Recent zbMATH articles in MSC 37Chttps://zbmath.org/atom/cc/37C2022-09-13T20:28:31.338867ZWerkzeugRate-induced tipping phenomena in compartment models of epidemicshttps://zbmath.org/1491.340622022-09-13T20:28:31.338867Z"Merker, Jochen"https://zbmath.org/authors/?q=ai:merker.jochen"Kunsch, Benjamin"https://zbmath.org/authors/?q=ai:kunsch.benjaminSummary: The aim of this chapter is to explore non-autonomous compartment models of epidemics, like, e.g., SIR models with time-dependent transmission and recovery rates as parameters, and particularly the occurrence of rate-induced tipping phenomena. Specifically, we are interested in the question, whether there can exist parameter paths that do not cross any bifurcation points, but yet give rise to tipping if the parameters vary over time. From literature, it is known that such rate-induced tipping occurs, e.g., in two-dimensional models of ecosystems or predator-prey systems. We show in this chapter that rate-induced tipping can also occur in compartment models of epidemics. Thus, regarding the Covid-19 crisis, not only the measures established in a lockdown and the moment of the lockdown, but also the rate by which lockdown measures are implemented may have a drastic influence on the number of infectious.
For the entire collection see [Zbl 1470.92006].Generalization of the Filippov method for systems with a large periodic inputhttps://zbmath.org/1491.340632022-09-13T20:28:31.338867Z"Morel, C."https://zbmath.org/authors/?q=ai:morel.cristina"Morel, J.-Y."https://zbmath.org/authors/?q=ai:morel.jean-yves"Danca, M. F."https://zbmath.org/authors/?q=ai:danca.maris-f|danca.marius-florinSummary: Using the Filippov method, the stability of a nominal cyclic steady state of a nonlinear dynamic system (a buck dc-dc converter) is investigated. A common approach to this study is based upon a complete clock period, and assumes that the input is from a regulated dc power supply. In reality, this is usually not the case: converters are mostly fed from a rectified and filtered source. This dc voltage will then contain ripples (i.e. the peak-to-peak input voltage is not zero). Therefore, we consider the input as a sinusoidal voltage. Its frequency is chosen as a submultiple \(T\) of the converter's clock and our objective is to analyze, clarify and predict some of the nonlinear behaviors that these circuits may exhibit, when the input voltage frequency changes in time. This input frequency's parameter \(T\) determines the number of the switching instants over a whole clock cycle, obtained as Newton-Raphson solutions. Then, for the considered buck converter, we develop a mathematical model in a compact form of the Jacobian matrix with a variable dimension proportional to the input voltage harmonics. Finally, the Floquet multipliers of the monodromy matrix are used to predict the system stability. Numerical examples illustrate how these multipliers cross the unit cycle causing various bifurcations.A temperature-dependent mathematical model of malaria transmission with stage-structured mosquito population dynamicshttps://zbmath.org/1491.340672022-09-13T20:28:31.338867Z"Traoré, Bakary"https://zbmath.org/authors/?q=ai:traore.bakary"Barro, Moussa"https://zbmath.org/authors/?q=ai:barro.moussa"Sangaré, Boureima"https://zbmath.org/authors/?q=ai:sangare.boureima"Traoré, Sado"https://zbmath.org/authors/?q=ai:traore.sadoSummary: In this paper, we formulate a temperature-dependent model for malaria transmission dynamics which includes immature stages of mosquitoes. The model is constructed by using ordinary differential equations with some parameters which are periodic functions. Two thresholds dynamics associated to the model have been derived: the vector reproduction ratio \(\mathcal{R}_v\) and the basic reproduction ratio \(\mathcal{R}_0\). Through a rigorous analysis via theories and methods of dynamical systems, we prove that the global behavior of the model depends strongly on these two parameters. More precisely, we show that if \(\mathcal{R}_v\) is greater than one and \(\mathcal{R}_0\) is less than one then, the disease-free periodic equilibrium is globally attractive. If \(\mathcal{R}_v\) is greater than one and \(\mathcal{R}_0\) is greater than one, the disease remains persistent and the system admits at least one positive periodic solution. Finally, using the reported monthly mean temperature for Burkina Faso, numerical simulations are carried out to illustrate our mathematical results.Conditional stability and periodicity of solutions to evolution equationshttps://zbmath.org/1491.340722022-09-13T20:28:31.338867Z"Nguyen, Thieu Huy"https://zbmath.org/authors/?q=ai:nguyen-thieu-huy."Vu, Thi Ngoc Ha"https://zbmath.org/authors/?q=ai:vu-thi-ngoc-ha.The authors first prove the existence and uniqueness of periodic mild solutions to a first-order linear nonautonomous differential equation in a Banach space. The key assumption is that the time dependent linear operators governing the equation generate a strongly continuous, exponentially bounded evolution family. A related semilinear evolution equation is next considered. Finally, applications of the theory to exponentially dichotomic evolution families, and nonautonomous damped wave equations are given.
