\frac{h''}{(1+h'^2)^{3/2}}+\lambda y=0,\quad h(0)=0,\quad h'(0)=-\alpha.
\]
Then, denoting by \(B\) the beta function, and based on some estimates found in [\textit{M. Ehrnström} et al., Methods Appl. Anal. 20, No. 1, 33--46 (2013; Zbl 1291.34042)], the author is able to prove the existence of a \(\lambda^\ast>0\) such that for all
\[
\lambda\in \left(\lambda^\ast,\tfrac{1}{2\pi^2}B^2(3/4,1/2)\right]
\]
there exists a stationary \(2\pi\)-periodic solution of the Muskat problem that does not self-intersect. Moreover, if \(\lambda<\lambda^\ast\), no periodic solutions can arise. Finally, some numerical experiments are done analyzing the length of the interval \((\lambda^\ast,\frac{1}{2\pi^2}B^2(3/4,1/2))\).
Reviewer: Eduardo Muñoz-Hernández (Madrid)Periodic solutions of \(p\)-Laplacian differential equations with jumping nonlinearity across half-eigenvalueshttps://zbmath.org/1521.340422023-11-13T18:48:18.785376Z"Shen, Tengfei"https://zbmath.org/authors/?q=ai:shen.tengfei"Liu, Wenbin"https://zbmath.org/authors/?q=ai:liu.wenbin.1Using a topological degree approach, via a continuation theorem by Manásevich and Mawhin for periodic \(p\)-Laplacian equations, the authors prove the existence of \(T\)-periodic solutions of
\[
(\phi_{p}(u'))' + a_{+}(t) \phi_{p}(u^{+}) - a_{-}(t) \phi_{p}(u^{-})+ W_{i}(t,u,u')=e(t), \quad i=0,1,
\]
where \(W_{0}(t,u,u') = g_{0}(u) + g_{1}(t, u)\) and \(W_{1}(t, u, u') = \bar{g}_0(u) + \bar{g}_1(t, u')\) are continuous and \(T\)-periodic in \(t\), \(a_{\pm}, e\) are continuous and \(T\)-periodic with \(a_{\pm}\) positive, \(\phi_{p}(s)= |s|^{p-2}s\), \(\phi_{p}(0)=0\), with \(p>1\), and \(u^{\pm}\) denote the positive/negative part of \(u\). The main results generalise previous contributions on this subject.
Reviewer: Guglielmo Feltrin (Udine)A non-existence result for periodic solutions of the relativistic pendulum with frictionhttps://zbmath.org/1521.340432023-11-13T18:48:18.785376Z"Torres, Pedro J."https://zbmath.org/authors/?q=ai:torres.pedro-joseThe author considers the so-called relativistic pendulum equation
\[
\left(\frac{x'}{\sqrt{1-\frac{x'^2}{c^2}}}\right)'+kx'+a\sin x=p(t),
\]
where \(c > 0\) is the speed of light in the vacuum, \(k \ge 0\) is a possible viscous friction coefficient, \(a > 0\) and \(p(t)\) is a continuous and \(T\)-periodic forcing term with zero mean value.
He first reports a non-existence result for the the \(T\)-periodic problem associated with the classical Newtonian equation
\[
x''+kx'+a\sin x=p(t),
\]
and then proves that, for the same coefficients, also the relativistic pendulum equation has no \(T\)-periodic solutions, provided that \(c\) is large enough.
Reviewer: Alessandro Fonda (Trieste)Existence of periodic solutions for the forced pendulum equations of variable lengthhttps://zbmath.org/1521.340442023-11-13T18:48:18.785376Z"Yang, Hujun"https://zbmath.org/authors/?q=ai:yang.hujun"Han, Xiaoling"https://zbmath.org/authors/?q=ai:han.xiaolingIn this paper, the authors study the forced pendulum equations of variable length
\[
x''+kx'+a(t)\sin x=e(t),
\]
where \(a(t)\), \(e(t)\) are continuous \(T\)-periodic functions, \(k\) is a constant. They prove the existence of \(T\)-periodic solutions of the given equation under suitable assumptions on the functions \(a(t)\), \(e(t)\) by using Mawhin's continuation theorem.
Reviewer: Zaihong Wang (Beijing)Chaotic behavior in diffusively coupled systemshttps://zbmath.org/1521.340452023-11-13T18:48:18.785376Z"Nijholt, Eddie"https://zbmath.org/authors/?q=ai:nijholt.eddie"Pereira, Tiago"https://zbmath.org/authors/?q=ai:pereira.tiago-leite"Queiroz, Fernando C."https://zbmath.org/authors/?q=ai:queiroz.fernando-c"Turaev, Dmitry"https://zbmath.org/authors/?q=ai:turaev.dmitry-vSummary: We study emergent oscillatory behavior in networks of diffusively coupled nonlinear ordinary differential equations. Starting from a situation where each isolated node possesses a globally attracting equilibrium point, we give, for an arbitrary network configuration, general conditions for the existence of the diffusive coupling of a homogeneous strength which makes the network dynamics chaotic. The method is based on the theory of local bifurcations we develop for diffusively coupled networks. We, in particular, introduce the class of the so-called versatile network configurations and prove that the Taylor coefficients of the reduction to the center manifold for any versatile network can take any given value.Slow travelling wave solutions of the nonlocal Fisher-KPP equationhttps://zbmath.org/1521.340462023-11-13T18:48:18.785376Z"Billingham, John"https://zbmath.org/authors/?q=ai:billingham.johnThe article is concerned with the spike behaviors of slow travelling wave solutions of the nonlocal Fisher-KPP equation. Here slow travelling wave solutions mean that the speed of the travelling wave solutions is sufficiently small. This is a more difficult problem compared to the fast travelling wave solutions. In fact, the fast travelling wave solutions are small perturbations of the travelling wave solution of the local Fisher-KPP equation.
By using the formal method of matched asymptotic expansions and numerical methods, this article is devoted to revealing the number and spacing of the spikes associated with the slow travelling wave solutions, which depend crucially on the behaviour of the kernel in the nonlocal Fisher-KPP equation. Under certain kernels, finite and infinite number of spikes can be generated.
Reviewer: Jianhe Shen (Fuzhou)Stationary fronts and pulses for multistable equations with saturating diffusionhttps://zbmath.org/1521.340472023-11-13T18:48:18.785376Z"Garrione, Maurizio"https://zbmath.org/authors/?q=ai:garrione.maurizio"Sovrano, Elisa"https://zbmath.org/authors/?q=ai:sovrano.elisaAuthors' abstract: We deal with stationary solutions of a reaction-diffusion equation with flux-saturated diffusion and multistable reaction term, in dependence on a positive parameter \(\varepsilon\). Motivated by previous numerical results obtained by \textit{A. Kurganov} and \textit{P. Rosenau} [Nonlinearity 19, No. 1, 171--193 (2006; Zbl 1094.35063)], we investigate stationary solutions of front and pulse-type and discuss their qualitative features. We study the limit of such solutions for \(\varepsilon\rightarrow0\), showing that, in spite of the wide variety of profiles that can be constructed, there is essentially a unique configuration in the limit for both stationary fronts and pulses. We finally discuss some variational features that include the case where the solutions having continuous energy may not be global minimizers of the associated action functional.On the ``Traveling pulses'' of the limit of the FitzHugh-Nagumo equation when \(\varepsilon \to 0\)https://zbmath.org/1521.340482023-11-13T18:48:18.785376Z"Llibre, Jaume"https://zbmath.org/authors/?q=ai:llibre.jaume"Valls, Claudia"https://zbmath.org/authors/?q=ai:valls.claudiaConsider the existence of traveling pulses for the FitzHugh-Nagumo system
\[
u_t =u_{xx}+u(u-a)(1-u) +w, \quad w_t=\varepsilon(u-\gamma w)
\]
in case \(\varepsilon =0 \). This problem is equivalent to the existence of homoclinic orbits of the planar autonomous system
\[
\frac{dx}{ds} = y, \quad \frac{dy}{ds} =-cy-x(x-a)(1-x)+w \tag{1}
\]
where \(c\) is the speed of the pulse, \(a\) and \(w\) are real parameters. The authors determine the phase potrait of system (1) in the Poincaré disc for different parameter values by investigating the equilibria of (1) at infinity.
Reviewer: Klaus R. Schneider (Berlin)Multiple families of bounded solutions near perturbed homoclinic orbits, application to a nonlinear wave equationhttps://zbmath.org/1521.340492023-11-13T18:48:18.785376Z"Soleimani, L."https://zbmath.org/authors/?q=ai:soleimani.leila"RabieiMotlagh, O."https://zbmath.org/authors/?q=ai:rabieimotlagh.omidSummary: We consider a planar functional differential equation with a perturbed homoclinic orbit. We apply the exponential dichotomy of the variational equation and find a bifurcation function for the corresponding homoclinic bifurcation. By using the Lyapunov-Schmidt reduction method and the Malgrange preparation theorem, we find the roots of the bifurcation function and provide new sufficient conditions for the existence of bounded solutions near the homoclinic orbit. We show that the perturbed system may have multiple families of bounded solutions with chaotic motions bifurcating from the homoclinic orbit. In particular, we find three distinct families of bounded solutions. As far as we know, this is rare for planar systems (see part (iii) of theorem 1.2). Finally, as a numerical simulation, we apply the results to a perturbed wave equation and find a bounded solution near a perturbed homoclinic orbit. The results are more accessible for application in comparison to the former results, and they can be applied to various PDEs and RDEs. In addition, they do not have the limitations of previous results.Rate-induced tipping: thresholds, edge states and connecting orbitshttps://zbmath.org/1521.340502023-11-13T18:48:18.785376Z"Wieczorek, Sebastian"https://zbmath.org/authors/?q=ai:wieczorek.sebastian-m"Xie, Chun"https://zbmath.org/authors/?q=ai:xie.chun"Ashwin, Peter"https://zbmath.org/authors/?q=ai:ashwin.peterThe phenomenon of rate-induced tipping (briefly R-tipping) arises when a temporal variation of input parameters in a dynamical system interacts with the time scales of the system and yield genuine nonautonomous instabilities. These instabilities appear as the input varies at some critical rates. In general it cannot be understood by applying classical autonomous bifurcation theory to the frozen system with a fixed time-constant input.
In the paper at hand, the authors develop an ambient mathematical framework for R-tipping in finite-dimensional nonautonomous dynamical systems having an autonomous future limit. They concentrate on R-tipping via loss of tracking of base attractors being equilibria in the frozen system, due to crossing denoted as regular R-tipping thresholds. According to the authors, ``These thresholds are anchored at infinity by regular R-tipping edge states: compact normally hyperbolic invariant sets of the autonomous future limit system that have one unstable direction, orientable stable manifold, and lie on a basin boundary.''
R-tipping and critical rates for nonautonomous systems are introduced in terms of special solutions that converge to a compact invariant set of the autonomous future limit system which is not an attractor. The paper concentrates on the situation when the limit set is a regular edge state. Introducing the concept of edge tails, R-tipping into reversible, irreversible, and degenerate cases is rigorously classified. As central idea the autonomous dynamics of the future limit system is used to analyse R-tipping in the nonautonomous system. Here, the original nonautonomous system is compactified in order to include the limiting autonomous dynamics. Finally, considering regular R-tipping edge states that are equilibria, allows the authors to derive two results. First, sufficient conditions for the occurrence of R-tipping in terms of easily verifiable properties of the frozen system and input variation are given. Second, necessary and sufficient conditions for the occurrence of reversible and irreversible R-tipping in terms of computationally verifiable (heteroclinic) connections to regular R-tipping edge states in the autonomous compactified system are provided.
Reviewer: Christian Pötzsche (Klagenfurt)Existence of homoclinic solutions for a class of damped vibration problemshttps://zbmath.org/1521.340512023-11-13T18:48:18.785376Z"Xu, Huijuan"https://zbmath.org/authors/?q=ai:xu.huijuan"Jiang, Shan"https://zbmath.org/authors/?q=ai:jiang.shan"Liu, Guanggang"https://zbmath.org/authors/?q=ai:liu.guanggangSummary: In this paper, we consider the existence of homoclinic solution for a class of damped vibration problem
\[
\ddot{x}(t) + (q(t)I_{N\times N}+B)\dot{x}(t)+\left(\frac{1}{2}q(t)B - A(t)\right) x(t)+H_x(t, x(t)) = f(t).
\]
For every \(k\in\mathbb{N}\), we obtain the \(2kT\)-periodic solution \(x_k\) by a standard minimizing argument. By taking the limit of \(\{x_k\}\), we get a solution \(x_0\) of this problem. We prove that \(x_0\) satisfies \(x_0\rightarrow0\) and \(\dot{x}_0\rightarrow 0\) as \(t\rightarrow \pm\infty\), and therefore \(x_0\) is a homoclinic solution of the problem.On solvability of homological equations in the problem of center manifold approximationhttps://zbmath.org/1521.340522023-11-13T18:48:18.785376Z"Fazlytdinov, M. F."https://zbmath.org/authors/?q=ai:fazlytdinov.marat-flyurovichSummary: The article considers the questions of the homological equations solvability that arise in the problem of center manifold approximations in the neighborhood of a non-hyperbolic equilibrium point. The specificity of the problem lies in the solutions of functional equations, the solutions of which are homogeneous functions defined in one and taking a value in another subspace. The spectral properties of the operators that determine the homological equations in this problem are obtained. Based on the obtained properties, the solvability of the given equations is proved. Applications of results on the solvability of homological equations in the problem of a center manifold approximations are given. The article deals with homological equations obtained in problems with continuous and discrete time.On periodic solutions to a nonlinear dynamical system from one-dimensional cold plasma modelhttps://zbmath.org/1521.340532023-11-13T18:48:18.785376Z"Astashova, I."https://zbmath.org/authors/?q=ai:astashova.irina-v"Bellikova, K."https://zbmath.org/authors/?q=ai:bellikova.kConsider the planar polynomial system
\[
\begin{array}{l} \frac{{dx}}{{dt}}= y- x^2, \\
\frac{{dy}}{{dt}} = x-\gamma x y \end{array}\tag{1}
\]
depending on the real parameter \( \gamma \). The authors aim to determine points in the phase plane to which there belongs a closed orbit. For this purpose they apply a transformation in the halfplane \(x > 0\) mapping (1) into a linear system. By this way they determine the exact periodic solution and the corresponding period.
Reviewer's remark: Obviously there is no closed isolated orbit (limit cycle) of (1).
