We introduce the notion of angular values for deterministic linear difference equations and random linear cocycles. We measure the principal angles between subspaces of fixed dimension as they evolve under nonautonomous or random linear dynamics. The focus is on long-term averages of these principal angles, which we call angular values: we demonstrate relationships between different types of angular values and prove their existence for random dynamical systems. For one-dimensional subspaces in two-dimensional systems our angular values agree with the classical theory of rotation numbers for orientation-preserving circle homeomorphisms if the matrix has positive determinant and does not rotate vectors by more than $\fracπ{2}$. Because our notion of angular values ignores orientation by looking at subspaces rather than vectors, our results apply to dynamical systems of any dimension and to subspaces of arbitrary dimension. The second part of the paper delves deeper into the theory of the autonomous case. We explore the relation to (generalized) eigenspaces, provide some explicit formulas for angular values, and set up a general numerical algorithm for computing angular values via Schur decompositions.
Let $(X,f_{1,\infty})$ be a nonautonomous dynamical system. In this paper we summarize known definitions of periodic points for general nonautonomous dynamical systems and propose a new, very natural, definition of asymptotic periodicity. Moreover, this definition is resistant to changes of a beginning of the sequence generating the nonautonomous system. We show the relations among these definitions and discuss their properties. We prove that for pointwise convergent nonautonomous systems topological transitivity together with dense set of asymptotically periodic points imply sensitivity. We a…
The goal of this article is to study how combinatorial equivalence implies topological conjugacy. For that, we introduce the concept of kneading sequences for nonautonomous discrete dynamical systems and show that these sequences are a complete invariant for topological conjugacy classes.
We study chaotic properties of uniformly convergent nonautonomous dynamical systems. We show that, contrary to the autonomous systems on the compact interval, positivity of topological sequence entropy and occurrence of Li-Yorke chaos are not equivalent, more precisely, neither of the two possible implications is true.
We consider nonautonomous discrete dynamical systems $\{ f_n\}_{n\ge 1}$, where every $f_n$ is a surjective continuous map $[0,1]\to [0,1]$ such that $f_n$ converges uniformly to a map $f$. We show, among others, that if $f$ is chaotic in the sense of Li and Yorke then the nonautonomous system $ \{ f_n\}_{n\ge 1}$ is Li-Yorke chaotic as well, and that the same is true for distributional chaos. If $f$ has zero topological entropy then the nonautonomous system inherits its infinite $ω$-limit sets.
Decomposition of state spaces into dynamically different components is helpful for the understanding of dynamical behaviors of complex systems. A Conley type decomposition theorem is proved for nonautonomous dynamical systems defined on a non-compact but separable state space. Namely, the state space can be decomposed into a chain recurrent part and a gradient-like part. This result applies to both nonautonomous ordinary differential equations on Euclidean space (which is only locally compact), and nonautonomous partial differential equations on infinite dimensional function space (which is no…
Different notions of entropy play a fundamental role in the classical theory of dynamical systems. Unlike many other concepts used to analyze autonomous dynamics, both measure-theoretic and topological entropy can be extended quite naturally to discrete-time nonautonomous dynamical systems given in the process formulation. This paper provides an overview of the author's work on this subject. Also an example is presented that has not appeared before in the literature.
In this paper we study the problem of almost periodicity of solutions for dissipative differential equations (Bronshtein's conjecture). We give a positive answer to this conjecture for monotone almost periodic systems of differential/difference equations.