Reviewer: Sergiu Aizicovici (Verona)Conjugate points in \(\mathcal{D}_\mu^s(S^2)\)https://zbmath.org/1491.353272022-09-13T20:28:31.338867Z"Benn, J."https://zbmath.org/authors/?q=ai:benn.jamesSummary: Rossby-Haurwitz waves on the sphere \(S^2\) form a set of exact time-dependent solutions to the Euler equations of hydrodynamics and generate a family of non-stationary geodesics of the \(L^2\) metric in the volume preserving diffeomorphism group of \(S^2\). Restricting to a particular subset of Rossby-Haurwitz waves, this article shows that under certain conditions on the physical characteristics of the waves each corresponding geodesic contains conjugate points. In addition, a physical interpretation of conjugate points is given and links the result to the stability analysis of meteorological Rossby-Haurwitz waves.On a higher integral invariant for closed magnetic lines, revisitedhttps://zbmath.org/1491.353292022-09-13T20:28:31.338867Z"Akhmet'ev, Peter M."https://zbmath.org/authors/?q=ai:akhmetev.petr-mSummary: We recall a definition of an asymptotic invariant of classical link, which is called \(M\)-invariant. \(M\)-invariant is a special Massey integral, this integral has an ergodic form and is generalized for magnetic fields with open magnetic lines in a bounded \(3D\)-domain. We present a proof that this integral is well defined. A combinatorial formula for \(M\)-invariant using the Conway polynomial is presented. The \(M\)-invariant is a higher invariant, it is not a function of pairwise linking numbers of closed magnetic lines. We discuss applications of \(M\)-invariant for MHD.Homoclinic solutions of periodic discrete Schrödinger equations with local superquadratic conditionshttps://zbmath.org/1491.354052022-09-13T20:28:31.338867Z"Lin, Genghong"https://zbmath.org/authors/?q=ai:lin.genghong"Yu, Jianshe"https://zbmath.org/authors/?q=ai:yu.jian-sheGlobal dynamics and diffusion in the rational standard maphttps://zbmath.org/1491.370012022-09-13T20:28:31.338867Z"Cincotta, Pablo M."https://zbmath.org/authors/?q=ai:cincotta.pablo-m"Simó, Carles"https://zbmath.org/authors/?q=ai:simo.carlesSummary: In this paper we study the dynamics of the Rational Standard Map, which is a generalization of the Standard Map. It depends on two parameters, the usual \(K\) and a new one, \(0 \leq \mu < 1\), that breaks the entire character of the perturbing function. By means of analytical and numerical methods it is shown that this system presents significant differences with respect to the classical Standard Map. In particular, for relatively large values of \(K\) the integer and semi-integer resonances are stable for some range of \(\mu\) values. Moreover, for \(K\) not small and near suitable values of \(\mu\), its dynamics could be assumed to be well represented by a nearly integrable system. On the other hand, periodic solutions or accelerator modes also show differences between this map and the standard one. For instance, in case of \(K \approx 2\pi\) accelerator modes exist for \(\mu\) less than some critical value but also within very narrow intervals when \(0.9 < \mu < 1\). Big differences for the domains of existence of rotationally invariant curves (much larger, for \(\mu\) moderate, or much smaller, for \(\mu\) close to 1 than for the standard map) appear. While anomalies in the diffusion are observed, for large values of the parameters, the system becomes close to an ergodic one.Strong Birkhoff ergodic theorem for subharmonic functions with irrational shift and its application to analytic quasi-periodic cocycleshttps://zbmath.org/1491.370052022-09-13T20:28:31.338867Z"Tao, Kai"https://zbmath.org/authors/?q=ai:tao.kaiSummary: In this paper, we first prove the strong Birkhoff Ergodic Theorem for subharmonic functions with the irrational shift on the Torus. Then, we apply it to the analytic quasi-periodic Jacobi cocycles and show that for suitable frequency and coupling number, if the Lyapunov exponent of these cocycles is positive at one point, then it is positive on an interval centered at this point and Hölder continuous in \(E \) on this interval. What's more, if the coupling number of the potential is large, then the Lyapunov exponent is always positive for all irrational frequencies and Hölder continuous in \(E \) for all finite Liouville frequencies. For the Schrödinger cocycles, a special case of the Jacobi ones, its Lyapunov exponent is also Hölder continuous in the frequency and the lengths of the intervals where the Hölder condition of the Lyapunov exponent holds only depend on the coupling number.Markovian properties of continuous group actions: algebraic actions, entropy and the homoclinic grouphttps://zbmath.org/1491.370102022-09-13T20:28:31.338867Z"Barbieri, Sebastián"https://zbmath.org/authors/?q=ai:barbieri.sebastian"García-Ramos, Felipe"https://zbmath.org/authors/?q=ai:garcia-ramos.felipe"Li, Hanfeng"https://zbmath.org/authors/?q=ai:li.hanfengSummary: We provide a unifying approach which links results on algebraic actions by \textit{D. Lind} and \textit{K. Schmidt} [J. Am. Math. Soc. 12, No. 4, 953--980 (1999; Zbl 0940.22004); Russ. Math. Surv. 70, No. 4, 657--714 (2015; Zbl 1357.37041); translation from Usp. Mat. Nauk 70, No. 4, 77--142 (2015)], \textit{N.-P. Chung} and \textit{H. Li} [Invent. Math. 199, No. 3, 805--858 (2015; Zbl 1320.37009)], and a topological result by \textit{T. Meyerovitch} [Ergodic Theory Dyn. Syst. 39, No. 9, 2570--2591 (2019; Zbl 1431.37030)]