Reviewer: Klaus R. Schneider (Berlin)Stability and Turing patterns of a predator-prey model with Holling type II functional response and Allee effect in predatorhttps://zbmath.org/1521.340542023-11-13T18:48:18.785376Z"Chen, Lu"https://zbmath.org/authors/?q=ai:chen.lu.4"Yang, Feng"https://zbmath.org/authors/?q=ai:yang.feng"Song, Yong-li"https://zbmath.org/authors/?q=ai:song.yongliSummary: In this paper, we are concerned with a predator-prey model with Holling type II functional response and Allee effect in predator. We first mathematically explore how the Allee effect affects the existence and stability of the positive equilibrium for the system without diffusion. The explicit dependent condition of the existence of the positive equilibrium on the strength of Allee effect is determined. It has been shown that there exist two positive equilibria for some modulate strength of Allee effect. The influence of the strength of the Allee effect on the stability of the coexistence equilibrium corresponding to high predator biomass is completely investigated and the analytically critical values of Hopf bifurcations are theoretically determined. We have shown that there exists stability switches induced by Allee effect. Finally, the diffusion-driven Turing instability, which can not occur for the original system without Allee effect in predator, is explored, and it has been shown that there exists diffusion-driven Turing instability for the case when predator spread slower than prey because of the existence of Allee effect in predator.Global dynamics of a mosquito population suppression model with seasonal switchinghttps://zbmath.org/1521.340552023-11-13T18:48:18.785376Z"Chen, Yining"https://zbmath.org/authors/?q=ai:chen.yining"Wang, Yufeng"https://zbmath.org/authors/?q=ai:wang.yufeng"Yu, Jianshe"https://zbmath.org/authors/?q=ai:yu.jian-she"Zheng, Bo"https://zbmath.org/authors/?q=ai:zheng.bo.1"Zhu, Zhongcai"https://zbmath.org/authors/?q=ai:zhu.zhongcaiSummary: In this paper, we establish and analyze a mosquito population suppression model with seasonal switching. Under the assumption that the ratio of sexually active \textit{Wolbachia}-infected males and wild mosquitoes is kept at a constant level during the favorable seasons, we give a rather complete description on the dynamics including the stability of the origin, existence, stability and semi-stability of a unique, exact two periodic solutions, and so on. We obtain sufficient conditions for the origin to be globally asymptotically stable, for the model to have a unique semi-stable periodic solution, and to have exactly two periodic solutions among which one is stable and the other is unstable, respectively. Numerical examples to support our theoretical results and brief discussions are also provided.Impacts of inclusion of time delay in efforts on the dynamics of forestry biomass, concentration of greenhouse gases and elevated temperaturehttps://zbmath.org/1521.340562023-11-13T18:48:18.785376Z"Devi, Sapna"https://zbmath.org/authors/?q=ai:devi.sapna"Fatma, Reda"https://zbmath.org/authors/?q=ai:fatma.reda"Gupta, Nivedita"https://zbmath.org/authors/?q=ai:gupta.niveditaSummary: To understand the impacts of efforts applied to increase the density of forestry biomass on the dynamics of forestry biomass, concentration of greenhouse gases and elevated temperature a nonlinear mathematical model is proposed and analysed. The mathematical model involves four dynamical variables namely: the density of forestry biomass, efforts applied to increase the density of forestry biomass, concentration of greenhouse gases and elevated environmental temperature. Since, there is always a time lag between implementation of efforts and its outcome, therefore we extend our model by introducing time delay in efforts. It is found that as delay in efforts crosses a critical value, the delay model loses its stability and undergoes Hopf bifurcation. The stability and direction of Hopf bifurcation are analysed using normal form method and central manifold theory. Numerical simulations are performed to verify and validate our analytical results. It is observed that if efforts are implemented for appropriate time, the density of forestry biomass can be conserved but implementation of efforts with longer time delay has destabilizing effect on the system. Increasing rate of atmospheric temperature due to greenhouse gases, decreases the density of forestry biomass but this decrease can also be maintained by implementation of efforts. Therefore, implementation of efforts for sufficient period of time, plays a very vital role in increasing the density of forestry biomass and decreasing the concentration of greenhouse gases and elevated temperature.Application of non-singular kernel in a tumor model with strong Allee effecthttps://zbmath.org/1521.340572023-11-13T18:48:18.785376Z"Khajanchi, Subhas"https://zbmath.org/authors/?q=ai:khajanchi.subhas"Sardar, Mrinmoy"https://zbmath.org/authors/?q=ai:sardar.mrinmoy"Nieto, Juan J."https://zbmath.org/authors/?q=ai:nieto.juan-joseSummary: We obtain the analytical solutions in implicit form of a tumor cell population differential equation with strong Allee effect. We consider the ordinary case and then a fractional version. Some particular cases are plotted.Mixed-mode oscillations in coupled Fitzhugh-Nagumo oscillators: blow-up analysis of cusped singularitieshttps://zbmath.org/1521.340582023-11-13T18:48:18.785376Z"Kristiansen, Kristian Uldall"https://zbmath.org/authors/?q=ai:uldall-kristiansen.k"Pedersen, Morten Gram"https://zbmath.org/authors/?q=ai:pedersen.morten-gramSummary: In this paper, we use geometric singular perturbation theory and blowup as our main technical tool to study the mixed-mode oscillations (MMOs) that occur in two coupled FitzHugh-Nagumo units with symmetric and repulsive coupling. In particular, we demonstrate that the MMOs in this model are not due to generic folded singularities, but rather due to singularities at a cusp -- not a fold -- of the critical manifold. Using blowup, we determine the number of small-amplitude oscillations (SAOs) analytically, showing -- as for the folded nodes -- that they are determined by the Weber equation and the ratio of eigenvalues. We also show that the model undergoes a saddle-node bifurcation in the desingularized reduced problem, which -- although resembling a folded saddle-node (type II) at this level -- also occurs on a cusp, and not a fold. We demonstrate that this bifurcation is associated with the emergence of an invariant cylinder, the onset of SAOs, as well as SAOs of increasing amplitude. We relate our findings with numerical computations and find excellent agreement.Periodic orbits in the Muthuswamy-Chua simplest chaotic circuithttps://zbmath.org/1521.340592023-11-13T18:48:18.785376Z"Messias, Marcelo"https://zbmath.org/authors/?q=ai:messias.marcelo"Reinol, Alisson C."https://zbmath.org/authors/?q=ai:reinol.alisson-cThe authors consider the Muthuswamy-Chua system
\[
\begin{aligned}
& \dot{x}=y/C, \\
& \dot{y}=-\left( x+\beta \left( {{z}^{2}}-1 \right)y \right)/L, \\
& \dot{z}=-y-\alpha z+yz;\quad (x,y,z)\in {{\mathbb{R}}^{3}}, \\
\end{aligned}
\]
where \(C=1\), \(L=3\), \(\alpha ,\beta \in \mathbb{R}\) are system parameters.
For \(\alpha =0\) and sufficiently small \(\beta >0\), the authors prove the existence and uniqueness of a stable periodic orbit around each equilibrium point \(\left( 0,0,z \right)\) with \(\left| z \right|<1\). For \(\alpha >0\) and \(\beta >0\) sufficiently small, the authors prove the existence of a stable periodic orbit in the phase space of the system tending to an ellipse on the cylinder \({{x}^{2}}+3{{y}^{2}}-4=0\) when \(\alpha \to 0\) and \(\beta \to 0\).
These results are proved using averaging theory and based on the existence of first integrals and invariant algebraic surfaces for the system under consideration.
Reviewer: Eduard Musafirov (Grodno)Algorithmic criteria for the validity of quasi-steady state and partial equilibrium models: the Michaelis-Menten reaction mechanismhttps://zbmath.org/1521.340602023-11-13T18:48:18.785376Z"Patsatzis, Dimitris G."https://zbmath.org/authors/?q=ai:patsatzis.dimitris-g"Goussis, Dimitris A."https://zbmath.org/authors/?q=ai:goussis.dimitris-aSummary: We present ``on the fly'' algorithmic criteria for the accuracy and stability (non-stiffness) of reduced models constructed with the quasi-steady state and partial equilibrium approximations. The criteria comprise those introduced in [\textit{DimitrisA. Goussis}, Combust. Theory Model. 16, No. 5, 869--926 (2012; Zbl 1516.80009)] that addressed the case where each fast time scale is due to one reaction and a new one that addresses the case where a fast time scale is due to more than one reactions. The development of these criteria is based on the ability to approximate accurately the fast and slow subspaces of the tangent space. Their validity is assessed on the basis of the Michaelis-Menten reaction mechanism, for which extensive literature is available regarding the validity of the existing various reduced models. The criteria predict correctly the regions in both the parameter and phase spaces where each of these models is valid. The findings are supported by numerical computations at indicative points in the parameter space. Due to their algorithmic character, these criteria can be readily employed for the reduction of large and complex mathematical models.Mathematical insights into the influence of interventions on sexually transmitted diseaseshttps://zbmath.org/1521.340612023-11-13T18:48:18.785376Z"Zhang, Kai"https://zbmath.org/authors/?q=ai:zhang.kai.8"Xue, Ling"https://zbmath.org/authors/?q=ai:xue.ling"Li, Xuezhi"https://zbmath.org/authors/?q=ai:li.xuezhi"He, Daihai"https://zbmath.org/authors/?q=ai:he.daihaiSummary: We establish a mathematical model to analyze what factors cause the epidemics of sexually transmitted diseases (STDs) and how to eliminate or mitigate them. According to the level of prevention awareness, we divide the susceptible population into two groups of individuals, whose behavior, population size, and recruitment rate are affected by the interventions. First, the threshold, \(R_0\), of STDs model is obtained. If \(R_0 < 1\), the disease-free equilibrium is globally asymptotically stable. We also obtain the conditions for switching the equilibrium state of the model among disease-free equilibrium, endemic equilibrium, and limit cycle. Second, the threshold and transcritical bifurcation show that interventions for high-risk sexual behaviors of high-risk susceptible individuals can eliminate STDs. Additionally, sex education, influenced by the size of infected individuals and by interventions, can effectively cut down the size of STDs. Third, extending the survival time of the infected individual may prolong the time to end STDs unless they reject high-risk sexual behavior. Fourth, we analyze the existence, stability, and direction of Hopf bifurcation, which may explain the periodic oscillation in the size of infected population.Invariant synchrony and anti-synchrony subspaces of weighted networkshttps://zbmath.org/1521.340622023-11-13T18:48:18.785376Z"Nijholt, Eddie"https://zbmath.org/authors/?q=ai:nijholt.eddie"Sieben, Nándor"https://zbmath.org/authors/?q=ai:sieben.nandor"Swift, James W."https://zbmath.org/authors/?q=ai:swift.james-wSummary: The internal state of a cell in a coupled cell network is often described by an element of a vector space. Synchrony or anti-synchrony occurs when some of the cells are in the same or the opposite state. Subspaces of the state space containing cells in synchrony or anti-synchrony are called polydiagonal subspaces. We study the properties of several types of polydiagonal subspaces of weighted coupled cell networks. In particular, we count the number of such subspaces and study when they are dynamically invariant. Of special interest are the evenly tagged anti-synchrony subspaces in which the number of cells in a certain state is equal to the number of cells in the opposite state. Our main theorem shows that the dynamically invariant polydiagonal subspaces determined by certain types of couplings are either synchrony subspaces or evenly tagged anti-synchrony subspaces. A special case of this result confirms a conjecture about difference-coupled graph network systems.Synchronization of coupled phase oscillators with stochastic disturbances and the cycle space of the graphhttps://zbmath.org/1521.340632023-11-13T18:48:18.785376Z"Xi, Kaihua"https://zbmath.org/authors/?q=ai:xi.kaihua"Wang, Zhen"https://zbmath.org/authors/?q=ai:wang.zhen.13"Cheng, Aijie"https://zbmath.org/authors/?q=ai:cheng.aijie"Lin, Hai Xiang"https://zbmath.org/authors/?q=ai:lin.haixiang"van Schuppen, Jan H."https://zbmath.org/authors/?q=ai:van-schuppen.jan-h"Zhang, Chenghui"https://zbmath.org/authors/?q=ai:zhang.chenghuiSummary: The synchronization stability of a complex network system of coupled phase oscillators is discussed. In case the network is affected by disturbances, a stochastic linearized system of the coupled phase oscillators may be used to determine the fluctuations of phase differences in the lines between the nodes and to identify the vulnerable lines that may lead to desynchronization. The main result is the derivation of the asymptotic variance matrices of the phase differences which characterizes the severity of the fluctuations. It is found that the cycle space of the graph of the system plays a role in this characterization. With theory of the cycle space, the effect of forming small cycles on the fluctuations is evaluated. It is proven that adding a new line or increasing the coupling strength of a line affects the fluctuations in the lines in any cycle including this line, while it does not affect the fluctuations in the other lines. In particular, if the phase differences at the synchronous state are not changed by these actions, then the affected fluctuations reduce.Stabilization of the motion of nonautonomous polynomial systemshttps://zbmath.org/1521.340642023-11-13T18:48:18.785376Z"Martynyuk, A. A."https://zbmath.org/authors/?q=ai:martynyuk.anatoly-a"Chernienko, V. O."https://zbmath.org/authors/?q=ai:chernienko.v-o(no abstract)Asymptotic stability investigation of the zero solution for a class of nonlinear nonstationary systems by the averaging methodhttps://zbmath.org/1521.340652023-11-13T18:48:18.785376Z"Aleksandrov, Alexander Yur'evich"https://zbmath.org/authors/?q=ai:aleksandrov.aleksandr-yurevichSummary: A system of nonlinear differential equations is considered that describes the interaction of two coupled subsystems, one of these subsystems is linear, and the other is nonlinear and homogeneous with an order of homogeneity greater than one. It is assumed that this system is affected by nonstationary perturbations with zero mean values. Using the averaging method, sufficient conditions are determined under which perturbations do not disturb the asymptotic stability of the zero solution. The derivation of these conditions is based on the use of a special construction of the nonstationary Lyapunov function which takes into account the structure of the acting perturbations. In addition, we consider the case where there is a constant delay in the right-hand sides of the system. An original approach to the construction of the Lyapunov-Krasovskii functional for such a system is proposed. Using this functional, conditions are found that guarantee the preservation of the asymptotic stability for any positive delay.Traveling pulses and their bifurcation in a diffusive Rosenzweig-MacArthur system with a small parameterhttps://zbmath.org/1521.340662023-11-13T18:48:18.785376Z"Hou, Xiaojie"https://zbmath.org/authors/?q=ai:hou.xiaojie"Li, Yi"https://zbmath.org/authors/?q=ai:li.yiSummary: Conditions for the long term coexistence of the prey and predator populations of a diffusive Rosenzweig-MacArthur model are studied. The coexistence is represented by traveling pulses, which approach a coexistence equilibrium state as the moving coordinate approaches to infinities. Three different pulses, according to their speeds, are analyzed by regular perturbation as well as geometric singular perturbation methods. We further show that the pulses are connected by a bifurcation curve (surface) in parametric space. The paper concludes with several numerical simulations.A generically singular type of saddle-node bifurcation that occurs for relativistic shock waveshttps://zbmath.org/1521.340672023-11-13T18:48:18.785376Z"Pellhammer, Valentin"https://zbmath.org/authors/?q=ai:pellhammer.valentinSummary: This paper is concerned with two-parameter families of planar dynamical systems in which a saddle-node bifurcation interacts with a singular perturbation. We identify bifurcation curves along which the non-saddle point switches from node to focus and establish the existence of heteroclinic connections in either case. As an application, the result is used to show the existence of oscillatory shock profiles in a model of relativistic fluid dynamics.Construction of mean-square Lyapunov-basins for random ordinary differential equationshttps://zbmath.org/1521.340682023-11-13T18:48:18.785376Z"Rupp, Florian"https://zbmath.org/authors/?q=ai:rupp.florian-h-hPertinent theory of random ordinary differential equations (RODE), including mean square stability, is given. Then a search algorithm is presented that is designed to generate an approximate level set of a local Lyapunov function for a system of RODE. This facilitates the estimation of the basin of attraction of an equilibrium point of the RODE system.
To provide an example, the algorithm is applied to the \(2\times 2\) system of RODE:
\begin{align*}
\dot{X}_t & =X_t\cdot\left(-1+4X^2_t+\frac{1}{4} Y^2_t+ c_1\xi_t^{(1)}\right)+\frac{1}{8} Y^3_t\\
\dot{Y}_t & = Y_t\cdot\left(-1+\frac{5}{2} X^2_t+ \frac{3}{2}Y^2_t+c_2\xi^{(2)}_t\right)- 6X^3_t
\end{align*}
where \(\xi^{(1)}_t\), \(\xi^{(2)}_t\) are independent Ornstein-Uhlenbeck processes. The effect of noise intensity on the size of the computed basin is used to determine the intensity at which mean-square asymptotically stability is lost, thus identifying the intensity where stochastic bifurcation occurs.