This work focuses on angular values of nonautonomous dynamical systems which have been introduced for general random and (non)autonomous dynamical systems in a previous publication [W.-J. Beyn, G. Froyland, and T. Hüls, SIAM J. Appl. Dyn. Syst., 21 (2022), pp. 1245--1286]. The angular value of dimension $s$ measures the maximal average rotation which an $s$-dimensional subspace of the phase space experiences through the dynamics of a discrete-time linear system. Our main results relate the notion of angular value to the well-known dichotomy (or Sacker--Sell) spectrum and its associated spectra…
We introduce the notion of metric entropy for a nonautonomous dynamical system given by a sequence of probability spaces and a sequence of measure-preserving maps between these spaces. This notion generalizes the classical concept of metric entropy established by Kolmogorov and Sinai, and is related via a variational inequality to the topological entropy of nonautonomous systems as defined by Kolyada, Misiurewicz and Snoha. Moreover, it shares several properties with the classical notion of metric entropy. In particular, invariance with respect to appropriately defined isomorphisms, a power ru…
In this article, we study the forward dynamical behavior of nonautonomous lattice systems. We first construct a family of sets $\{\mathcal{A}_\varepsilon(σ)\}_{σ\in Σ}$ in arbitrary small neighborhood of a global attractor of the skew-product flow generated by a general nonautonomous lattice system, which is forward invariant and uniformly forward attracts any bounded subset of the phase space. Moreover, under some suitable conditions, we further construct a family of sets $\{\mathcal{B}_\varepsilon(σ)\}_{σ\in Σ}$ such that it uniformly forward exponentially attracts bounded subsets of the pha…
The paper develops and studies a very general notion of dichotomy, referred to as "nonuniform $(h,k,μ,ν)$-dichotomy". The new notion contains as special cases most versions of dichotomy existing in the literature. The paper then provides corresponding new versions of robustness, Hartman-Grobman theorem, and stable manifold theorem for nonautonomous dynamical systems in Banach spaces in term of the nonuniform $(h,k,μ,ν)$-dichotomy.
This paper studies the dynamics of families of monotone nonautonomous neutral functional differential equations with nonautonomous operator, of great importance for their applications to the study of the long-term behavior of the trajectories of problems described by this kind of equations, such us compartmental systems and neural networks among many others. Precisely, more general admissible initial conditions are included in the study to show that the solutions are asymptotically of the same type as the coefficients of the neutral and non-neutral part.
In this paper, we study the Poisson stability (in particular, stationarity, periodicity, quasi-periodicity, Bohr almost periodicity, almost automorphy, recurrence in the sense of Birkhoff, Levitan almost periodicity, pseudo periodicity, almost recurrence in the sense of Bebutov, pseudo recurrence, Poisson stability) of motions for monotone nonautonomous dynamical systems and of solutions for some classes of monotone nonautonomous evolution equations (ODEs, FDEs and parabolic PDEs). As a byproduct, some of our results indicate that all the trajectories of monotone systems converge to the above …
We study nonautonomous discrete dynamical systems with randomly perturbed trajectories. We suppose that such a system is generated by a sequence of continuous maps which converges uniformly to a map $f$. We give conditions, under which a recurrent point of a (standard) autonomous discrete dynamical system generated by the limit function $f$ is also recurrent for the nonautonomous system with randomly perturbed trajectories. We also provide a necessary condition for a nonautonomous discrete dynamical system to be nonchaotic in the sense of Li and Yorke with respect to small random perturbations…
Let $\mathcal{M}(X)$ be the space of Borel probability measures on a compact metric space $X$ endowed with the weak$^\ast$-topology. In this paper, we prove that if the topological entropy of a nonautonomous dynamical system $(X,\{f_n\}_{n=1}^{+\infty})$ vanishes, then so does that of its induced system $(\mathcal{M}(X),\{f_n\}_{n=1}^{+\infty})$; moreover, once the topological entropy of $(X,\{f_n\}_{n=1}^{+\infty})$ is positive, that of its induced system $(\mathcal{M}(X),\{f_n\}_{n=1}^{+\infty})$ jumps to infinity. In contrast to Bowen's inequality, we construct a nonautonomous dynamical sys…
We describe a mathematical formalism and numerical algorithms for identifying and tracking slowly mixing objects in nonautonomous dynamical systems. In the autonomous setting, such objects are variously known as almost-invariant sets, metastable sets, persistent patterns, or strange eigenmodes, and have proved to be important in a variety of applications. In this current work, we explain how to extend existing autonomous approaches to the nonautonomous setting. We call the new time-dependent slowly mixing objects coherent sets as they represent regions of phase space that disperse very slowly …
We introduce a new concept of finite-time entropy which is a local version of the classical concept of metric entropy. Based on that, a finite-time version of Pesin's entropy formula and also an explicit formula of finite-time entropy for $2$-D systems are derived. We also discuss about how to apply the finite-time entropy field to detect special dynamical structures such as Lagrangian coherent structures.