that relates entropy to the set of asymptotic pairs. In order to do this we introduce a series of Markovian properties and, under the assumption that they are satisfied, we prove several results that relate topological entropy and asymptotic pairs (the homoclinic group in the algebraic case). As new applications of our method, we give a characterization of the homoclinic group of any finitely presented expansive algebraic action of (1) any elementary amenable group with an upper bound on the orders of finite subgroups or (2) any left orderable amenable group, using the language of independence entropy pairs.Stability in a grouphttps://zbmath.org/1491.370122022-09-13T20:28:31.338867Z"Conant, Gabriel"https://zbmath.org/authors/?q=ai:conant.gabrielSummary: We develop local stable group theory directly from topological dynamics, and extend the main tools in this subject to the setting of stability ``in a model.'' Specifically, given a group \(G\), we analyze the structure of sets \(A \subseteq G\) such that the bipartite relation \(xy\in A\) omits infinite half-graphs. Our proofs rely on the characterization of model-theoretic stability via Grothendieck's ``double-limit'' theorem (as shown by \textit{I. Ben Yaacov} [Bull. Symb. Log. 20, No. 4, 491--496 (2014; Zbl 1345.03058)]), and the work of \textit{R. Ellis} and \textit{M. Nerurkar} [Trans. Am. Math. Soc. 313, No. 1, 103--119 (1989; Zbl 0674.54026)]
on weakly almost periodic \(G\)-flows.Katok-Hasselblatt-kinematic expansive flowshttps://zbmath.org/1491.370132022-09-13T20:28:31.338867Z"Huynh, Hien Minh"https://zbmath.org/authors/?q=ai:huynh.hien-minhSummary: In this paper we introduce a new notion of expansive flows, which is the combination of expansivity in the sense of \textit{A. Katok} and \textit{B. Hasselblatt} [Introduction to the modern theory of dynamical systems. Cambridge: Cambridge Univ. Press (1995; Zbl 0878.58020)]
and kinematic expansivity, named KH-kinematic expansivity. We present new properties of several variations of expansivity. A new hierarchy of expansive flows is given.Preservation of expansivity in hyperspace dynamical systemshttps://zbmath.org/1491.370142022-09-13T20:28:31.338867Z"Koo, Namjip"https://zbmath.org/authors/?q=ai:koo.namjip"Lee, Hyunhee"https://zbmath.org/authors/?q=ai:lee.hyunheeSummary: In this paper we study the preservation of various notions of expansivity in discrete dynamical systems and the induced map for \(n\)-fold symmetric products and hyperspaces. Then we give a characterization of a compact metric space admitting hyper \(N\)-expansive homeomorphisms via the topological dimension. More precisely, we show that \(C^0\)-generically, any homeomorphism on a compact manifold is not hyper \(N\)-expansive for any \(N\in \mathbb{N} \). Also we give some examples to illustrate our results.Expansive flows on uniform spaceshttps://zbmath.org/1491.370152022-09-13T20:28:31.338867Z"Ku, Se-Hyun"https://zbmath.org/authors/?q=ai:ku.se-hyunSummary: In this paper we study several dynamical properties on uniform spaces. We define expansive flows on uniform spaces and provide some equivalent ways of defining expansivity. We also define the concept of expansive measures for flows on uniform spaces. We prove for flows on compact uniform spaces that every expansive measure vanishes along the orbits and has no singularities in the support. We also prove that every expansive measure for flows on uniform spaces is aperiodic and is expansive with respect to time-\( T \) map. Furthermore we show that every expansive measure for flows on compact uniform spaces maintains expansive under topological equivalence.Coboundaries of commuting Borel automorphismshttps://zbmath.org/1491.370162022-09-13T20:28:31.338867Z"Sanadhya, Shrey"https://zbmath.org/authors/?q=ai:sanadhya.shreySummary: We show that if \(S, T\) are two commuting automorphisms of a standard Borel space such that they generate a free Borel \(\mathbb{Z}^2\)-action then \(S\) and \(T\) do not have same sets of real valued bounded coboundaries. We also prove a weaker form of Rokhlin Lemma for Borel \(\mathbb{Z}^d\)-actions.Maximal invariance of topologically almost continuous iterative dynamicshttps://zbmath.org/1491.370212022-09-13T20:28:31.338867Z"Kahng, Byungik"https://zbmath.org/authors/?q=ai:kahng.byungikSummary: It is known that the maximal invariant set of a continuous iterative dynamical system in a compact Hausdorff space is equal to the intersection of its forward image sets, which we will call the \textit{first minimal image set}. In this article, we investigate the corresponding relation for a class of discontinuous self maps that are on the verge of continuity, or \textit{topologically almost continuous endomorphisms}. We prove that the iterative dynamics of a topologically almost continuous endomorphisms yields a chain of minimal image sets that attains a unique transfinite \textit{length}, which we call the \textit{maximal invariance order}, as it stabilizes itself at the maximal invariant set. We prove the converse, too. Given ordinal number \(\xi \), there exists a topologically almost