Reviewer: Melvin D. Lax (Long Beach)Periodic boundary value problem for impulsive evolution equations with noncompact semigrouphttps://zbmath.org/1521.340692023-11-13T18:48:18.785376Z"Ma, Weifeng"https://zbmath.org/authors/?q=ai:ma.weifeng"Li, Yongxiang"https://zbmath.org/authors/?q=ai:li.yongxiang.1|li.yongxiangIn this paper the authors study the following periodic boundary value problem of a first order semilinear impulsive evolution equation in a Banach space \(E\)
\[
\begin{cases}
u'(t)+Au(t)=f(t,u(t)), ~~ t\in J=[0,\omega], ~~ t\ne t_k,\\
\Delta u|_{t=t_k}=I_k(u(t_k)), ~~ k=1,2,\dots, m,\\
u(0)=u(\omega),
\end{cases}
\]
where \(A : D(A)\subset E\to E\) is a linear operator and \(-A\) generates a \(C_0\)-semigroup \(T(t)\), \((t\ge 0)\), \(0=t_0<t_1<\cdots<t_m<t_{m+1}=\omega\), \(f : J \times E \to E \) and \(I_k : E \to E\) \((k = 1,2,\dots, m)\) are continuous, \(\Delta u|_{t=t_k}=u(t_k^+) - u(t_k^-)\), where \(u(t_k^+)\) and \(u(t_k^-)\) represent the right and left limits of \(u(t)\) at \(t =t_k\), respectively. Existence and uniqueness results of mild solution are obtained via Sadovskii's fixed point theorem under growth conditions and measure of noncompactness conditions on \(f\) and \(I_k\), without assuming the compactness of the semigroup. An example illustrating the obtained results is also presented.
Reviewer: Sotiris K. Ntouyas (Ioannina)Correction to: ``Novel algebraic criteria on global Mittag-Leffler synchronization for FOINNs with the Caputo derivative and delay''https://zbmath.org/1521.340702023-11-13T18:48:18.785376Z"Cheng, Yuhong"https://zbmath.org/authors/?q=ai:cheng.yuhong"Zhang, Hai"https://zbmath.org/authors/?q=ai:zhang.hai.1"Zhang, Weiwei"https://zbmath.org/authors/?q=ai:zhang.weiwei"Zhang, Hongmei"https://zbmath.org/authors/?q=ai:zhang.hongmeiSeveral typesetting errors in equations in the authors' paper [ibid. 68, No. 5, 3527--3544 (2022; Zbl 1514.34127)] have been now corrected. Original article has been updated.Uniform stability of recovering Sturm-Liouville-type operators with frozen argumenthttps://zbmath.org/1521.340712023-11-13T18:48:18.785376Z"Kuznetsova, Maria"https://zbmath.org/authors/?q=ai:kuznetsova.maria-n|kuznetsova.mariya-andreevnaSummary: In the last years, there has been considerable interest in inverse spectral problems for functional-differential operators with frozen argument. Until now, however, no aspects of their stability have been studied. One of the difficulties here is caused by the non-standard characterization of the spectra of such operators. In the present paper, we eliminate this gap and introduce a natural metric that allows us to obtain the Lipschitz stability on each ball of finite radius. Along with the previous results, this brings a final missing piece to the well-posedness of the inverse problem under consideration.On implicit impulsive conformable fractional differential equations with infinite delay in \(b\)-metric spaceshttps://zbmath.org/1521.340722023-11-13T18:48:18.785376Z"Krim, Salim"https://zbmath.org/authors/?q=ai:krim.salim"Salim, Abdelkrim"https://zbmath.org/authors/?q=ai:salim.abdelkrim"Abbas, Saïd"https://zbmath.org/authors/?q=ai:abbas.said"Benchohra, Mouffak"https://zbmath.org/authors/?q=ai:benchohra.mouffakIn this paper, the authors prove some existence results for a class of conformable implicit fractional differential equations with instantaneous impulses and infinite delay in b-metric spaces. The results are obtained by using the \(\omega\)-\(\psi\)-Geraghty type contraction and the fixed point theory. An example is provided to illustrate the theory.
The authors adopt the definitions of conformable fractional integral and fractional derivative from the paper [\textit{T. Abdeljawad}, J. Comput. Appl. Math. 279, 57--66 (2015; Zbl 1304.26004)]. Regarding this paper the readers attentions are drawn to the criticism in the following papers [\textit{A. A. Abdelhakim}, Fract. Calc. Appl. Anal. 22, No. 2, 242--254 (2019; Zbl 1426.26007); \textit{A. A. Abdelhakim} and \textit{J. A. T. Machado}, Nonlinear Dyn. 95, No. 4, 3063--3073 (2019; Zbl 1437.26006)]. Other researchers are also made comments on this notion.
Reviewer's remark: Several authors have introduced ``new definitions of fractional derivatives'', such us: conformable fractional derivative, deformable derivative or \(\alpha\)-derivative, M-fractional derivative, generalized fractional derivative, and so on. In fact that these concepts do not bring any novelty and have no physical relevance, they also create a certain confusion by using the term ``fractional'', these concepts having nothing to do with the classical Riemann-Liouville concept of the fractional derivative. Moreover, analyzing the definition of these ``new concepts'', it is easy to observe that anyone can introduce his own concept of fractional derivative without any physical significance.
Reviewer: Krishnan Balachandran (Coimbatore)On implicit boundary value problems with deformable fractional derivative and delay in \(b\)-metric spaceshttps://zbmath.org/1521.340732023-11-13T18:48:18.785376Z"Salim, Abdelkrim"https://zbmath.org/authors/?q=ai:salim.abdelkrim"Krim, Salim"https://zbmath.org/authors/?q=ai:krim.salim"Benchohra, Mouffak"https://zbmath.org/authors/?q=ai:benchohra.mouffakSummary: We demonstrate various existence and uniqueness results for a class of deformable implicit fractional differential equations with delay in \(b\)-metric spaces with boundary conditions. We base our arguments on some suitable fixed point theorems. In the last section, we provide different examples to illustrate our obtained results.Dynamics of a predator-prey model with distributed delay to represent the conversion process or maturationhttps://zbmath.org/1521.340742023-11-13T18:48:18.785376Z"Teslya, Alexandra"https://zbmath.org/authors/?q=ai:teslya.alexandra"Wolkowicz, Gail S. K."https://zbmath.org/authors/?q=ai:wolkowicz.gail-s-kSummary: Distributed delay is included in a simple predator-prey model in the prey-to-predator biomass conversion term. The delayed term includes a delay-dependent ``discount'' factor that ensures the predators that do not survive the delay interval, do not contribute to growth of the predator population. A simple model was chosen so that without delay all solutions converge to a globally asymptotically stable equilibrium in order to show the possible effects of delay on the dynamics. If the co-existence equilibrium does not exist, the dynamics of the system is identical to its non-delayed analog. However, with delay, there is a delay-dependent threshold for the existence of the co-existence equilibrium. When the co-existence equilibrium exists, unlike the dynamics of the model without delay, a much wider range of dynamics is possible, including a strange attractor and bi-stability, although the system is uniformly persistent. A bifurcation theory approach is taken, using both the mean delay and the predator death rate as bifurcation parameters. We consider the gamma and the uniform distributions as delay kernels and show that the ``discounting'' term ensures that the Hopf bifurcations occur in pairs, as was observed in the analogous system with discrete delay (i.e., using the Dirac delta distribution). We show that there are certain features common to all distributions, although the model with different kernels can have a significantly different range of dynamics. In particular, the number of bi-stabilities, the sequence of bifurcations, the criticality of the Hopf bifurcations, and the size of the stability regions can differ. Also, the width of the interval over which the delay history is nonzero seems to have a significant effect on the range of dynamics. Thus, ignoring the delay and/or not choosing the right delay kernel might result in inaccurate modelling predictions.Stochastic bifurcations of nonlinear vibroimpact system with time delay and fractional derivative excited by Gaussian white noisehttps://zbmath.org/1521.340752023-11-13T18:48:18.785376Z"Wu, Hao"https://zbmath.org/authors/?q=ai:wu.hao.14"Wang, Qiubao"https://zbmath.org/authors/?q=ai:wang.qiubao"Zhang, Congqing"https://zbmath.org/authors/?q=ai:zhang.congqing"Han, Zikun"https://zbmath.org/authors/?q=ai:han.zikun"Tian, Ruilan"https://zbmath.org/authors/?q=ai:tian.ruilanSummary: This paper investigates the stochastic dynamics in the nonlinear vibroimpact system with time delay and fractional derivative under Gaussian white noise excitation. Firstly, based on the definition of Caputo-type fractional derivative and the method of non-smooth transformation, the original system is transformed into an equivalent delayed stochastic vibroimpact system without fractional derivative. Then, by using the stochastic averaging method, the stationary density function of the stochastic \(I t \hat{o}\) equation is obtained. The effectiveness of the proposed method was validated by comparing the consistency between the original system and the optimized system without fractional derivative or the term of time delay. At last, we also explore the stochastic P-bifurcation induced by the power spectral density of two uncorrelated noises, time delay, fractional order, and restitution coefficient of the system.Novel insight into a single-species metapopulation model with time delayshttps://zbmath.org/1521.340762023-11-13T18:48:18.785376Z"Zhang, Xiangming"https://zbmath.org/authors/?q=ai:zhang.xiangming"Hou, Mengmeng"https://zbmath.org/authors/?q=ai:hou.mengmengSummary: Complex metapopulation dynamics research has a profound impact on our understanding of the relationship between species and their habitats. In this paper, the dynamical behaviors of the single-species metapopulation model with reproductive and reaction time delays based on Levins' model are investigated by analyzing stability charts, rightmost characteristic roots, and bifurcation diagrams of the positive equilibrium. Finally, the theoretical results are compared with the numerical results.A growth estimate for the monodromy matrix of a canonical systemhttps://zbmath.org/1521.340772023-11-13T18:48:18.785376Z"Pruckner, Raphael"https://zbmath.org/authors/?q=ai:pruckner.raphael"Woracek, Harald"https://zbmath.org/authors/?q=ai:woracek.haraldSummary: We investigate the spectrum of 2-dimensional canonical systems in the limit circle case. It is discrete and, by the Krein-de Branges formula, cannot be more dense than the integers. But in many cases it will be more sparse. The spectrum of a particular selfadjoint realisation coincides with the zeroes of one entry of the monodromy matrix of the system. Classical function theory thus establishes an immediate connection between the growth of the monodromy matrix and the distribution of the spectrum.
We prove a general and flexible upper estimate for the monodromy matrix, use it to prove a bound for the case of a continuous Hamiltonian, and construct examples which show that this bound is sharp. The first two results run along the lines of earlier work by R. Romanov, but significantly improve upon these results. This is seen even on the rough scale of exponential order.On the spectrum of the differential operators of even order with periodic matrix coefficientshttps://zbmath.org/1521.340782023-11-13T18:48:18.785376Z"Veliev, O. A."https://zbmath.org/authors/?q=ai:veliev.oktay-alishThe paper deals with the differential operator \(L\) generated in the space \(L_2^m(\mathbb{R})\) of vector-valued functions by the formally self-adjoint differential expression \[(-i)^{2\nu}y^{(2\nu)}(x)+\displaystyle\sum_{k=2}^{2\nu}P_k(x)y^{(2\nu-k)}(x),\] where \(\nu>1\) and \(P_k(x)\) is a \(m\times m\) matrix with summable entries \(p_{k,i,j}\) which satisfy the periodicity conditions \(p_{k,i,j}(x+1)=p_{k,i,j}(x)\) for all \(i=1,2,\ldots,m\) and \(j=1,2,\ldots,m\), for \(k=2,3,\ldots,2\nu\). The author investigates the band functions, Bloch functions and the spectrum of operator \(L\).
Reviewer: Rodica Luca (Iaşi)On completeness of weak eigenfunctions for multi-interval Sturm-Liouville equations with boundary-interface conditionshttps://zbmath.org/1521.340792023-11-13T18:48:18.785376Z"Olgar, Hayati"https://zbmath.org/authors/?q=ai:olgar.hayatiThis manuscript investigates eigenvalues and weak eigenfunctions for a new type of Sturm-Liouville problem. The problem under consideration differs from the standard Sturm-Liouville problems (SLP) in that the Sturm-Liouville equation is defined on a finite number of disjoint subintervals and the boundary conditions are specified not only at the end points, but also at a finite number of internal points. For a self-adjoint interpretation of the considered SLP the author defines some self-adjoint linear operators in such a way that the considered multi-interval SLP can be interpreted as an operator-pencil equation in a suitable Hilbert space. First, the concept of weak solutions (eigenfunctions) is defined. Then some important properties of the eigenvalues and the corresponding weak eigenfunctions are found. In particular, it is proved that the spectrum is discrete and the system of weak eigenfunctions forms a Riesz basis in the corresponding Hilbert space.
Reviewer: Hüseyin Tuna (Burdur)The fundamental gap of a kind of sub-elliptic operatorhttps://zbmath.org/1521.340802023-11-13T18:48:18.785376Z"Sun, Hongli"https://zbmath.org/authors/?q=ai:sun.hongli"Yang, Donghui"https://zbmath.org/authors/?q=ai:yang.donghuiSummary: In this paper the minimum fundamental gap of a kind of sub-elliptic operator is concerned, we deal with the existence and uniqueness of weak solution for that. We verify that the minimization fundamental gap problem can be achieved by some function, and characterize the optimal function by adopting the differential of eigenvalues.High energy asymptotics for the perturbed anharmonic oscillatorhttps://zbmath.org/1521.340812023-11-13T18:48:18.785376Z"Fedosova, Ksenia"https://zbmath.org/authors/?q=ai:fedosova.ksenia"Nursultanov, Medet"https://zbmath.org/authors/?q=ai:nursultanov.medetThe paper deals with the spectral properties of Sturm-Liouville operators with potentials which are perturbations of that for anharmonic oscillators. The authors study the asymptotics of the eigenvalues of such operators. The emphasis is made on the way how the reduced smoothness of the perturbation affects the eigenvalue asymptotics. Namely, the perturbations are assumed to be bounded, compactly supported, piecewise Hölder continuous functions. Particularly it is proved that in the case where the anharmonic potential is \(|x|^\alpha\), it is possible to write down three explicit asymptotic terms and that the Hölder exponent affects only the third term. In order to demonstrate more explicitly the effect of smoothness, an example is constructed in the case \(\alpha=2\) with the perturbations of the Weierstrass type. Another asymptotic result deals with more general potentials, where the obtained asymptotic terms turn out to be less explicit. Finally, the asymptotic formulas are established for the eigenvalues of the operators on the half-line, with the Dirichlet or Neumann boundary conditions.
Reviewer: Dmitry Shepelsky (Kharkov)Asymptotic distribution of resonances for matrix Schrödinger operator in one dimensionhttps://zbmath.org/1521.340822023-11-13T18:48:18.785376Z"Abdelmoula, Salmine"https://zbmath.org/authors/?q=ai:abdelmoula.salmine"Baklouti, Hamadi"https://zbmath.org/authors/?q=ai:baklouti.hamadiSummary: Using some classical theorems on entire functions, we determine the asymptotic for the counting function of the resonances for \(2\times 2\) matrix Schrödinger operator in one dimension for potentials with compact support.A question of Gol'dberg and Ostrovskii concerning linear differential equations with coefficients of completely regular growthhttps://zbmath.org/1521.340832023-11-13T18:48:18.785376Z"Bergweiler, Walter"https://zbmath.org/authors/?q=ai:bergweiler.walterIn this paper, the author answers a question of Gol'dberg and Ostrovskii. He shows that a linear differential equation whose coefficients are entire functions of completely regular growth may have an entire solution of finite order which is not of completely regular growth.