A numerical technique used to solve boundary value problems is modified to find periodic steady-state solutions of nonautonomous dynamical systems. The technique uses a matrix representation of the time derivative obtained through trigonometric interpolation of a periodic function. Such a differentiation matrix yields exact values for the derivative of a trigonometric polynomial and therefore, can be used as the main ingredient of a numerical method to solve nonlinear dynamical systems. We apply this technique to obtain some limit cycles and bifurcation points of a sinusoidally driven pendulum…
In this paper we study the dynamics of a general non-autonomous dynamical system generated by a family of continuous self maps on a compact space $X$. We derive necessary and sufficient conditions for the system to exhibit complex dynamical behavior. In the process we discuss properties like transitivity, weakly mixing, topologically mixing, minimality, sensitivity, topological entropy and Li-Yorke chaoticity for the non-autonomous system. We also give examples to prove that the dynamical behavior of the non-autonomous system in general cannot be characterized in terms of the dynamical behavio…
Attractor-repeller decompositions of isolated invariant sets give rise to so-called connecting homomorphisms. These homomorphisms reveal information on the existence and structure of connecting trajectories of the underlying dynamical system. To give a meaningful generalization of this general principle to nonautonomous problems, the nonautonomous homology Conley index is expressed as a direct limit. Moreover, it is shown that a nontrivial connecting homomorphism implies, on the dynamical systems level, a sort of uniform connectedness of the attractor-repeller decomposition.
In this paper we advance the entropy theory of discrete nonautonomous dynamical systems that was initiated by Kolyada and Snoha in 1996. The first part of the paper is devoted to the measure-theoretic entropy theory of general topological systems. We derive several conditions guaranteeing that an initial probability measure, when pushed forward by the system, produces an invariant measure sequence whose entropy captures the dynamics on arbitrarily fine scales. In the second part of the paper, we apply the general theory to the nonstationary subshifts of finite type, introduced by Fisher and Ar…
The analysis of the long-term behavior of the mathematical model of a neural network constitutes a suitable framework to develop new tools for the dynamical description of nonautonomous state-dependent delay equations (SDDEs). The concept of global attractor is given, and some results which establish properties ensuring its existence and providing a description of its shape, are proved. Conditions for the exponential stability of the global attractor are also studied. Some properties of comparison of solutions constitute a key in the proof of the main results, introducing methods of monotonici…
In this paper, we focus on some properties, calculations and estimations of topological entropy for a nonautonomous dynamical system $(X,f_{0,\infty})$ generated by a sequence of continuous self-maps $f_{0,\infty}=\{f_n\}_{n=0}^{\infty}$ on a compact uniform space $X$. We obtain the relations of topological entropy among $(X, f_{0,\infty})$, its $k$-th product system and its $n$-th iteration system. We confirm that the entropy of $(X, f_{0,\infty})$ equals to that of $f_{0,\infty}$ restricted to its non-wandering set provided that $f_{0,\infty}$ is equi-continuous. We prove that the entropy of…
We present a notion of almost periodicity wich can be applied to random dynamical systems as well as almost periodic stochastic differential equations in Hilbert spaces (abstract stochastic partial differential equations). This concept allows for improvements of known results of almost periodicity in distribution, for general random processes and for solutions to stochastic differential equations.