continuous endomorphism \(f\) on a compact Hausdorff space \(X\) with the maximal invariance order \(\xi \). We also discuss some further results regarding the maximal invariance order as more layers of topological restrictions are added.Locating Ruelle-Pollicott resonanceshttps://zbmath.org/1491.370282022-09-13T20:28:31.338867Z"Butterley, Oliver"https://zbmath.org/authors/?q=ai:butterley.oliver"Kiamari, Niloofar"https://zbmath.org/authors/?q=ai:kiamari.niloofar"Liverani, Carlangelo"https://zbmath.org/authors/?q=ai:liverani.carlangeloThis main purpose of this paper is the study of the spectrum of transfer operators associated to various dynamical systems. Historically, the research on this topic mainly focused on studying the peripheral spectrum and on establishing the spectral gap. Here the authors want to understand the point spectrum much deeper. In particular, they propose a unitary approach based on a general philosophy: to study the commutator between some type of differentiation and the transfer operator. They consider various settings where new information can be obtained along the proposed path. These settings include affine expanding Markov maps, uniformly expanding Markov maps, nonuniformly expanding or simply monotone maps and hyperbolic diffeomorphisms.
Reviewer: Jialun Li (Zürich)Existence and uniqueness of global Koopman eigenfunctions for stable fixed points and periodic orbitshttps://zbmath.org/1491.370292022-09-13T20:28:31.338867Z"Kvalheim, Matthew D."https://zbmath.org/authors/?q=ai:kvalheim.matthew-d"Revzen, Shai"https://zbmath.org/authors/?q=ai:revzen.shaiSummary: We consider \(C^1\) dynamical systems having an attracting hyperbolic fixed point or periodic orbit and prove existence and uniqueness results for \(C^k\) (actually \(C_{\mathrm{loc}}^{k, \alpha}\)) linearizing semiconjugacies -- of which Koopman eigenfunctions are a special case -- defined on the entire basin of attraction. Our main results both generalize and sharpen Sternberg's \(C^k\) linearization theorem for hyperbolic sinks, and in particular our corollaries include uniqueness statements for Sternberg linearizations and Floquet normal forms. Using our main results we also prove new existence and uniqueness statements for \(C^k\) Koopman eigenfunctions, including a complete classification of \(C^\infty\) eigenfunctions assuming a \(C^\infty\) dynamical system with semisimple and nonresonant linearization. We give an intrinsic definition of ``principal Koopman eigenfunctions'' which generalizes the definition of Mohr and Mezić for linear systems, and which includes the notions of ``isostables'' and ``isostable coordinates'' appearing in work by Ermentrout, Mauroy, Mezić, Moehlis, Wilson, and others. Our main results yield existence and uniqueness theorems for the principal eigenfunctions and isostable coordinates and also show, e.g., that the (a priori non-unique) ``pullback algebra'' defined in [\textit{R. Mohr} and \textit{I. Mezić}, Koopman principle eigenfunctions and linearization of diffeomorphisms, Preprint, \url{arXiv:1611.01209} (2016)] is unique under certain conditions. We also discuss the limit used to define the ``faster'' isostable coordinates in [\textit{D. Wilson} and \textit{B. Ermentrout}, J. Math. Biol. 76, No. 1--2, 37--66 (2018; Zbl 1392.92007); \textit{B. Monga} et al., Biol. Cybern. 113, No. 1--2, 11--46 (2019; Zbl 1411.92122)] in light of our main results.Multi-directional and saturated chaotic attractors with many scrolls for fractional dynamical systemshttps://zbmath.org/1491.370302022-09-13T20:28:31.338867Z"Goufo, Emile Franc Doungmo"https://zbmath.org/authors/?q=ai:doungmo-goufo.emile-francSummary: Chaotic dynamical attractors are themselves very captivating in Science and Engineering, but systems with multi-dimensional and saturated chaotic attractors with many scrolls are even more fascinating for their multi-directional features. In this paper, the dynamics of a Caputo three-dimensional saturated system is successfully investigated by means of numerical techniques. The continuity property for the saturated function series involved in the model preludes suitable analytical conditions for existence and stability of the solution to the model. The Haar wavelet numerical method is applied to the saturated system and its convergence is shown thanks to error analysis. Therefore, the performance of numerical approximations clearly reveals that the Caputo model and its general initial conditions display some chaotic features with many directions. Such a chaos shows attractors with many scrolls and many directions. Then, the saturated Caputo system is indeed chaotic in the standard integer case (Caputo derivative order \(\alpha = 1)\) and this chaos remains in the fractional case \((\alpha = 0.9)\). Moreover the dynamics of the system change depending on the parameter \(\alpha\), leading to an important observation that the saturated system is likely to be regulated or controlled via such a parameter.On the equidistribution of unstable curves for pseudo-Anosov diffeomorphisms of compact surfaceshttps://zbmath.org/1491.370312022-09-13T20:28:31.338867Z"Forni, Giovanni"https://zbmath.org/authors/?q=ai:forni.giovanniSummary: We prove that the asymptotics of ergodic integrals along an invariant foliation of a toral Anosov diffeomorphism, or of a pseudo-Anosov diffeomorphism on a compact orientable surface of higher genus, is determined (up to a logarithmic error) by the action of the diffeomorphism on the cohomology of the surface. As a consequence of our argument and of the results of \textit{P. Giulietti} and \textit{C. Liverani} [J. Eur. Math. Soc. (JEMS) 21, No. 9, 2793--2858 (2019; Zbl 1425.37005)]