Reviewer: Karima Hamani (Mostaganem)Perturbations of dynamical systems on simple time scaleshttps://zbmath.org/1521.340842023-11-13T18:48:18.785376Z"Pilyugin, S. Yu."https://zbmath.org/authors/?q=ai:pilyugin.sergei-yuSummary: We study perturbations of dynamical systems in Banach spaces for which time varies on simple time scales consisting of families of isolated segments of the real axis. On a segment of the time scale, the system is governed by an ordinary differential equation; the transfer of a trajectory from a segment to the next one is determined by a map of the Banach space. The main problem which we study is the following one: given a trajectory of the original system, can we find a close trajectory of a perturbed system? We study perturbations applying the so-called multiscale approach: it is assumed that there exists a countable family of projections of the phase space and the smallness conditions are imposed on the projections of perturbations. To find a solution close to a specified solution of the unperturbed system, we introduce a generalization of the Perron method.Velocity averaging for diffusive transport equations with discontinuous fluxhttps://zbmath.org/1521.351122023-11-13T18:48:18.785376Z"Erceg, M."https://zbmath.org/authors/?q=ai:erceg.marko"Mišur, M."https://zbmath.org/authors/?q=ai:misur.marin"Mitrović, D."https://zbmath.org/authors/?q=ai:mitrovic.darkoSummary: We consider a diffusive transport equation with discontinuous flux and prove the velocity averaging result under non-degeneracy conditions. In order to achieve the result, we introduce a new variant of micro-local defect functionals which are able to `recognise' changes of the type of the equation. As a corollary, we show the existence of a weak solution for the Cauchy problem for non-linear degenerate parabolic equation with discontinuous flux. We also show existence of strong traces at \(t=0\) for so-called quasi-solutions to degenerate parabolic equations under non-degeneracy conditions on the diffusion term.Lagrangian solutions to the transport-Stokes systemhttps://zbmath.org/1521.351172023-11-13T18:48:18.785376Z"Inversi, Marco"https://zbmath.org/authors/?q=ai:inversi.marcoSummary: In this paper we consider the transport-Stokes system, which describes the sedimentation of a particles in a viscous fluid in inertialess regime. We show existence of Lagrangian solutions to the Cauchy problem with \(L^1\) initial data. We prove uniqueness of solutions as a corollary of a stability estimate with respect to the 1-Wasserstein distance for solutions with initial data in a Yudovich-type refinement of \(L^3\), with finite first moment. Moreover, we describe the evolution starting from axisymmetric initial data. Our approach is purely Lagrangian.On global dynamics of type-\(K\) competitive Kolmogorov differential systemshttps://zbmath.org/1521.370262023-11-13T18:48:18.785376Z"Hou, Zhanyuan"https://zbmath.org/authors/?q=ai:hou.zhanyuanSummary: This paper deals with global asymptotic behaviour of the dynamics for \(N\)-dimensional type-\(K\) competitive Kolmogorov systems of differential equations defined in the first orthant. It is known that the backward dynamics of such systems is type-\(K\) monotone. Assuming the system is dissipative and the origin is a repeller, it is proved that there exists a compact invariant set \(\Sigma\) which separates the basin of repulsion of the origin and the basin of repulsion of infinity and attracts all the non-trivial orbits. There are two closed sets \(S_H\) and \(S_V\), their restriction to the interior of the first orthant are \((N-1)\)-dimensional hypersurfaces, such that the asymptotic dynamics of the type-K system in the first orthant can be described by a system on either \(S_H\) or \(S_V\): each trajectory in the interior of the first orthant is asymptotic to one in \(S_H\) and one in \(S_V\). Geometric and asymptotic features of the global attractor \(\Sigma\) are investigated. It is proved that the partition \(\Sigma=\Sigma_H\cup\Sigma_0\cup\Sigma_V\) holds such that \(\Sigma_H\cup\Sigma_0\subset S_H\) and \(\Sigma_V\cup\Sigma_0\subset S_V\). Thus, \(\Sigma_0\) contains all the \(\omega\)-limit sets for all interior trajectories of any type-\(K\) subsystems and the closure \(\overline{\Sigma_H\cup\Sigma_V}\) as a subset of \(\Sigma\) is invariant and the upper boundary of the basin of repulsion of the origin. This \(\Sigma\) has the same asymptotic feature as the modified carrying simplex for a competitive system: every nontrivial trajectory below \(\Sigma\) is asymptotic to one in \(\Sigma\) and the \(\omega\)-limit set is in \(\Sigma\) for every other nontrivial trajectory.Plenty of hyperbolicity on a class of linear homogeneous jerk differential equationshttps://zbmath.org/1521.370302023-11-13T18:48:18.785376Z"Bessa, Mário"https://zbmath.org/authors/?q=ai:bessa.marioThe author considers \(3\times 3\) partially hyperbolic linear differential systems over an ergodic flow \(X^t\) which are derived from the linear homogeneous differential equation \[\dddot{x}+ \beta(X^{t})\dot{x}+ \gamma(X^{t})x=0.\]
In the main result of the paper it is shown that if there is a zero Lyapunov exponent along the central direction, then there exists a perturbation of the parameters \(\beta\) and \(\gamma\) such that the corresponding system has a non-zero Lyapunov exponent.
Reviewer: Miguel Paternain (Montevideo)Limit cycle bifurcations of near-Hamiltonian systems with multiple switching curves and applicationshttps://zbmath.org/1521.370562023-11-13T18:48:18.785376Z"Liu, Wenye"https://zbmath.org/authors/?q=ai:liu.wenye"Han, Maoan"https://zbmath.org/authors/?q=ai:han.maoanSummary: In the present paper, we are devoted to the study of limit cycle bifurcations in piecewise smooth near-Hamiltonian systems with multiple switching curves, obtaining a formula of the first order Melnikov function in general case. As an application, we give lower bounds of the number of limit cycles for a piecewise smooth near-Hamiltonian system with a closed switching curve passing through the origin under piecewise polynomial perturbations.Construction of localized particular solutions of chains with three independent variableshttps://zbmath.org/1521.370692023-11-13T18:48:18.785376Z"Kuznetsova, M. N."https://zbmath.org/authors/?q=ai:kuznetsova.mariya-nikolaevna|kuznetsova.maria-nSummary: We consider differential-difference chains with three independent variables of the form \(u^j_{n+1,x} = F(u^j_{n,x}, u^{j+1}_n, u^j_n, u^j_{n+1}, u^{j-1}_{n+1})\). An effective approach to the study and classification of equations with three independent variables is the method based on Darboux-integrable reductions. Using the Darboux-integrable reductions, we construct localized particular solutions of chains with three independent variables.Nonlinear diffusion in multi-patch logistic modelhttps://zbmath.org/1521.371012023-11-13T18:48:18.785376Z"Elbetch, Bilel"https://zbmath.org/authors/?q=ai:elbetch.bilel"Moussaoui, Ali"https://zbmath.org/authors/?q=ai:moussaoui.aliSummary: We examine a multi-patch model of a population connected by nonlinear asymmetrical migration, where the population grows logistically on each patch. Utilizing the theory of cooperative differential systems, we prove the global stability of the model. In cases of perfect mixing, where migration rates approach infinity, the total population follows a logistic law with a carrying capacity that is distinct from the sum of carrying capacities and is influenced by migration terms. Furthermore, we establish conditions under which fragmentation and nonlinear asymmetrical migration can lead to a total equilibrium population that is either greater or smaller than the sum of carrying capacities. Finally, for the two-patch model, we classify the model parameter space to determine if nonlinear dispersal is beneficial or detrimental to the sum of two carrying capacities.Jump phenomena of the \(n\)-th eigenvalue of discrete Sturm-Liouville problems with application to the continuous casehttps://zbmath.org/1521.390022023-11-13T18:48:18.785376Z"Ren, Guojing"https://zbmath.org/authors/?q=ai:ren.guojing"Zhu, Hao"https://zbmath.org/authors/?q=ai:zhu.haoSummary: In this paper, we characterize jump phenomena of the \(n\)-th eigenvalue of self-adjoint discrete Sturm-Liouville problems in any dimension. For a fixed Sturm-Liouville equation, we completely characterize jump phenomena of the \(n\)-th eigenvalue. For a fixed boundary condition, unlike in the continuous case, the \(n\)-th eigenvalue exhibits jump phenomena and we describe the singularity under a non-degenerate assumption. Compared with the continuous case in [\textit{X. Hu} et al., J. Differ. Equations 266, No. 7, 4106--4136 (2019; Zbl 1458.34053)] and [\textit{Q. Kong} et al., J. Differ. Equations 156, No. 2, 328--354 (1999; Zbl 0932.34081)], the jump set here is involved with coefficients of the Sturm-Liouville equations. This, along with arbitrariness of the dimension, causes difficulty when dividing the jump areas. We study the singularity by partitioning and analysing the local coordinate systems, and provide a Hermitian matrix which can determine the areas' division. To prove the asymptotic behaviour of the \(n\)-th eigenvalue, we generalize the method developed in [\textit{H. Zhu} and \textit{Y. Shi}, J. Differ. Equations 260, No. 7, 5987--6016 (2016; Zbl 1383.39011)] to any dimension. As an application, by transforming the continuous Sturm-Liouville problem of Atkinson type to a discrete one, we determine the number of eigenvalues and obtain complete characterization of jump phenomena of the \(n\)-th eigenvalue for the Atkinson type.Close turning points and the Harper operatorhttps://zbmath.org/1521.390202023-11-13T18:48:18.785376Z"Fedotov, A. A."https://zbmath.org/authors/?q=ai:fedotov.alexei-a|fedotov.andrey-aFrom the text: We discuss the spectrum of the Harper operator in the semiclassical approximation.Weakly perturbed impulsive boundary-value problem for integrodifferential systems in the resonance casehttps://zbmath.org/1521.450042023-11-13T18:48:18.785376Z"Bondar, I. A."https://zbmath.org/authors/?q=ai:bondar.i-a"Strakh, O. P."https://zbmath.org/authors/?q=ai:strakh.o-pSummary: We establish conditions for the existence of solutions of weakly perturbed impulsive boundary-value problems for systems of integrodifferential equations and determine the structure of these solutions. The sufficient condition for the existence of solutions of these problems are investigated with the help of the theory of orthoprojectors and pseudoinverse Moore-Penrose matrices.The continuous Redner-Ben-Avraham-Kahng coagulation system: well-posedness and asymptotic behaviourhttps://zbmath.org/1521.450072023-11-13T18:48:18.785376Z"Verma, Pratibha"https://zbmath.org/authors/?q=ai:verma.pratibha"Giri, Ankik Kumar"https://zbmath.org/authors/?q=ai:giri.ankik-kumar"Da Costa, F. P."https://zbmath.org/authors/?q=ai:da-costa.fernando-pestanaThe authors study the following continuous model of a coagulation:
\[
\frac{\partial \phi}{\partial t}=\int_{0}^\infty a(\psi+\rho,\rho) \phi(\psi+\rho,t)\phi(\rho,t)d\rho -\int_{0}^\infty a(\psi,\rho) \phi(\psi,t)\phi(\rho,t)d\rho,
\]
with initial condition
\[
\phi(\psi,0)=\phi^{in}(\psi)\geq 0,
\]
where \(\phi(\sigma, t)\) denotes the concentration of particles of volume \(\sigma\in [0,\infty)\) at time \(t \geq 0\). The non-negative quantity \(a(\psi,\rho)\) represents the coagulation rate at which particles of volume \(\psi\) and particles of volume \(\rho\) interact to produce particles of volume \(|\psi-\rho|\).
The main goal of the paper is to obtain sufficient conditions for existence and uniqueness of solutions for the above model.
Reviewer: Leonid Berezansky (Be'er Sheva)Random Hamiltonians with arbitrary point interactions in one dimensionhttps://zbmath.org/1521.470712023-11-13T18:48:18.785376Z"Damanik, David"https://zbmath.org/authors/?q=ai:damanik.david"Fillman, Jake"https://zbmath.org/authors/?q=ai:fillman.jake"Helman, Mark"https://zbmath.org/authors/?q=ai:helman.mark"Kesten, Jacob"https://zbmath.org/authors/?q=ai:kesten.jacob"Sukhtaiev, Selim"https://zbmath.org/authors/?q=ai:sukhtaiev.selimSummary: We consider disordered Hamiltonians given by the Laplace operator subject to arbitrary random self-adjoint singular perturbations supported on random discrete subsets of the real line. Under minimal assumptions on the type of disorder, we prove the following dichotomy: Either every realization of the random operator has purely absolutely continuous spectrum or spectral and exponential dynamical localization hold. In particular, we establish Anderson localization for Schrödinger operators with Bernoulli-type random singular potential and singular density.Approximation formula for a propagator in symmetrically normed idealshttps://zbmath.org/1521.470722023-11-13T18:48:18.785376Z"Akhymbek, Meiram"https://zbmath.org/authors/?q=ai:akhymbek.meiram-erkanatuly"Zanin, Dmitriy"https://zbmath.org/authors/?q=ai:zanin.dmitriy-vSummary: We extend a result of \textit{V. A. Zagrebnov} [Math. Nachr. 295, No. 6, 1233--1245 (2022; \url{doi:10.1002/mana.202000019})] and prove an approximation formula for a propagator of an abstract Cauchy problem for non-autonomous evolution equation in the norm of an arbitrary Banach ideal.On the regularity and simplicity of a class of fractional elliptic operatorshttps://zbmath.org/1521.470782023-11-13T18:48:18.785376Z"Li, Yulong"https://zbmath.org/authors/?q=ai:li.yulongSummary: This work studies the spectral problem of a class of fractional elliptic operators
\[
\begin{cases}
-\alpha DI_{a+}^{1-\mu} Du-\beta DI_{b-}^{1-\mu}Du = \lambda u,\; x \in (a, b) \\
u(a) = u(b) = 0, \\
0<\mu <1,\; 0<\alpha, \beta <1,\; \alpha +\beta = 1.
\end{cases}
\]
With regularity results, we mainly prove that each eigenvalue \(\lambda \in \mathbb{C}\) must have \(|\mathrm{Arg} \lambda |\leq \arctan (|\alpha -\beta| \tan\frac{(1-\mu)\pi}{2})\) and that if \(\Im (\lambda) = 0\), then it must be simple. In particular, we point out that, when \(\alpha = \beta = \frac{1}{2}\), the problem is essentially equivalent to the spectral problem of 1-D fractional Laplacian, and with this connection, we provide a promising approach to link the way of calculating analytic eigenfunctions of the 1-D fractional Laplacian to a new type of singular integral equation involving Hilbert transform.Applications of the Bielecki renorming techniquehttps://zbmath.org/1521.470912023-11-13T18:48:18.785376Z"Bessenyei, Mihály"https://zbmath.org/authors/?q=ai:bessenyei.mihaly"Páles, Zsolt"https://zbmath.org/authors/?q=ai:pales.zsoltIn this interesting and well-written paper, the authors discuss variants of the renorming technique which makes a non-contractive operator contractive after a suitable change of the metric and seems to go back to \textit{A. Bielecki} [Bull. Acad. Pol. Sci., Cl. III 4, 261--264 (1956; Zbl 0070.08103)]. In the second half, the authors illustrate the abstract results by means of Uryson-Fredholm integral equations, Uryson-Volterra integral equations, and Presić-type functional equations involving restrictable operators (also called operators ``with memory'').
Reviewer: Jürgen Appell (Würzburg)Existence and uniqueness of fixed points for monotone operators in partially ordered Banach spaces and applicationshttps://zbmath.org/1521.470942023-11-13T18:48:18.785376Z"Khazou, Mohamed"https://zbmath.org/authors/?q=ai:khazou.mohamed"Taoudi, Mohamed Aziz"https://zbmath.org/authors/?q=ai:taoudi.mohamed-azizSummary: In this paper, we employ partial order method, cone theory, and the techniques of measure of weak noncompactness to prove several new theorems on the existence and the uniqueness of fixed points or coupled fixed points for operators satisfying some monotonicity assumptions. Our conclusions generalize and improve several well-known results. As an application, we investigate the existence of a unique solution for a class nonlinear second-order ordinary differential equations. We also discuss the existence of a unique solution for a system of integral equations.Differential inclusions with mixed semicontinuity properties in a Banach spacehttps://zbmath.org/1521.471012023-11-13T18:48:18.785376Z"Tolstonogov, A. A."https://zbmath.org/authors/?q=ai:tolstonogov.alexander-aThe paper studies differential inclusions of the form \[ x'\in F(t,x)+G(t,x),\quad x(0)=x_0, \] where \(X\) is a separable Banach space and \(F,G:[0,a]\times X\to \mathcal{P}(X)\) are set-valued maps.