This paper provides a dynamical frame to study non-autonomous parabolic partial differential equations with finite delay. Assuming monotonicity of the linearized semiflow, conditions for the existence of a continuous separation of type II over a minimal set are given. Then, practical criteria for the uniform or strict persistence of the systems above a minimal set are obtained.
We investigate the dynamical behavior of pull-back trajectories for nonautonomous stochastic feedback systems with multiplicative noise. We proved that there exists a random periodic solution of this system and all pull-back trajectories converge to this random periodic solution as time goes to infinitely almost surely. Our results can be applied to nonautonomous stochastic Goodwin negative feedback system, nonautonomous stochastic Othmer-Tyson positive feedback system and nonautonomous stochastic competitive systems etc.
In this article, we present data-driven reduced-order modeling for nonautonomous dynamical systems in multiscale media using Koopman operators. Different from the case of autonomous dynamical systems, the Koopman operator family of nonautonomous dynamical systems significantly depend on a time pair. In order to effectively estimate the time-dependent Koopman operators, a moving time window is used to decompose the snapshot data, and the extended dynamic mode decomposition method is applied to computing the Koopman operators in each local temporal domain. Many physical properties in multiscale …
We develop a general compactification framework to facilitate analysis of nonlinear nonautonomous ODEs where nonautonomous terms decay asymptotically. The strategy is to compactify the problem: the phase space is augmented with a bounded but open dimension and then extended at one or both ends by gluing in flow-invariant subspaces that carry autonomous dynamics of the limit systems from infinity. We derive the weakest decay conditions possible for the compactified system to be continuously differentiable on the extended phase space. This enables us to use equilibria and other compact invariant…
In this work we study permanence of hyperbolicity for autonomous differential equations under nonautonomous random/stochastic perturbations. For the linear case, we study robustness and existence of exponential dichotomies for nonautonomous random dynamical systems. Next, we establish a result on the persistence of hyperbolic equilibria for nonlinear differential equations. We show that for each nonautonomous random perturbation of an autonomous semilinear problem with a hyperbolic equilibrium there exists a bounded \textit{random hyperbolic solution} for the associated nonlinear nonautonomous…
Inspired by an example of Grebogi et al [1], we study a class of model systems which exhibit the full two-step scenario for the nonautonomous Hopf bifurcation, as proposed by Arnold [2]. The specific structure of these models allows a rigorous and thorough analysis of the bifurcation pattern. In particular, we show the existence of an invariant 'generalised torus' splitting off a previously stable central manifold after the second bifurcation point. The scenario is described in two different settings. First, we consider deterministically forced models, which can be treated as continuous skew p…
We study stochastic resonance in an over-damped approximation of the stochastic Duffing oscillator from a random dynamical systems point of view. We analyse this problem in the general framework of random dynamical systems with a nonautonomous forcing. We prove the existence of a unique global attracting random periodic orbit and a stationary periodic measure. We use the stationary periodic measure to define an indicator for the stochastic resonance.
When extending bifurcation theory of dynamical systems to nonautonomous problems, it is a central observation that hyperbolic equilibria persist as bounded entire solutions under small temporally varying perturbations. In this paper, we abandon the smallness assumption and aim to investigate the global structure of the entity of all such bounded entire solutions in the situation of nonautonomous difference equations. Our tools are global implicit function theorems based on an ambient degree theory for Fredholm operators due to Fitzpatrick, Pejsachowicz and Rabier. For this we yet have to restr…
We study the effect of external forcing on the saddle-node bifurcation pattern of interval maps. By replacing fixed points of unperturbed maps by invariant graphs, we obtain direct analogues to the classical result both for random forcing by measure-preserving dynamical systems and for deterministic forcing by homeomorphisms of compact metric spaces. Additional assumptions like ergodicity or minimality of the forcing process then yield further information about the dynamics. The main difference to the unforced situation is that at the critical bifurcation parameter, two alternatives exist. In …