on horospherical averages, toral Anosov diffeomorphisms have no Ruelle resonances in the open interval \((1, e^{h_{\text{top}}})\) .Thermodynamic formalism for transient dynamics on the real linehttps://zbmath.org/1491.370322022-09-13T20:28:31.338867Z"Gröger, M."https://zbmath.org/authors/?q=ai:groger.maik"Jaerisch, J."https://zbmath.org/authors/?q=ai:jaerisch.johannes"Kesseböhmer, M."https://zbmath.org/authors/?q=ai:kessebohmer.marcThe authors consider skew-periodic \(\mathbb{Z}\)-extensions of expanding interval maps. More specifically, they introduce a new thermodynamic formalism to study the dimension gap, which is the difference between the Hausdorff dimension of the domain of the system and the Hausdorff dimension of the uniformly recurrent set. By defining the fibre-induced pressure, which is a generalization of Gurevich pressure, the authors prove that a system has a dimension gap if and only if the drift is not zero. The transient dynamics is further studied through the Hausdorff dimension of level sets, called uniformly \(\alpha\)-escaping sets.
Reviewer: Steve Pederson (Atlanta)Narrow equidistribution and counting of closed geodesics on noncompact manifoldshttps://zbmath.org/1491.370342022-09-13T20:28:31.338867Z"Schapira, Barbara"https://zbmath.org/authors/?q=ai:schapira.barbara"Tapie, Samuel"https://zbmath.org/authors/?q=ai:tapie.samuelSummary: We prove the equidistribution of (weighted) periodic orbits for the geodesic flow on noncompact negatively curved manifolds toward equilibrium states in the narrow topology, i.e. in the dual of bounded continuous functions. We deduce an exact asymptotic counting for periodic orbits (weighted or not), which was previously known only for geometrically finite manifolds.Combinatorial vs. classical dynamics: recurrencehttps://zbmath.org/1491.370412022-09-13T20:28:31.338867Z"Mrozek, Marian"https://zbmath.org/authors/?q=ai:mrozek.marian"Srzednicki, Roman"https://zbmath.org/authors/?q=ai:srzednicki.roman"Thorpe, Justin"https://zbmath.org/authors/?q=ai:thorpe.justin"Wanner, Thomas"https://zbmath.org/authors/?q=ai:wanner.thomasSummary: Establishing the existence of periodic orbits is one of the crucial and most intricate topics in the study of dynamical systems, and over the years, many methods have been developed to this end. On the other hand, finding closed orbits in discrete contexts, such as graph theory or in the recently developed field of combinatorial dynamics, is straightforward and computationally feasible. In this paper, we present an approach to study classical dynamical systems as given by semiflows or flows using techniques from combinatorial topological dynamics. More precisely, we present a general existence theorem for periodic orbits of semiflows which is based on suitable phase space decompositions, and indicate how combinatorial techniques can be used to satisfy the necessary assumptions. In this way, one can obtain computer-assisted proofs for the existence of periodic orbits and even certain chaotic behavior.Pointwise periodic maps with quantized first integralshttps://zbmath.org/1491.370432022-09-13T20:28:31.338867Z"Cima, Anna"https://zbmath.org/authors/?q=ai:cima.anna"Gasull, Armengol"https://zbmath.org/authors/?q=ai:gasull.armengol"Mañosa, Víctor"https://zbmath.org/authors/?q=ai:manosa.victor"Mañosas, Francesc"https://zbmath.org/authors/?q=ai:manosas.francescSummary: We describe the global dynamics of some pointwise periodic piecewise linear maps in the plane that exhibit interesting dynamic features. For each of these maps we find a first integral. For these integrals the set of values are discrete, thus quantized. Furthermore, the level sets are bounded sets whose interior is formed by a finite number of open tiles of certain regular or uniform tessellations. The action of the maps on each invariant set of tiles is described geometrically.Symmetry \& critical points for a model shallow neural networkhttps://zbmath.org/1491.370492022-09-13T20:28:31.338867Z"Arjevani, Yossi"https://zbmath.org/authors/?q=ai:arjevani.yossi"Field, Michael"https://zbmath.org/authors/?q=ai:field.michael-jSummary: Using methods based on the analysis of real analytic functions, symmetry and equivariant bifurcation theory, we obtain sharp results on families of critical points of spurious minima that occur in optimization problems associated with fitting two-layer ReLU networks with \(k\) hidden neurons. The main mathematical result proved is to obtain power series representations of families of critical points of spurious minima in terms of \(1/\sqrt{k}\) (coefficients independent of \(k\)). We also give a path based formulation that naturally connects the critical points with critical points of an associated linear, but highly singular, optimization problem. These critical points closely approximate the critical points in the original problem.