\(F\) has nonempty closed values, is measurable in the first variable, is Lipschitz in the second variable, and has a linear growth property. \(G\) has closed values, satisfies a certain measurability condition, and has the following property: for any \(t\in [0,a]\) and \(x\in X\), \(G(t,.)\) has a closed graph at \(x\) and \(G(t,x)\) is convex or the restriction of \(G(t,\cdot)\) to a neighborhood of \(x\) is lower semicontinuous.
Under these assumptions, the existence of solutions of this problem is proved.
Reviewer: Aurelian Cernea (Bucureşti)Stepanov ergodic perturbations for nonautonomous evolution equations in Banach spaceshttps://zbmath.org/1521.471142023-11-13T18:48:18.785376Z"Dianda, Abdoul Aziz Kalifa"https://zbmath.org/authors/?q=ai:dianda.abdoul-aziz-kalifa"Ezzinbi, Khalil"https://zbmath.org/authors/?q=ai:ezzinbi.khalil"Khalil, Kamal"https://zbmath.org/authors/?q=ai:khalil.kamalSummary: We prove the existence and uniqueness of \(\mu\)-pseudo almost automorphic solutions for a class of semilinear nonautonomous evolution equations of the form: \(u'(t)=A(t)u(t)+f(t,u(t))\), \( t\in \mathbb{R}\) where \((A(t))_{t\in \mathbb{R}}\) is a family of closed densely defined linear operators acting on a Banach space \(X\), generating a strongly continuous evolution family that have an exponential dichotomy on \(\mathbb{R}\). The nonlinear term \(f : \mathbb{R} \times X \longrightarrow X\) is assumed to be only \(\mu\)-pseudo almost automorphic in Stepanov's sense in \(t\) and Lipschitz continuous with respect to the second variable. To illustrate our theoretical results, we provide an application to a reaction-diffusion equation on \(\mathbb{R}\) with time-dependent parameters.Error analysis of modified Runge-Kutta-Nyström methods for nonlinear second-order delay boundary value problemshttps://zbmath.org/1521.650662023-11-13T18:48:18.785376Z"Zhang, Chengjian"https://zbmath.org/authors/?q=ai:zhang.chengjian"Wang, Siyi"https://zbmath.org/authors/?q=ai:wang.siyi"Tang, Changyang"https://zbmath.org/authors/?q=ai:tang.changyangSummary: This paper is concerned with the numerical solutions of nonlinear second-order boundary value problems with time-variable delay. By adapting Runge-Kutta-Nyström (RKN) methods and combining Lagrange interpolation, a class of modified RKN (MRKN) methods are suggested for solving the problems. Under some suitable conditions, MRKN methods are proved to be convergent of order \(\min\{p, q\}\), where \(p\), \(q\) are the local orders of MRKN methods and Lagrange interpolation, respectively. Numerical experiments further confirm the computational effectiveness and accuracy of MRKN methods.Asymptotic behavior of an adapted implicit discretization of slowly damped second order dynamical systemshttps://zbmath.org/1521.650672023-11-13T18:48:18.785376Z"Horsin, Thierry"https://zbmath.org/authors/?q=ai:horsin.thierry"Jendoubi, Mohamed Ali"https://zbmath.org/authors/?q=ai:jendoubi.mohamed-aliSummary: In the context of damped second order linear dynamical systems, we study the asymptotic behavior of a time discretization of a slowly damped differential equation. We prove that this discretization can be constructed by means of a variable time step that gives rise to the same asymptotic behaviour as for the system in continuous time.Application of compact local integrated RBF (CLI-RBF) for solving transient forward and backward heat conduction problems with continuous and discontinuous sourceshttps://zbmath.org/1521.650682023-11-13T18:48:18.785376Z"Abbaszadeh, Mostafa"https://zbmath.org/authors/?q=ai:abbaszadeh.mostafa"Ebrahimijahan, Ali"https://zbmath.org/authors/?q=ai:ebrahimijahan.ali"Dehghan, Mehdi"https://zbmath.org/authors/?q=ai:dehghan.mehdi(no abstract)Numerical solution of the one dimensional Schrödinger equation using a basis set of scaled and shifted sinc functions on a finite intervalhttps://zbmath.org/1521.650692023-11-13T18:48:18.785376Z"Braun, Moritz"https://zbmath.org/authors/?q=ai:braun.moritzSummary: In this contribution we define a basis set consisting of shifted and scaled sinc functions on the interval \([x_{\min}, x_{\max}]\). We use this set to numerically solve the Schrödinger equation by using the variational method for the harmonic oscillator and the Morse potential. Good agreement with and rapid convergence to the analytic energy values and wave function is shown.Connectionist learning models for application problems involving differential and integral equationshttps://zbmath.org/1521.681432023-11-13T18:48:18.785376Z"Mall, Susmita"https://zbmath.org/authors/?q=ai:mall.susmita"Jeswal, Sumit Kumar"https://zbmath.org/authors/?q=ai:jeswal.sumit-kumar"Chakraverty, Snehashish"https://zbmath.org/authors/?q=ai:chakraverty.snehashishSummary: Artificial intelligence has different research areas such as artificial neural network (ANN), genetic algorithm, support vector machine, fuzzy concept, etc. Functional link artificial neural network (FLANN) is a class of higher-order single-layer ANN models. This chapter describes the general formulation of differential equations (DEs) and integral equations using FLANN, and discusses formulations of ordinary differential equations. It includes the architecture of Laguerre Neural Network (LgNN), general formulation for DEs, and its learning algorithm with gradient computation of the ANN parameters with respect to its inputs. The chapter proposes an ANN-based method for solving a system of Fredholm integral equations. A boundary value problem, a nonlinear Lane-Emden equation, and an application problem of Duffing oscillator equation are also investigated to show the efficiency and powerfulness of the proposed ANN-based method.
For the entire collection see [Zbl 1439.74003].ODE-RU: a dynamical system view on recurrent neural networkshttps://zbmath.org/1521.681452023-11-13T18:48:18.785376Z"Meng, Pinchao"https://zbmath.org/authors/?q=ai:meng.pinchao"Wang, Xinyu"https://zbmath.org/authors/?q=ai:wang.xinyu.1"Yin, Weishi"https://zbmath.org/authors/?q=ai:yin.weishiSummary: The core of the demonstration of this paper is to interpret the forward propagation process of machine learning as a parameter estimation problem of nonlinear dynamical systems. This process is to establish a connection between the Recurrent Neural Network and the discrete differential equation, so as to construct a new network structure: ODE-RU. At the same time, under the inspiration of the theory of ordinary differential equations, we propose a new forward propagation mode. In a large number of simulations and experiments, the forward propagation not only shows the trainability of the new architecture, but also achieves a low training error on the basis of main-taining the stability of the network. For the problem requiring long-term memory, we specifically study the obstacle shape reconstruction problem using the backscattering far-field features data set, and demonstrate the effectiveness of the proposed architecture using the data set. The results show that the network can effectively reduce the sensitivity to small changes in the input feature. And the error generated by the ordinary differential equation cyclic unit network in inverting the shape and position of obstacles is less than \(10^{-2}\).On global positional stabilization of a single-link manipulator with a nonlinear elastic jointhttps://zbmath.org/1521.700102023-11-13T18:48:18.785376Z"Khoroshun, A. S."https://zbmath.org/authors/?q=ai:khoroshun.a-s(no abstract)The parametric analysis of Volterra-Salnikov nonlinear mathematical modelhttps://zbmath.org/1521.700302023-11-13T18:48:18.785376Z"Tsybenova, Svetlana B."https://zbmath.org/authors/?q=ai:tsybenova.svetlana-bSummary: Parametric analysis of Volter-Salnikov nonlinear mathematical model is made. The parametric dependencies of the steady states, bifurcation curves, parametric and phase portraits, temporary dependencies are constructed.Weak solution of longitudinal waves in carbon nanotubeshttps://zbmath.org/1521.741142023-11-13T18:48:18.785376Z"Nicolescu, Adrian Eracle"https://zbmath.org/authors/?q=ai:nicolescu.adrian-eracle"Bobe, Alexandru"https://zbmath.org/authors/?q=ai:bobe.alexandruSummary: This paper studies weak solutions of the axial oscillations of the carbon nanotube modeled as an elastic nanobeam surrounded by an elastic medium. The model used combines Eringen's theory of non-local elasticity and the equations of axial oscillations proposed by Aydogdu. Based on the Sturm-Liouville problem associated with this case, the weak solutions of the longitudinal waves in the carbon nanotube are determined.Logarithmic quantum dynamical bounds for arithmetically defined ergodic Schrödinger operators with smooth potentialshttps://zbmath.org/1521.810682023-11-13T18:48:18.785376Z"Jitomirskaya, Svetlana"https://zbmath.org/authors/?q=ai:jitomirskaya.svetlana-ya"Powell, Matthew"https://zbmath.org/authors/?q=ai:powell.matthew-jSummary: We present a method for obtaining power-logarithmic bounds on the growth of the moments of the position operator for one-dimensional ergodic Schrödinger operators. We use Bourgain's semialgebraic method to obtain such bounds for operators with multifrequency shift or skew-shift underlying dynamics with arithmetic conditions on the parameters.
For the entire collection see [Zbl 1506.46001].Hybrid quantum-classical dynamics of pure-dephasing systemshttps://zbmath.org/1521.810862023-11-13T18:48:18.785376Z"Manfredi, Giovanni"https://zbmath.org/authors/?q=ai:manfredi.giovanni"Rittaud, Antoine"https://zbmath.org/authors/?q=ai:rittaud.antoine"Tronci, Cesare"https://zbmath.org/authors/?q=ai:tronci.cesareSummary: We consider the interaction dynamics of a classical oscillator and a quantum two-level system for different pure-dephasing Hamiltonians of the type \(\widehat{H}(p, q) = H_C(p, q)\mathbf{1} + H_I(p, q)\widehat{\sigma}_z\). This type of systems represents a severe challenge for popular hybrid quantum-classical descriptions. For example, in the case of the common Ehrenfest model, the classical density evolution is shown to decouple entirely from the pure-dephasing quantum dynamics. We focus on a recently proposed hybrid wave equation that is based on Koopman's wavefunction description of classical mechanics. This model retains quantum-classical correlations whenever a coupling potential is present. Here, several benchmark problems are considered and the results are compared with those arising from fully quantum dynamics. A good agreement is found for a series of study cases involving harmonic oscillators with linear and quadratic coupling, as well as time-varying coupling parameters. In all these cases the classical evolution coincides exactly with the oscillator dynamics resulting from the fully quantum description. In the special case of time-independent coupling involving a classical oscillator with varying frequency, the quantum Bloch rotation exhibits peculiar features that escape from the hybrid description. In addition, nonlinear corrections to the harmonic Hamiltonian lead to an overall growth of decoherence at long times, which is absent in the fully quantum treatment.Upper bound for the diameter of a tree in the quantum graph theoryhttps://zbmath.org/1521.810912023-11-13T18:48:18.785376Z"Boyko, O. P."https://zbmath.org/authors/?q=ai:boyko.o-p"Martynyuk, O. M."https://zbmath.org/authors/?q=ai:martynyuk.olga-m"Pivovarchik, V. M."https://zbmath.org/authors/?q=ai:pivovarchik.vyacheslav-nSummary: We study two Sturm-Liouville spectral problems on an equilateral tree with continuity and Kirchhoff conditions at the internal vertices and Neumann conditions at the pendant vertices (first problem) and with Dirichlet conditions at the pendant vertices (second problem). The spectrum of each of these problems consists of infinitely many normal (isolated Fredholm) eigenvalues. It is shown that if we know the asymptotics of eigenvalues, then it is possible to estimate the diameter of a tree from above for each of these problems.Topological recursion and uncoupled BPS structures. II: Voros symbols and the \(\tau\)-functionhttps://zbmath.org/1521.813942023-11-13T18:48:18.785376Z"Iwaki, Kohei"https://zbmath.org/authors/?q=ai:iwaki.kohei"Kidwai, Omar"https://zbmath.org/authors/?q=ai:kidwai.omarSummary: We continue our study of the correspondence between BPS structures and topological recursion in the uncoupled case, this time from the viewpoint of quantum curves. For spectral curves of hypergeometric type, we show the Borel-resummed Voros symbols of the corresponding quantum curves solve Bridgeland's ``BPS Riemann-Hilbert problem''. In particular, they satisfy the required jump property in agreement with the generalized definition of BPS indices \(\Omega\) in our previous work. Furthermore, we observe the Voros coefficients define a closed one-form on the parameter space, and show that (log of) Bridgeland's \(\tau\)-function encoding the solution is none other than the corresponding potential, up to a constant. When the quantization parameter is set to a special value, this agrees with the Borel sum of the topological recursion partition function \(Z_\mathrm{TR}\), up to a simple factor.