The mathematical theory is used to derive results on the original problem in neural nets. For example, precise estimates for several quantities that show that not all spurious minima are alike. In particular, we show that while the loss function at certain types of spurious minima decays to zero like \(k^{-1}\), in other cases the loss converges to a strictly positive constant.Bifurcations, symmetries and the notion of fixed subspacehttps://zbmath.org/1491.370502022-09-13T20:28:31.338867Z"Cicogna, Giampaolo"https://zbmath.org/authors/?q=ai:cicogna.giampaoloThe paper contains an exposition of some basic results about the bifurcation analysis of \[ F(\lambda,x )=0, \] where \(F:\mathbb R \times V \to V\) is a smooth equivariant map and \(V\) is a representation of a finite or compact Lie group \(G\). A Lyapunov-Schmidt reduction is presented and the reduction to the fixed point subspace of some subgroup \(H\subseteq G\) is achieved. Then some results are described, such as an equivariant bifurcation lemma, a Krasnosielskii-type bifurcation theorem, the bifurcation by change of the Brouwer degree and a principle of exchange of stability. Simple examples are presented in order to illustrate various applications.
Reviewer: Zdzisław Dzedzej (Gdańsk)Second-order fast-slow dynamics of non-ergodic Hamiltonian systems: thermodynamic interpretation and simulationhttps://zbmath.org/1491.370542022-09-13T20:28:31.338867Z"Klar, Matthias"https://zbmath.org/authors/?q=ai:klar.matthias"Matthies, Karsten"https://zbmath.org/authors/?q=ai:matthies.karsten"Reina, Celia"https://zbmath.org/authors/?q=ai:reina.celia"Zimmer, Johannes"https://zbmath.org/authors/?q=ai:zimmer.johannesSummary: A class of fast-slow Hamiltonian systems with potential \(U_\varepsilon\) describing the interaction of non-ergodic fast and slow degrees of freedom is studied. The parameter \(\varepsilon\) indicates the typical timescale ratio of the fast and slow degrees of freedom. It is known that the Hamiltonian system converges for \(\varepsilon \to 0\) to a homogenised Hamiltonian system. We study the situation where \(\varepsilon\) is small but positive.
First, we rigorously derive the second-order corrections to the homogenised (slow) degrees of freedom. They can be decomposed into explicitly given terms that oscillate rapidly around zero and terms that trace the average motion of the corrections, which are given as the solution to an inhomogeneous linear system of differential equations.
Then, we analyse the energy of the fast degrees of freedom expanded to second-order from a thermodynamic point of view. In particular, we define and expand to second-order a temperature, an entropy and external forces and show that they satisfy to leading-order, as well as on average to second-order, thermodynamic energy relations akin to the first and second law of thermodynamics.
Finally, we analyse for a specific fast-slow Hamiltonian system the second-order asymptotic expansion of the slow degrees of freedom from a numerical point of view. Their approximation quality for short and long time frames and their total computation time are compared with those of the solution to the original fast-slow Hamiltonian system of similar accuracy.Topological dynamics of volume-preserving maps without an equatorial heteroclinic curvehttps://zbmath.org/1491.370722022-09-13T20:28:31.338867Z"Arenson, Joshua G."https://zbmath.org/authors/?q=ai:arenson.joshua-g"Mitchell, Kevin A."https://zbmath.org/authors/?q=ai:mitchell.kevin-aSummary: Understanding the topological structure of phase space for dynamical systems in higher dimensions is critical for numerous applications, including the computation of chemical reaction rates and transport of objects in the solar system. Many topological techniques have been developed to study maps of two-dimensional (2D) phase spaces, but extending these techniques to higher dimensions is often a major challenge or even impossible. Previously, one such technique, homotopic lobe dynamics (HLD), was generalized to analyze the stable and unstable manifolds of hyperbolic fixed points for volume-preserving maps in three dimensions. This prior work assumed the existence of an equatorial heteroclinic intersection curve, which was the natural generalization of the 2D case. The present work extends the previous analysis to the case where no such equatorial curve exists, but where intersection curves, connecting fixed points may exist. In order to extend HLD to this case, we shift our perspective from the invariant manifolds of the fixed points to the invariant manifolds of the invariant circle formed by the fixed-point-to-fixed-point intersections. The output of the HLD technique is a symbolic description of the minimal underlying topology of the invariant manifolds. We demonstrate this approach through a series of examples.Reactive islands for three degrees-of-freedom Hamiltonian systemshttps://zbmath.org/1491.370732022-09-13T20:28:31.338867Z"Krajňák, Vladimír"https://zbmath.org/authors/?q=ai:krajnak.vladimir"García-Garrido, Víctor J."https://zbmath.org/authors/?q=ai:garcia-garrido.victor-j"Wiggins, Stephen"https://zbmath.org/authors/?q=ai:wiggins.stephenSummary: We develop the geometrical, analytical, and computational framework for reactive island theory for three degrees-of-freedom time-independent Hamiltonian systems. In this setting, the dynamics occurs in a 5-dimensional energy surface in phase space and is governed by four-dimensional stable and unstable manifolds of a three-dimensional normally hyperbolic invariant sphere. The stable and unstable manifolds have the geometrical structure of spherinders and we provide the means to investigate the ways in which these spherinders and their intersections determine the dynamical evolution of trajectories. This geometrical picture is realized through the computational technique of Lagrangian descriptors. In a set of trajectories, Lagrangian descriptors allow us to identify the ones closest to a stable or unstable manifold. Using an approximation of the manifold on a surface of section we are able to calculate the flux between two regions of the energy surface.Deep learning of conjugate mappingshttps://zbmath.org/1491.370742022-09-13T20:28:31.338867Z"Bramburger, Jason J."https://zbmath.org/authors/?q=ai:bramburger.jason-j"Brunton, Steven L."https://zbmath.org/authors/?q=ai:brunton.steven-l"Nathan Kutz, J."https://zbmath.org/authors/?q=ai:kutz.j-nathanSummary: Despite many of the most common chaotic dynamical systems being continuous in time, it is through discrete time mappings that much of the understanding of chaos is formed. Henri Poincaré first made this connection by tracking consecutive iterations of the continuous flow with a lower-dimensional, transverse subspace. The mapping that iterates the dynamics through consecutive intersections of the flow with the subspace is now referred to as a Poincaré map, and it is the primary method available for interpreting and classifying chaotic dynamics. Unfortunately, in all but the simplest systems, an explicit form for such a mapping remains outstanding. This work proposes a method for obtaining explicit Poincaré mappings by using deep learning to construct an invertible coordinate transformation into a conjugate representation where the dynamics are governed by a relatively simple chaotic mapping. The invertible change of variable is based on an autoencoder, which allows for dimensionality reduction, and has the advantage of classifying chaotic systems using the equivalence relation of topological conjugacies. Indeed, the enforcement of topological conjugacies is the critical neural network regularization for learning the coordinate and dynamics pairing. We provide expository applications of the method to low-dimensional systems such as the Rössler and Lorenz systems, while also demonstrating the utility of the method on infinite-dimensional systems, such as the Kuramoto-Sivashinsky equation.Observer-invariant time derivatives on moving surfaceshttps://zbmath.org/1491.530162022-09-13T20:28:31.338867Z"Nitschke, Ingo"https://zbmath.org/authors/?q=ai:nitschke.ingo"Voigt, Axel"https://zbmath.org/authors/?q=ai:voigt.axelObserver-invariance is regarded as a minimum requirement for an appropriate definition of time derivatives. We derive various time derivatives systematically from a spacetime setting, where observer-invariance is a special case of a covariance principle and covered by Ricci-calculus. The analysis is considered for tangential $n$-tensor fields on moving surfaces and provides formulations which are applicable for numerical computations. For various special cases, e.g., vector fields $(n = 1)$ and symmetric and trace-less tensor fields $(n = 2)$ we compare material and convected derivatives and demonstrate the different underlying physics.