For Part I, see [the authors, Adv. Math. 398, Article ID 108191, 54 p. (2022; Zbl 1486.81157)].Research on multi-topic network public opinion propagation model with time delay in emergencieshttps://zbmath.org/1521.913012023-11-13T18:48:18.785376Z"Zhang, Jing"https://zbmath.org/authors/?q=ai:zhang.jing.199"Wang, Xiaoli"https://zbmath.org/authors/?q=ai:wang.xiaoli.3"Xie, Yanxi"https://zbmath.org/authors/?q=ai:xie.yanxi"Wang, Meihua"https://zbmath.org/authors/?q=ai:wang.meihuaSummary: In this paper, three multi-topic network public opinion propagation models, including a without time delay model when each group transforms into new topic propagators (model I); a time delay model when potential propagators transform into new topic propagators (model II); and a time delay model when the initial topic propagators transform into new topic propagators (model III), are established and the basic reproductive number with equilibrium points are found. In addition, stability theory is used to analyze the stability of the public opinion eliminating equilibrium point and the public opinion spreading equilibrium point of the three models. Meanwhile, in order to effectively deal with the negative impact of multi-topic public opinion, an optimal control analysis of the multi-topic network public opinion propagation model is carried out. Finally, a simulation analysis is conducted. It is found that the time delay in model II has no effect on the stability of the public opinion equilibrium points, while the influence of the time delay in model III causes the public opinion eliminating equilibrium point and the public opinion spreading equilibrium point to exhibit Hopf bifurcation.Investigation of financial bubble mathematical model under fractal-fractional Caputo derivativehttps://zbmath.org/1521.913462023-11-13T18:48:18.785376Z"Li, Bo"https://zbmath.org/authors/?q=ai:li.bo.19|li.bo.38|li.bo.36|li.bo.5|li.bo.13|li.bo.10|li.bo.9|li.bo.32|li.bo.16|li.bo.6|li.bo.3|li.bo.2|li.bo.15|li.bo.37|li.bo|li.bo.26|li.bo.12|li.bo.21|li.bo.7"Zhang, Tongxin"https://zbmath.org/authors/?q=ai:zhang.tongxin"Zhang, Chao"https://zbmath.org/authors/?q=ai:zhang.chao|zhang.chao.2|zhang.chao.6|zhang.chao.5|zhang.chao.3|zhang.chao.1|zhang.chao.11|zhang.chao.10|zhang.chao.12|zhang.chao.16(no abstract)Comparison between an exact and a heuristic neural mass model with second-order synapseshttps://zbmath.org/1521.920272023-11-13T18:48:18.785376Z"Clusella, Pau"https://zbmath.org/authors/?q=ai:clusella.pau"Köksal-Ersöz, Elif"https://zbmath.org/authors/?q=ai:ersoz.elif-koksal"Garcia-Ojalvo, Jordi"https://zbmath.org/authors/?q=ai:garcia-ojalvo.jordi"Ruffini, Giulio"https://zbmath.org/authors/?q=ai:ruffini.giulioSummary: Neural mass models (NMMs) are designed to reproduce the collective dynamics of neuronal populations. A common framework for NMMs assumes heuristically that the output firing rate of a neural population can be described by a static nonlinear transfer function (NMM1). However, a recent exact mean-field theory for quadratic integrate-and-fire (QIF) neurons challenges this view by showing that the mean firing rate is not a static function of the neuronal state but follows two coupled nonlinear differential equations (NMM2). Here we analyze and compare these two descriptions in the presence of second-order synaptic dynamics. First, we derive the mathematical equivalence between the two models in the infinitely slow synapse limit, i.e., we show that NMM1 is an approximation of NMM2 in this regime. Next, we evaluate the applicability of this limit in the context of realistic physiological parameter values by analyzing the dynamics of models with inhibitory or excitatory synapses. We show that NMM1 fails to reproduce important dynamical features of the exact model, such as the self-sustained oscillations of an inhibitory interneuron QIF network. Furthermore, in the exact model but not in the limit one, stimulation of a pyramidal cell population induces resonant oscillatory activity whose peak frequency and amplitude increase with the self-coupling gain and the external excitatory input. This may play a role in the enhanced response of densely connected networks to weak uniform inputs, such as the electric fields produced by noninvasive brain stimulation.Interplay of p53 and XIAP protein dynamics orchestrates cell fate in response to chemotherapyhttps://zbmath.org/1521.920562023-11-13T18:48:18.785376Z"Abukwaik, Roba"https://zbmath.org/authors/?q=ai:abukwaik.roba"Vera-Siguenza, Elias"https://zbmath.org/authors/?q=ai:vera-siguenza.elias"Tennant, Daniel A."https://zbmath.org/authors/?q=ai:tennant.daniel-a"Spill, Fabian"https://zbmath.org/authors/?q=ai:spill.fabianSummary: Chemotherapeutic drugs are used to treat almost all types of cancer, but the intended response, i.e., elimination, is often incomplete, with a subset of cancer cells resisting treatment. Two critical factors play a role in chemoresistance: the p53 tumour suppressor gene and the X-linked inhibitor of apoptosis (XIAP). These proteins have been shown to act synergistically to elicit cellular responses upon DNA damage induced by chemotherapy, yet, the mechanism is poorly understood. This study introduces a mathematical model characterising the apoptosis pathway activation by p53 before and after mitochondrial outer membrane permeabilisation upon treatment with the chemotherapy Doxorubicin (Dox). \textit{``In-silico''} simulations show that the p53 dynamics change dose-dependently. Under medium to high doses of Dox, p53 concentration ultimately stabilises to a high level regardless of XIAP concentrations. However, caspase-3 activation may be triggered or not depending on the XIAP induction rate, ultimately determining whether the cell will perish or resist. Consequently, the model predicts that failure to activate apoptosis in some cancer cells expressing wild-type p53 might be due to heterogeneity between cells in upregulating the XIAP protein, rather than due to the p53 protein concentration. Our model suggests that the interplay of the p53 dynamics and the XIAP induction rate is critical to determine the cancer cells' therapeutic response.A model-based strategy for the COVID-19 vaccine roll-out in the Philippineshttps://zbmath.org/1521.920582023-11-13T18:48:18.785376Z"Escosio, Rey Audie S."https://zbmath.org/authors/?q=ai:escosio.rey-audie-s"Cawiding, Olive R."https://zbmath.org/authors/?q=ai:cawiding.olive-r"Hernandez, Bryan S."https://zbmath.org/authors/?q=ai:hernandez.bryan-s"Mendoza, Renier G."https://zbmath.org/authors/?q=ai:mendoza.renier-g"Mendoza, Victoria May P."https://zbmath.org/authors/?q=ai:mendoza.victoria-may-p"Mohammad, Rhudaina Z."https://zbmath.org/authors/?q=ai:mohammad.rhudaina-z"Pilar-Arceo, Carlene P. C."https://zbmath.org/authors/?q=ai:pilar-arceo.carlene-p-c"Salonga, Pamela Kim N."https://zbmath.org/authors/?q=ai:salonga.pamela-kim-n"Suarez, Fatima Lois E."https://zbmath.org/authors/?q=ai:suarez.fatima-lois-e"Sy, Polly W."https://zbmath.org/authors/?q=ai:sy.polly-wee"Vergara, Thomas Herald M."https://zbmath.org/authors/?q=ai:vergara.thomas-herald-m"de los Reyes, Aurelio A."https://zbmath.org/authors/?q=ai:de-los-reyes.aurelio-a-vSummary: COVID-19 has affected millions of people worldwide, causing illness and death, and disrupting daily life while imposing a significant social and economic burden. Vaccination is an important control measure that significantly reduces mortality if properly and efficiently distributed. In this work, an age-structured model of COVID-19 transmission, incorporating an unreported infectious compartment, is developed. Three age groups are considered: \textit{young} (0--19 years), \textit{adult} (20--64 years), and \textit{elderly} (65+ years). The transmission rate and reporting rate are determined for each group by utilizing the number of COVID-19 cases in the National Capital Region in the Philippines. Optimal control theory is employed to identify the best vaccine allocation to different age groups. Further, three different vaccination periods are considered to reflect phases of vaccination priority groups: the first, second, and third account for the inoculation of the elderly, adult and elderly, and all three age groups, respectively. This study could guide in making informed decisions in mitigating a population-structured disease transmission under limited resources.No sensitivity to functional forms in the Rosenzweig-MacArthur model with strong environmental stochasticityhttps://zbmath.org/1521.920672023-11-13T18:48:18.785376Z"Barraquand, Frédéric"https://zbmath.org/authors/?q=ai:barraquand.fredericSummary: The classic Rosenzweig-MacArthur predator-prey model has been shown to exhibit, like other coupled nonlinear ordinary differential equations (ODEs) from ecology, worrying sensitivity to model structure. This sensitivity manifests as markedly different community dynamics arising from saturating functional responses with nearly identical shapes but different mathematical expressions. Using a stochastic differential equation (SDE) version of the Rosenzweig-MacArthur model with the three functional responses considered by \textit{G. F. Fussmann} and \textit{B. Blasius} [Biol. Lett. 1, No. 1, 9--12 (2005; \url{doi:10.1098/rsbl.2004.0246})], I show that such sensitivity seems to be solely a property of ODEs or stochastic systems with weak noise. SDEs with strong environmental noise have by contrast very similar fluctuations patterns, irrespective of the mathematical formula used. Although eigenvalues of linearized predator-prey models have been used as an argument for structural sensitivity, they can also be an argument against structural sensitivity. While the sign of the eigenvalues' real part is sensitive to model structure, its magnitude and the presence of imaginary parts are not, which suggests noise-driven oscillations for a broad range of carrying capacities. I then discuss multiple other ways to evaluate structural sensitivity in a stochastic setting, for predator-prey or other ecological systems.Dispersal-induced growth in a time-periodic environmenthttps://zbmath.org/1521.920702023-11-13T18:48:18.785376Z"Katriel, Guy"https://zbmath.org/authors/?q=ai:katriel.guyDispersal-induced growth (DIG) occurs when several populations with time-varying growth rates, each of which, when isolated, would become extinct, are able to persist and grow exponentially when dispersal among the populations is present. This work employs the results of \textit{S. Liu} et al. [Proc. Am. Math. Soc. 147, No. 12, 5291--5302 (2019; Zbl 1423.35278)] to provide a detailed analysis of the DIG phenomena and the conditions under which it arises. It is shown that occurrence of the DIG effect depends on an appropriate balance of three factors. In summary, this work provides a mathematical exploration of this surprising phenomenon, in the context of a deterministic model with periodic variation of growth rates, and characterizes the factors which are important in generating the DIG effect, and the corresponding conditions on the parameters involved.
Reviewer: Shengqiang Liu (Tianjin)A note on a prey-predator model with constant-effort harvestinghttps://zbmath.org/1521.920722023-11-13T18:48:18.785376Z"Lemos-Silva, Márcia"https://zbmath.org/authors/?q=ai:lemos-silva.marcia"Torres, Delfim F. M."https://zbmath.org/authors/?q=ai:torres.delfim-f-mSummary: We study a prey-predator model based on the classical Lotka-Volterra system with Leslie-Gower and Holling IV schemes and a constant-effort harvesting. Our goal is twofold: to present the model proposed by \textit{L. Cheng} and \textit{L. Zhang} [J. Inequal. Appl. 2021, Paper No. 68, 23 p. (2021; Zbl 1504.37094)], pointing out some inconsistencies; to analyse the number and type of equilibrium points of the model. We end by proving the stability of the meaningful equilibrium point, according to the distribution of the eigenvalues.
For the entire collection see [Zbl 1515.93013].Impact of resource distributions on the competition of species in stream environmenthttps://zbmath.org/1521.920762023-11-13T18:48:18.785376Z"Tung D. Nguyen"https://zbmath.org/authors/?q=ai:tung-d-nguyen."Wu, Yixiang"https://zbmath.org/authors/?q=ai:wu.yixiang"Tang, Tingting"https://zbmath.org/authors/?q=ai:tang.tingting"Veprauskas, Amy"https://zbmath.org/authors/?q=ai:veprauskas.amy"Zhou, Ying"https://zbmath.org/authors/?q=ai:zhou.ying.1|zhou.ying"Rouhani, Behzad Djafari"https://zbmath.org/authors/?q=ai:djafari-rouhani.behzad"Shuai, Zhisheng"https://zbmath.org/authors/?q=ai:shuai.zhishengSummary: Our earlier work in
[``Impact of resource distributions on the competition of species in stream environment'', Preprint, \url{arXiv:2306.05555}]
shows that concentrating resources on the upstream end tends to maximize the total biomass in a metapopulation model for a stream species. In this paper, we continue our research direction by further considering a Lotka-Volterra competition patch model for two stream species. We show that the species whose resource allocations maximize the total biomass has the competitive advantage.The dynamics of delayed models for interactive wild and sterile mosquito populationshttps://zbmath.org/1521.920782023-11-13T18:48:18.785376Z"Wang, Juan"https://zbmath.org/authors/?q=ai:wang.juan"Yue, Peixia"https://zbmath.org/authors/?q=ai:yue.peixia"Cai, Liming"https://zbmath.org/authors/?q=ai:cai.limingIn Chinese, specially in Guanzhou, the sterile insect technique (SIT) has been applied as an alternative method to reduce or eradicate mosquito-borne diseases. During the past decade, to explore the impact of the sterile mosquitoes on controlling the wild mosquito populations, Li had wrote several papers on this direction, for example, [\textit{J. Li}, J. Biol. Dyn. 11, 316--333 (2017; Zbl 1448.92217)]. In this paper, the authors further extended the work considered in that paper to delayed models. By performing mathematical analysis, the threshold dynamics of the proposed models are explored, respectively. In particular, Hopf bifurcation phenomena are observed as the delay τ is varying. This aspect of the study deserves to be expanded further because of the actual ecological context of the model.
Reviewer: Fengde Chen (Fuzhou)Impact of cross border reverse migration in Delhi-UP region of India during COVID-19 lockdownhttps://zbmath.org/1521.920832023-11-13T18:48:18.785376Z"Dwivedi, Shubhangi"https://zbmath.org/authors/?q=ai:dwivedi.shubhangi"Perumal, Saravana Keerthana"https://zbmath.org/authors/?q=ai:perumal.saravana-keerthana"Kumar, Sumit"https://zbmath.org/authors/?q=ai:kumar.sumit"Bhattacharyya, Samit"https://zbmath.org/authors/?q=ai:bhattacharyya.samit"Kumari, Nitu"https://zbmath.org/authors/?q=ai:kumari.nituSummary: The declaration of a nationwide lockdown in India led to millions of migrant workers, particularly from Uttar Pradesh (UP) and Bihar, returning to their home states without proper transportation and social distancing from cities such as Delhi, Mumbai, and Hyderabad. This unforeseen migration and social mixing accelerated the transmission of diseases across the country. To analyze the impact of reverse migration on disease progression, we have developed a disease transmission model for the neighboring Indian states of Delhi and UP. The model's essential mathematical properties, including positivity, boundedness, equilibrium points (EPs), and their linear stability, as well as computation of the basic reproduction number \((R_0)\), are studied. The mathematical analysis reveals that the model with active reverse migration cannot reach a disease-free equilibrium, indicating that the failure of restrictive mobility intervention caused by reverse migration kept the disease propagation alive. Further, PRCC analysis highlights the need for effective home isolation, better disease detection techniques, and medical interventions to curb the spread. The study estimates a significantly shorter doubling time for exponential growth of the disease in both regions. In addition, the occurrence of synchronous patterns between epidemic trajectories of the Delhi and UP regions accentuates the severe implications of migrant plight on UP's already fragile rural health infrastructure. By using COVID-19 incidence data, we quantify key epidemiological parameters, and our scenario analyses demonstrate how different lockdown plans might have impacted disease prevalence. Based on our observations, the transmission rate has the most significant impact on COVID-19 cases. This case study exemplifies the importance of carefully considering these issues before implementing lockdowns and social isolation throughout the country to combat future outbreaks.Correction to: ``Mathematical modeling of COVID-19 and dengue co-infection dynamics in Bangladesh: optimal control and data-driven analysis''https://zbmath.org/1521.920862023-11-13T18:48:18.785376Z"Hye, Md. Abdul"https://zbmath.org/authors/?q=ai:hye.md-abdul"Biswas, Md. Haider Ali"https://zbmath.org/authors/?q=ai:biswas.md-haider-ali"Uddin, Mohammed Forhad"https://zbmath.org/authors/?q=ai:uddin.mohammed-forhad"Saifuddin, Mohammad"https://zbmath.org/authors/?q=ai:saifuddin.mohammadFrom the text: The second author's name is incorrect. It should read Md. Haider Ali Biswas.
Correction to the authors' paper [ibid. 33, No. 2, 173--192 (2022; Zbl 1512.92105)].Fractional dynamics of the transmission phenomena of dengue infection with vaccinationhttps://zbmath.org/1521.920872023-11-13T18:48:18.785376Z"Jan, Rashid"https://zbmath.org/authors/?q=ai:jan.rashid"Boulaaras, Salah"https://zbmath.org/authors/?q=ai:boulaaras.salah-mahmoud"Alyobi, Sultan"https://zbmath.org/authors/?q=ai:alyobi.sultan"Rajagopal, Karthikeyan"https://zbmath.org/authors/?q=ai:rajagopal.karthikeyan"Jawad, Muhammad"https://zbmath.org/authors/?q=ai:jawad.muhammadAuthors' abstract: Dengue infection brings unimaginable damage in low-income countries across the world to public health, social, and economical fields. Scientists rely on modeling to preferable understand the transmission phenomena of dengue in order to forecast some preventive measures and to give better data for the development of vaccines and medications. In present work, we construct a compartmental model for dengue infection via fractional derivatives to show the influence of memory. The suggested system's solution is investigated for existence as well as uniqueness with the help of fractional calculus fundamental characteristics. The model is analyzed and the reproduction number is obtained through next-generation matrix technique, moreover, the sensitivity of the reproduction number is determined using the partial rank correlation coefficient (PRCC) method. We have shown that if \(R_0<1\), the disease-free steady-state of the dengue model is locally asymptotically stable (LAS) and globally asymptotically stable (GAS) in the absence of vaccine. In particular case, we have shown that the infection of dengue persists uniformly in the system for \(R_0\) greater than one. Furthermore, we quantitatively showed memory's impact on \(R_0\) by fluctuation of different factors. Our result predicted that the biting frequency, mosquito recruiting rate, and memory index are the most sensitive aspects in efficiently controlling new dengue illnesses, according to our findings. The effect of vaccination on the suggested system's threshold is also investigated. Novel numerical schemes are presented to conceptualize the time series of the system in different scenarios. The findings suggested that more accurate and precious results are obtained through non-integer derivative and that memory plays a beneficial role in the management of dengue disease. We demonstrated that manipulating the index of memory in the system may manage the system reproduction number and endemic level of dengue infection.
Reviewer: Jiaying Zhou (Shenzhen)Cost-effective optimal control analysis of a COVID-19 transmission model incorporating community awareness and waning immunityhttps://zbmath.org/1521.920882023-11-13T18:48:18.785376Z"Lamba, Sonu"https://zbmath.org/authors/?q=ai:lamba.sonu"Srivastava, Prashant K."https://zbmath.org/authors/?q=ai:srivastava.prashant-krSummary: This article presents a cost-effective optimal control analysis of interventions applied to a S2EI2RS type deterministic compartmental model of COVID-19, considering community awareness and immunity loss. We introduce two time-dependent controls, namely, home quarantine and treatment, to the model for defining an optimal control problem (OCP). In addition to some basic qualitative properties, we obtain the reproductive threshold \(R_0\) by using the next-generation method and see the impact of controls on it. We also investigate the effect of community awareness and waning immunity, when no controls are applied. The existence and characterization of optimal controls is proved to establish the optimality system, and the OCP is solved using the forward-backward sweep method. The results are simulated using MATLAB. Our comparative cost-effective analysis indicates that implementing both control strategies simultaneously, along with community awareness, is the most optimal and sustainable way to flatten COVID-19 curves in a short period of time than that of implementing single controls. This article offers valuable insights that can assist policymakers and public health experts in designing targeted and effective control measures for COVID-19 and future epidemics in the post-COVID era. Therefore, this piece of work could be a valuable contribution to the existing literature.Dynamical behavior of a stochastic HIV model with logistic growth and Ornstein-Uhlenbeck processhttps://zbmath.org/1521.920912023-11-13T18:48:18.785376Z"Liu, Qun"https://zbmath.org/authors/?q=ai:liu.qunSummary: In this paper, we investigate a stochastic human immunodeficiency virus (HIV) model with logistic growth and Ornstein-Uhlenbeck process, which is used to describe the pathogenesis and transmission dynamics of HIV in the population. We first validate that the stochastic system has a unique global solution with any initial value. Then we use a novel Lyapunov function method to establish sufficient conditions for the existence of a stationary distribution of the system, which shows the coexistence of all \(CD4^+\) T cells and free viruses. Especially, under some mild conditions which are used to ensure the local asymptotic stability of the quasi-chronic infection equilibrium of the stochastic system, we obtain the specific expression of covariance matrix in the probability density around the quasi-chronic infection equilibrium of the stochastic system. In addition, for completeness, we also obtain sufficient criteria for elimination of all infected \(CD4^+\) T cells and free virus particles. Finally, several examples together with comprehensive numerical simulations are conducted to support our analytic results.
{\copyright 2023 American Institute of Physics}Dynamical behavior of a stochastic dengue model with Ornstein-Uhlenbeck processhttps://zbmath.org/1521.920922023-11-13T18:48:18.785376Z"Liu, Qun"https://zbmath.org/authors/?q=ai:liu.qunSummary: We develop and study a stochastic dengue model with Ornstein-Uhlenbeck process, in which we assume that the transmission coefficients between vector and human satisfy the Ornstein-Uhlenbeck process. We first show that the stochastic system has a unique global solution with any initial value. Then we use a novel Lyapunov function method to establish sufficient criteria for the existence of a stationary distribution of the system, which indicates the persistence of the disease. In particular, under some mild conditions which are applied to ensure the local asymptotic stability of the endemic equilibrium of the deterministic system, we obtain the specific form of covariance matrix in the probability density around the quasi-positive equilibrium of the stochastic system. In addition, we also establish sufficient criteria for wiping out of the disease. Finally, several numerical simulations are performed to illustrate our theoretical conclusions.
{\copyright 2023 American Institute of Physics}Unequal effects of SARS-CoV-2 infections: model of SARS-CoV-2 dynamics in Cameroon (Sub-Saharan Africa) versus New York State (United States)https://zbmath.org/1521.920942023-11-13T18:48:18.785376Z"Siewe, Nourridine"https://zbmath.org/authors/?q=ai:siewe.nourridine"Yakubu, Abdul-Aziz"https://zbmath.org/authors/?q=ai:yakubu.abdul-azizSummary: Worldwide, the recent SARS-CoV-2 virus disease outbreak has infected more than 691,000,000 people and killed more than 6,900,000. Surprisingly, Sub-Saharan Africa has suffered the least from the SARS-CoV-2 pandemic. Factors that are inherent to developing countries and that contrast with their counterparts in developed countries have been associated with these disease burden differences. In this paper, we developed data-driven COVID-19 mathematical models of two `extreme': Cameroon, a developing country, and New York State (NYS) located in a developed country. We then identified critical parameters that could be used to explain the lower-than-expected COVID-19 disease burden in Cameroon versus NYS and to help mitigate future major disease outbreaks. Through the introduction of a `disease burden' function, we found that COVID-19 could have been much more severe in Cameroon than in NYS if the vaccination rate had remained very low in Cameroon and the pandemic had not ended.Discrete-time system of an intracellular delayed HIV model with CTL immune responsehttps://zbmath.org/1521.920962023-11-13T18:48:18.785376Z"Vaz, Sandra"https://zbmath.org/authors/?q=ai:vaz.sandra-c"Torres, Delfim F. M."https://zbmath.org/authors/?q=ai:torres.delfim-f-mSummary: In [Math. Comput. Sci. 12, No. 2, 111--127 (2018; Zbl 1403.34057)], a delayed model describing the dynamics of the human immunodeficiency virus (HIV) with cytotoxic T lymphocytes (CTL) immune response is investigated by \textit{K. Allali} et al. Here, we propose a discrete-time version of that model, which includes four nonlinear difference equations describing the evolution of uninfected, infected, free HIV viruses, and CTL immune response cells and includes intracellular delay. Using suitable Lyapunov functions, we prove the global stability of the disease free equilibrium point and of the two endemic equilibrium points. We finalize by making some simulations and showing, numerically, the consistence of the obtained theoretical results.
For the entire collection see [Zbl 1515.93013].A study on the transmission dynamics of COVID-19 considering the impact of asymptomatic infectionhttps://zbmath.org/1521.920972023-11-13T18:48:18.785376Z"Xu, Chuanqing"https://zbmath.org/authors/?q=ai:xu.chuanqing"Zhang, Zonghao"https://zbmath.org/authors/?q=ai:zhang.zonghao"Huang, Xiaotong"https://zbmath.org/authors/?q=ai:huang.xiaotong"Cheng, Kedeng"https://zbmath.org/authors/?q=ai:cheng.kedeng"Guo, Songbai"https://zbmath.org/authors/?q=ai:guo.songbai"Wang, Xiaojing"https://zbmath.org/authors/?q=ai:wang.xiaojing"Liu, Maoxing"https://zbmath.org/authors/?q=ai:liu.maoxing"Liu, Xiaoling"https://zbmath.org/authors/?q=ai:liu.xiaolingSummary: The COVID-19 epidemic has been spreading around the world for nearly three years, and asymptomatic infections have exacerbated the spread of the epidemic. To analyse and evaluate the role of asymptomatic infections in the spread of the epidemic, we establish an improved COVID-19 infectious disease dynamics model. We fit the epidemic data in the four time periods corresponding to the selected 614G, Alpha, Delta and Omicron variants and obtain the proportion of asymptomatic persons among the infected persons gradually increased and with the increase of the detection ratio, the cumulative number of cases has dropped significantly, but the decline in the proportion of asymptomatic infections is not obvious. Therefore, in view of the hidden transmission of asymptomatic infections, the cooperation between various epidemic prevention and control policies is required to effectively curb the spread of the epidemic.Effects of behaviour change on HFMD transmissionhttps://zbmath.org/1521.920982023-11-13T18:48:18.785376Z"Zhang, Tongrui"https://zbmath.org/authors/?q=ai:zhang.tongrui"Zhang, Zhijie"https://zbmath.org/authors/?q=ai:zhang.zhijie"Yu, Zhiyuan"https://zbmath.org/authors/?q=ai:yu.zhiyuan"Huang, Qimin"https://zbmath.org/authors/?q=ai:huang.qimin"Gao, Daozhou"https://zbmath.org/authors/?q=ai:gao.daozhouSummary: We propose a hand, foot and mouth disease (HFMD) transmission model for children with behaviour change and imperfect quarantine. The symptomatic and quarantined states obey constant behaviour change while others follow variable behaviour change depending on the numbers of new and recent infections. The basic reproduction number \(\mathcal{R}_0\) of the model is defined and shown to be a threshold for disease persistence and eradication. Namely, the disease-free equilibrium is globally asymptotically stable if \(\mathcal{R}_0\leq 1\) whereas the disease persists and there is a unique endemic equilibrium otherwise. By fitting the model to weekly HFMD data of Shanghai in 2019, the reproduction number is estimated at 2.41. Sensitivity analysis for \(\mathcal{R}_0\) shows that avoiding contagious contacts and implementing strict quarantine are essential to lower HFMD persistence. Numerical simulations suggest that strong behaviour change not only reduces the peak size and endemic level dramatically but also impairs the role of asymptomatic transmission.How to combat atmospheric carbon dioxide along with development activities? A mathematical modelhttps://zbmath.org/1521.920992023-11-13T18:48:18.785376Z"Misra, A. K."https://zbmath.org/authors/?q=ai:misra.arvind-k|misra.arvind-kumar|misra.amit-kumar|misra.aruna-kumari|misra.ankit-kumar|misra.abhishek-kumar|misra.arun-k"Jha, Anjali"https://zbmath.org/authors/?q=ai:jha.anjaliSummary: Depletion in forest density due to development activities is one of the reasons for the elevated level of carbon dioxide (\(\mathrm{CO}_2\)). In this scenario, the plantation of leafy trees, which absorb more \(\mathrm{CO}_2\) compared to regular trees, under a planned strategy may help to attain the desired level of atmospheric \(\mathrm{CO}_2\). We have formulated a non-linear mathematical model concerning the strategy of maintaining the atmospheric level of \(\mathrm{CO}_2\) along with development activities and analyzed the proportion of deforested land needed for the plantation of leafy trees. This strategy will ensure that the absorption of \(\mathrm{CO}_2\) remains at its previous level. To study the long-term behavior of the formulated model system, we employ the qualitative theory of differential equations. We have derived sufficient conditions under which all the considered dynamical variables settle to their equilibrium level. After analyzing the formulated model system, we find that the system undergoes Hopf and transcritical bifurcations with respect to the deforestation rate. Furthermore, numerical simulations have been carried out to support the analytically obtained results. Through simulation, we have determined the proportion of cleared land that should be utilized for the plantation of leafy trees to maintain the level of atmospheric \(\mathrm{CO}_2\).Role of ecotourism in conserving forest biomass: a mathematical modelhttps://zbmath.org/1521.921012023-11-13T18:48:18.785376Z"Pathak, Rachana"https://zbmath.org/authors/?q=ai:pathak.rachana"Bhadauria, Archana Singh"https://zbmath.org/authors/?q=ai:bhadauria.archana-singh"Chaudhary, Manisha"https://zbmath.org/authors/?q=ai:chaudhary.manisha"Verma, Harendra"https://zbmath.org/authors/?q=ai:verma.harendra"Mathur, Pankaj"https://zbmath.org/authors/?q=ai:mathur.pankaj"Agrawal, Manju"https://zbmath.org/authors/?q=ai:agrawal.manju-rani|agrawal.manju-lata"Singh, Ram"https://zbmath.org/authors/?q=ai:singh.ram-gopal|singh.ram-chandra|singh.ram-veer|singh.ram-mehar|singh.ram-nawal|singh.ram-binoy|singh.ram-milan|singh.ram-sakal|singh.ram-narayanSummary: Ecotourism is a form of tourism involving responsible travel to natural areas, conserving the environment, and improving the well-being of the local people. Its purpose may be to educate the traveler, to provide funds for ecological conservation, to directly benefit the economic development, and political empowerment of local communities. Ecotourism has come up as an important conservation strategy in the tropical areas where diversity of species and habitats are threatened because of the traditional forms of development. This study deals with a non-linear mathematical model with a novel idea for sustainable development of biomass with ecotourism which is imperative in the present scenario. Stability and bifurcation analysis of the model is done and it is observed from our study that the system predicts instability and exhibits bifurcation if ecotourism goes beyond a threshold value.The impact of predators of mosquito larvae on \textit{Wolbachia} spreading dynamicshttps://zbmath.org/1521.921032023-11-13T18:48:18.785376Z"Zhu, Zhongcai"https://zbmath.org/authors/?q=ai:zhu.zhongcai"Hui, Yuanxian"https://zbmath.org/authors/?q=ai:hui.yuanxian"Hu, Linchao"https://zbmath.org/authors/?q=ai:hu.linchaoSummary: Dengue fever creates more than 390 million cases worldwide yearly. The most effective way to deal with this mosquito-borne disease is to control the vectors. In this work we consider two weapons, the endosymbiotic bacteria \textit{Wolbachia} and predators of mosquito larvae, for combating the disease. As \textit{Wolbachia}-infected mosquitoes are less able to transmit dengue virus, releasing infected mosquitoes to invade wild mosquito populations helps to reduce dengue transmission. Besides this measure, the introduction of predators of mosquito larvae can control mosquito population. To evaluate the impact of the predators on \textit{Wolbachia} spreading dynamics, we develop a stage-structured five-dimensional model, which links the predator-prey dynamics with the \textit{Wolbachia} spreading. By comparatively analysing the dynamics of the models without and with predators, we observe that the introduction of the predators augments the number of coexistence equilibria and impedes \textit{Wolbachia} spreading. Some numerical simulations are presented to support and expand our theoretical results.Approximate controllability of non-instantaneous impulsive semi-linear neutral differential equations with non-local conditions and unbounded delayhttps://zbmath.org/1521.930132023-11-13T18:48:18.785376Z"Camacho, Oscar"https://zbmath.org/authors/?q=ai:camacho.oscar"Lalvay-Segovia, S."https://zbmath.org/authors/?q=ai:lalvay-segovia.s"Leiva, Hugo"https://zbmath.org/authors/?q=ai:leiva.hugo"Riera-Segura, L."https://zbmath.org/authors/?q=ai:riera-segura.lSummary: In this work, we study the approximate controllability for retarded neutral differential equations with unbounded delay, non-instantaneous impulses, and non-local conditions. First, we set the problem in a natural Banach phase space satisfying Hale-Kato axiomatic theory about the phase space for retarded ordinary equations with unbounded delay. Second, we assume some conditions on the nonlinear terms to achieve the approximate controllability using the technique developed by \textit{A. E. Bashirov} and \textit{N. Ghahramanlou} [``On partial approximate controllability of semilinear systems'', Cogent Eng. 1, No. 1, Article ID 965947, 13 p. (2014; \url{doi:10.1080/23311916.2014.965947})], which avoids the use of fixed point theorems.On the solvability of the Atangana-Baleanu fractional evolution equations: an integral contractor approachhttps://zbmath.org/1521.930142023-11-13T18:48:18.785376Z"Chaudhary, Renu"https://zbmath.org/authors/?q=ai:chaudhary.renu"Reich, Simeon"https://zbmath.org/authors/?q=ai:reich.simeonIn this paper the authors have studied the existence and controllability results for mild solutions to the Atangana-Baleanu fractional evolution equations in Banach spaces. The results are obtained by applying bounded integral contractors and a sequencing technique. Controllability results are established with out the condition of the induced inverse controllability operator, and the corresponding nonlinear function need not satisfy a Lipschitz condition. Further the authors discussed the trajectory controllability results. An application is given to illustrate the results.