Reviewer: Maria Aparecida Soares Ruas (São Carlos)What does a vector field know about volume?https://zbmath.org/1491.570272022-09-13T20:28:31.338867Z"Geiges, Hansjörg"https://zbmath.org/authors/?q=ai:geiges.hansjorgA vector field \(X\) without zeros on a manifold \(M\) is called geodesible if there is a Riemannian metric on \(M\) such that the flow lines of \(X\) are unit speed geodesics -- equivalently, if there exists a one-form \(\alpha\) invariant under the flow of \(X\) such that \(\alpha(X) = 1\). It was observed by \textit{C. B. Croke} and \textit{B. Kleiner} [J. Differ. Geom. 39, No. 3, 659--680 (1994; Zbl 0807.53035)] that if \(M^{2n+1}\) is closed and orientable, the integral \(\int_M \alpha\wedge (d\alpha)^n\) is independent of the form \(\alpha\); in the paper at hand this number is hence called the volume of \(X\). In particular, the volume of a closed contact manifold is determined by the Reeb vector field alone. This raised the question, posed to the author by Claude Viterbo, whether there exist nondiffeomorphic contact forms on the same manifold with the same Reeb vector field. This paper contains a variety of results motivated by this question (which is answered in the affirmative by giving explicit examples as Boothby-Wang bundles, using a construction of \textit{D. McDuff} [Invent. Math. 89, 13--36 (1987; Zbl 0625.53040)] of cohomologous, non-diffeomorphic symplectic forms) as well as the above notion of volume. Amongst others, the question whether one can compute the volume of a geodesible vector field explicitly from the vector field alone is answered positively in some special cases, such as Seifert fibered \(3\)-manifolds. Moreover, theorems of Gauß-Bonnet and Poincaré-Hopf are proved for \(2\)-dimensional orbifolds.
Reviewer: Oliver Goertsches (Marburg)Invariant Gibbs measures for the three-dimensional wave equation with a Hartree nonlinearity. I: measureshttps://zbmath.org/1491.600952022-09-13T20:28:31.338867Z"Bringmann, Bjoern"https://zbmath.org/authors/?q=ai:bringmann.bjornThe author of this study shows the Gibbs measure invariance for a three-dimensional wave equation with a Hartree nonlinearity. The singularity of the Gibbs measure with respect to the Gaussian free field is the primary innovation. In both measure-theoretic and dynamical aspects, the singularity has various repercussions. In this study, the author primarily creates and investigates the Gibbs measure. The technique used in this research is based on previous work by Barashkov and Gubinelli for the \(\Phi^4_3\)-model. In addition, the author creates new techniques to cope with the Hartree interaction's nonlocality. Depending on the regularity of the interaction potential, the exact threshold between singularity and absolute continuity of the Gibbs measure are also calculated.
Reviewer: Udhayakumar Ramalingam (Vellore)Reflection of a self-propelling rigid disk from a boundaryhttps://zbmath.org/1491.700142022-09-13T20:28:31.338867Z"Ei, Shin-Ichiro"https://zbmath.org/authors/?q=ai:ei.shin-ichiro"Mimura, Masayasu"https://zbmath.org/authors/?q=ai:mimura.masayasu"Miyaji, Tomoyuki"https://zbmath.org/authors/?q=ai:miyaji.tomoyukiSummary: A system of ordinary differential equations that describes the motion of a self-propelling rigid disk is studied. In this system, the disk moves along a straight-line and reflects from a boundary. Interestingly, numerical simulation shows that the angle of reflection is greater than that of incidence. The purpose of this study is to present a mathematical proof for this attractive phenomenon. Moreover, the reflection law is numerically investigated. Finally, existence and asymptotic stability of a square-shaped closed orbit for billiards in square table with inelastic reflection law are discussed.On the slope of the curvature power spectrum in non-attractor inflationhttps://zbmath.org/1491.830532022-09-13T20:28:31.338867Z"Özsoy, Ogan"https://zbmath.org/authors/?q=ai:ozsoy.ogan"Tasinato, Gianmassimo"https://zbmath.org/authors/?q=ai:tasinato.gianmassimo(no abstract)Control of homoclinic bifurcation in two-dimensional dynamical systems by a feedback law based on \(L^p\) spaceshttps://zbmath.org/1491.930422022-09-13T20:28:31.338867Z"Piccirillo, Vinícius"https://zbmath.org/authors/?q=ai:piccirillo.viniciusSummary: This paper proposes a homoclinic bifurcation control method in a planar system of nonlinear differential equations \((\dot{\mathbf{x}}=f(\mathbf{x}),\mathbf{x}\in\mathbb{R}^2,f:U\subset\mathbb{R}^2\longrightarrow\mathbb{R}^2)\). The feedback control law is formulated within the framework of Melnikov theory and \(L^p\) spaces, and it will be called as \(L^p\) control. Here it is proved that if \(\gamma_0(t)\) is the homoclinic orbit of the planar system, then \(f(\gamma_0(t))\in L^q\) space \((1\leq q\leq\infty)\). To avoid the transverse intersection of the stable and unstable manifolds of the hilltop saddle, a lot of control laws \((u(\mathbf{x})\in L^p\text{ space})\) have been developed, where each of them can be found by choosing one particular \(L^q\) space to \(f(\gamma_0(t))\), such that \(p\) and \(q\) are conjugate exponents, that is, \(\frac{1}{p}+\frac{1}{q}=1\). Furthermore, a procedure to find the control gains is presented. Numerical results show the efficiency of the proposed method in avoiding the homoclinic bifurcation of a classical Duffing system.