Remark: Several researchers have observed that the Atangana-Baleanu derivative lacks physical meaning and in this regard the readers can look into the following papers [\textit{K. Diethelm} et al., Fract. Calc. Appl. Anal. 23, No. 3, 610--634 (2020; Zbl 1474.26020); \textit{A. Giusti}, Nonlinear Dyn. 93, No. 3, 1757--1763 (2018; Zbl 1398.28001); \textit{E. C. de Oliveira} et al., ``On the mistake in defining fractional derivative using a non-singular kernel'', Preprint, \url{arXiv:1912.04422}; \textit{M. D. Ortigueira} et al., Fract. Calc. Appl. Anal. 22, No. 2, 255--270 (2019; Zbl 1427.93054)].
Reviewer: Krishnan Balachandran (Coimbatore)On the constrained and unconstrained controllability of semilinear Hilfer fractional systemshttps://zbmath.org/1521.930182023-11-13T18:48:18.785376Z"Sikora, Beata"https://zbmath.org/authors/?q=ai:sikora.beataSummary: In the paper finite-dimensional semilinear dynamical control systems described by fractional-order state equations with the Hilfer fractional derivative are discussed. The formula for a solution of the considered systems is presented and derived using the Laplace transform. Bounded nonlinear function \(f\) depending on a state and controls is used. New sufficient conditions for controllability without constraints are formulated and proved using Rothe's fixed point theorem and the generalized Darbo fixed point theorem. Moreover, the stability property is used to formulate constrained controllability criteria. An illustrative example is presented to give the reader an idea of the theoretical results obtained. A transient process in an electrical circuit described by a system of Hilfer type fractional differential equations is proposed as a possible application of the study.Controllability and observability for linear quaternion-valued impulsive differential equationshttps://zbmath.org/1521.930192023-11-13T18:48:18.785376Z"Suo, Leping"https://zbmath.org/authors/?q=ai:suo.leping"Fečkan, Michal"https://zbmath.org/authors/?q=ai:feckan.michal"Wang, JinRong"https://zbmath.org/authors/?q=ai:wang.jinrongAuthors' abstract: In this paper, the controllability and observability for linear quaternion-valued impulsive differential equations (QIDEs) are investigated. We mainly establish some sufficient and necessary conditions for state controllability and state observability of linear QIDEs. By virtue of the isomorphism between quaternion vector space and complex variables space as well as the adjoint matrix of quaternion matrix, all the theoretical results in the sense of complex-valued and quaternion-valued are equivalent to each other. Finally, the validity of theoretical results acquired is indicated by two instances shown.
Reviewer: Krishnan Balachandran (Coimbatore)Differential elimination for dynamical models via projections with applications to structural identifiabilityhttps://zbmath.org/1521.930342023-11-13T18:48:18.785376Z"Dong, Ruiwen"https://zbmath.org/authors/?q=ai:dong.ruiwen"Goodbrake, Christian"https://zbmath.org/authors/?q=ai:goodbrake.christian"Harrington, Heather A."https://zbmath.org/authors/?q=ai:harrington.heather-a"Pogudin, Gleb"https://zbmath.org/authors/?q=ai:pogudin.gleb-aThe research is motivated by the differential algebra approach development to assess the structural identifiability of a dynamical models. The main contribution is employment of the conventional representation of the state-space form of the dynamical system as input-output relation to design the new formalism to generate the differential-algebraic relations between the input and output variables of a dynamical system model. The algorithm for differential elimination using the projection-based representation is proposed and applied for assessing the structural identifiability. It is implemented in Julia language and available on GitHub and validated on 10 examples including SIR and pharmacokinetics models.
Reviewer: Denis Sidorov (Irkutsk)Fractional order \(\mathrm{PI}^\lambda\mathrm{D}^\mu\) A controller design based on Bode's ideal functionhttps://zbmath.org/1521.930572023-11-13T18:48:18.785376Z"Bettou, Khalfa"https://zbmath.org/authors/?q=ai:bettou.khalfa"Charef, Abdelfatah"https://zbmath.org/authors/?q=ai:charef.abdelfatahSummary: The fractional order proportional, integral, derivative and acceleration (PI\(^\lambda\)D\(^\mu\) A) controller is an extension of the classical PIDA controller with real rather than integer integration action order \(\lambda\) and differentiation action order \(\mu\). Because the orders \(\lambda\) and \(\mu\) are real numbers, they will provide more flexibility in the feedback control design for a large range of control systems. The Bode's ideal transfer function is largely adopted function in fractional control systems because of its iso-damping property which is an essential robustness factor.
In this paper an analytical design technique of a fractional order PI\(^\lambda\)D\(^\mu\) A controller is presented to achieve a desired closed loop system whose transfer function is the Bode's ideal function. In this design method, the values of the six parameters of the fractional order PI\(^\lambda\)D\(^\mu\) A controllers are calculated using only the measured step response of the process to be controlled. Some simulation examples for different third order motor models are presented to illustrate the benefits, the effectiveness and the usefulness of the proposed fractional order PI\(^\lambda\)D\(^\mu\) A controller tuning technique. The simulation results of the closed loop system obtained by the fractional order PI\(^\lambda\)D\(^\mu\) A controller are compared to those obtained by the classical PIDA controller with different design methods found in the literature. The simulation results also show a significant improvement in the closed loop system performances and robustness using the proposed fractional order PI\(^\lambda\)D\(^\mu\) A controller design.A new two-scroll 4-D hyperchaotic system with a unique saddle point equilibrium, its bifurcation analysis, circuit design and a control application to complete synchronizationhttps://zbmath.org/1521.930622023-11-13T18:48:18.785376Z"Vaidyanathan, Sundarapandian"https://zbmath.org/authors/?q=ai:vaidyanathan.sundarapandian"Moroz, Irene M."https://zbmath.org/authors/?q=ai:moroz.irene-m"Sambas, Aceng"https://zbmath.org/authors/?q=ai:sambas.acengSummary: In this work, we present new results for a two-scroll 4-D hyperchaotic system with a unique saddle point equilibrium at the origin. The bifurcation and multi-stability analysis for the new hyperchaotic system are discussed in detail. As a control application, we develop a feedback control based on integral sliding mode control (ISMC) for the complete synchronization of a pair of two-scroll hyperchaotic systems developed in this work. Numerical simulations using Matlab are provided to illustrate the hyperchaotic phase portraits, bifurcation diagrams and synchronization results. Finally, as an electronic application, we simulate the new hyperchaotic system using Multisim for real-world implementations.Generalized observer design of index one for descriptor systems with unknown inputshttps://zbmath.org/1521.930772023-11-13T18:48:18.785376Z"Kumar, Abhinav"https://zbmath.org/authors/?q=ai:kumar.abhinav"Gupta, Mahendra Kumar"https://zbmath.org/authors/?q=ai:gupta.mahendra-kumarSummary: Generalized observers are proposed to relax the existing conditions required to design Luenberger observers for rectangular linear descriptor systems with unknown inputs. The current work is focused on designing index one generalized observers, which can be naturally extended to higher indexes. Sufficient conditions in terms of system operators for the existence of generalized observers are given and proved. Orthogonal transformations are used to derive the results. A physical model is presented to show the usefulness of the proposed theory.Neural ODE control for classification, approximation, and transporthttps://zbmath.org/1521.930802023-11-13T18:48:18.785376Z"Ruiz-Balet, Domènec"https://zbmath.org/authors/?q=ai:ruiz-balet.domenec"Zuazua, Enrique"https://zbmath.org/authors/?q=ai:zuazua.enriqueSummary: We analyze neural ordinary differential equations (NODEs) from a control theoretical perspective to address some of the main properties and paradigms of deep learning (DL), in particular, data classification and universal approximation. These objectives are tackled and achieved from the perspective of the simultaneous control of systems of NODEs. For instance, in the context of classification, each item to be classified corresponds to a different initial datum for the control problem of the NODE, to be classified, all of them by the same common control, to the location (a subdomain of the Euclidean space) associated to each label. Our proofs are genuinely nonlinear and constructive, allowing us to estimate the complexity of the control strategies we develop. The nonlinear nature of the activation functions governing the dynamics of NODEs under consideration plays a key role in our proofs, since it allows deforming half of the phase space while the other half remains invariant, a property that classical models in mechanics do not fulfill. This very property allows us to build elementary controls inducing specific dynamics and transformations whose concatenation, along with properly chosen hyperplanes, allows us to achieve our goals in finitely many steps. The nonlinearity of the dynamics is assumed to be Lipschitz. Therefore, our results apply also in the particular case of the ReLU activation function. We also present the counterparts in the context of the control of neural transport equations, establishing a link between optimal transport and deep neural networks.Stabilization of oscillations of a controlled autonomous systemhttps://zbmath.org/1521.931242023-11-13T18:48:18.785376Z"Tkhai, V. N."https://zbmath.org/authors/?q=ai:tkhai.valentin-nikolaevichSummary: We consider a smooth autonomous system in general form that admits a non-degenerate periodic solution. A global family (with respect to the parameter \(h\)) of nondegenerate periodic solutions is constructed, the law of monotonic variation of the period on the family is derived, and the existence of a reduced second-order system is proved. For it, the problem of stabilizing the oscillation of the controlled system, distinguished by the value of the parameter \(h\), is solved. A smooth autonomous control is found, and an attracting cycle is constructed.Stability analysis of a class of variable fractional-order uncertain neutral-type systems with time-varying delayhttps://zbmath.org/1521.931332023-11-13T18:48:18.785376Z"Aghayan, Zahra Sadat"https://zbmath.org/authors/?q=ai:aghayan.zahra-sadat"Alfi, Alireza"https://zbmath.org/authors/?q=ai:alfi.alireza"Mousavi, Yashar"https://zbmath.org/authors/?q=ai:mousavi.yashar"Fekih, Afef"https://zbmath.org/authors/?q=ai:fekih.afefSummary: Variable fractional-order (VFO) differential equations are a beneficial tool for describing the nonlinear behavior of complex dynamical phenomena. In comparison with the constant FO derivatives, it describes the memory properties of such systems that can vary in the time domain and spatial location. This article investigates the stability and stabilization of VFO neutral systems in the presence of time-varying structured uncertainties and time-varying delay. FO Lyapunov theorem is adopted to achieve order-dependent and delay-dependent criteria for both nominal and uncertain VFO neutral delay systems. The obtained conditions are given in respect of linear matrix inequality by designing a delayed state feedback controller. Simulations verify the main results.A new stability criterion and its application to robust stability analysis for linear systems with distributed delayshttps://zbmath.org/1521.931342023-11-13T18:48:18.785376Z"Kudryakov, Dmitry A."https://zbmath.org/authors/?q=ai:kudryakov.dmitry-a"Alexandrova, Irina V."https://zbmath.org/authors/?q=ai:aleksandrova.irina-vladimirovnaThe paper considers a rather general time delay system of the form
\[ \dot{x}(t) = \sum_1^m\left(A_kx(t-h_k) + \int_{-h_k}^0G_k(\theta)x(t+\theta)d\theta\right)\tag{1}
\]
Let \(h=\max_k\;h_k\) and \(X = PC(-h,0;C^n)\). Let also \(S=\{\varphi\in X\mid\Vert \varphi\Vert_h=|\varphi(0)|\}\) and \(S(\tau,K)=\{\varphi\in C^1(-h,0;C^n)|\Vert \varphi\Vert_\tau=|\varphi(0)|, \Vert \varphi'\Vert_\tau \leq K|\varphi(0)|\}\) with
\[
\tau\geq h, K= \Vert A_0\Vert + \sum_1^m\left(\Vert A_i\Vert + \int_{-h_i}^0\Vert K_i(\theta)\Vert d\theta\right)
\]
The main result of the paper is to show that the negative definiteness of a Lyapunov-Krasovskii functional derivative along the solutions of (1) may be verified only on the set \(S\) or \(S(\tau,K)\) in order to obtain asymptotic stability.
Reviewer: Vladimir Răsvan (Craiova)The exponential input-to-state stability property: characterisations and feedback connectionshttps://zbmath.org/1521.931552023-11-13T18:48:18.785376Z"Guiver, Chris"https://zbmath.org/authors/?q=ai:guiver.chris"Logemann, Hartmut"https://zbmath.org/authors/?q=ai:logemann.hartmutThe aim of this paper is twofold. It can be viewed as a survey paper on I(nput)-to-S(tate)-S(tability) and also as containing new results on ``exponential'' ISS characterization and feedback connections. The basic model is given by
\[
\dot{x} = f(x,d(t))
\]
with \(d(t)\) -- an external input. Since exponential stability sends to an almost linear behavior, the locally Lipschitz assumption in both variables for \(f\) is natural. Also natural is the use of what is called ``G(lobal)-E(xponential)-S(tability)'' Lyapunov function for \(\dot{x}=g(x)\) i.e. a Lyapunov function satisfying
\[
\displaylines{0<a_1|z|^2\leq V(z)\leq a_2|z|^2 \cr (\nabla V(z),g(z))\leq - a_3V(z)<0}
\]
Also there are considered trajectories defined globally for \((0,\infty)\).
The main results of the paper are a ``round-about'' Theorem 3.4 on global exponential ISS and Theorem 4.3 on small gain conditions for global exponential ISS for feedback connections.
Reviewer: Vladimir Răsvan (Craiova)On determining the coefficients of a quadratic Lyapunov function with given properties in the case of multiple roots of the characteristic equationhttps://zbmath.org/1521.931632023-11-13T18:48:18.785376Z"Antonovskaya, O. G."https://zbmath.org/authors/?q=ai:antonovskaya.olga-gSummary: For continuous and discrete linear autonomous systems, we discuss the possibility of choosing the coefficients of the quadratic Lyapunov function that ensure the validity of the condition of sign-negativity of its first derivative (first difference) with a given margin in the case of multiple roots of the characteristic equation.Kramer-type sampling theorems associated with higher-order differential equationshttps://zbmath.org/1521.940172023-11-13T18:48:18.785376Z"Markett, Clemens"https://zbmath.org/authors/?q=ai:markett.clemensSummary: For many decades, \textit{H. P. Kramer}'s sampling theorem [J. Math. Phys., Mass. Inst. Techn. 38, 68--72 (1959; Zbl 0196.31702)] has been attracting enormous interest in view of its important applications in various branches. In this paper we present a new approach to a Kramer-type theory based on spectral differential equations of higher order on an interval of the real line. Its novelty relies partly on the fact that the corresponding eigenfunctions are orthogonal with respect to a scalar product involving a classical measure together with a point mass at a finite endpoint of the domain. In particular, a new sampling theorem is established, which is associated with a self-adjoint Bessel-type boundary value problem of fourth-order on the interval \([0,1]\). Moreover, we consider the Laguerre and Jacobi differential equations and their higher-order generalizations and establish the Green-type formulas of the differential operators as an essential key towards a corresponding sampling theory.Solutions of the van der Pol equationhttps://zbmath.org/1521.970242023-11-13T18:48:18.785376Z"D'Alessio, Serge"https://zbmath.org/authors/?q=ai:dalessio.sergeSummary: Presented in this paper are various solutions to the Van der Pol equation. Numerical solutions are utilized as an independent means of validating the various solutions discussed. A new solution in the form of a power series has been found. Although this solution is exact, its interval of convergence can only be estimated for a special case. Numerical experiments reveal that the power series solution can provide an exact solution over intervals where other approximate solutions are not valid. Thus, the new solution represents an additional solution that can complement other existing solutions. This work also emphasizes the importance and the role of computation. Although the power series solution along with the other approximate solutions mentioned are of theoretical interest, their restrictions limit their usefulness in real applications, and therefore numerical methods should also be considered.