In this work, we investigate the dynamics of a general non-autonomous system generated by a commutative family of homeomorphisms. In particular, we investigate properties such as periodicity, equicontinuity, minimality and transitivity for a general non-autonomous dynamical system. In \cite{sk2}, the authors derive necessary and sufficient conditions for a system to be minimal. We claim the result to be false and provide an example in support of our claim. Further, we correct the result to derive necessary and sufficient conditions for a non-autonomous system to be minimal. We prove that for an equicontinuous flow generated by a commutative family, while the system need not exhibit almost periodic points, if $x$ is almost periodic then every point in $\overline{\mathcal{O}_H(x)}$ is almost periodic. We further prove that in such a case, the set $\overline{\mathcal{O}_H(x)}$ is uniformly almost periodic and hence provide an analogous extension to a result known for the autonomous systems. We prove that a system generated by a commutative family is transitive if and only if it exhibits a point with dense orbit. We also prove that any minimal system generated by commutative family is either equicontinuous or has a dense set of sensitive points.
Answering an open question affirmatively it is shown that every ergodic invariant measure of a mean equicontinuous (i.e. mean-L-stable) system has discrete spectrum. Dichotomy results related to mean equicontinuity and mean sensitivity are obtained when a dynamical system is transitive or minimal. Localizing the notion of mean equicontinuity, notions of almost mean equicontinuity and almost Banach mean equicontinuity are introduced. It turns out that a system with the former property may have positive entropy and meanwhile a system with the later property must have zero entropy.
We discuss topological equicontinuity and even continuity in dynamical systems. In doing so we provide a classification of topologically transitive dynamical systems in terms of equicontinuity pairs, give a generalisation of the Auslander-Yorke Dichotomy for minimal systems and show there exists a transitive system with an even continuity pair but no equicontinuity point. We define what it means for a system to be eventually sensitive; we give a dichotomy for transitive dynamical systems in relation to eventual sensitivity. Along the way we define a property called splitting and discuss its re…
For nonautonomous linear difference equations, we introduce the notion of the so-called nonuniform dichotomy spectrum and prove a spectral theorem. Moreover, we introduce the notion of weak kinematical similarity and prove a reducibility result by the spectral theorem.
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…
Two different notions of μ-equicontinuity that apply to topological dynamical systems and probability measures were studied by Gilman (1987) and Huang-Lu-Ye (2011). One was used to classify measure preserving topological dynamical systems and the other for cellular automata. We show that if the probability space satisfies Lebesgue's density theorem and Vitali's covering theorem (for example a Cantor set or a subset of R^{d}) then both properties are equivalent. To do this we characterize when a function is Lusin measurable.
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.
In [Adv. Math., 267(2014), 277-306], Cheskidov and Lu develop a new framework of the evolutionary system that deals directly with the notion of a uniform global attractor due to Haraux, and by which a trajectory attractor is able to be defined for the original system under consideration. The notion of a trajectory attractor was previously established for a system without uniqueness by considering a family of auxiliary systems including the original one. In this paper, we further prove the existence of a notion of a strongly compact strong trajectory attractor if the system is asymptotically co…
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.
We prove that a linear nonautonomous differential system with nonuniform hyperbolicity on the half line can be expressed as diagonal system with a perturbation which is small enough. Moreover we show that the diagonal terms are contained in the nonuniform exponential dichotomy spectrum. For this purpose we introduce the concepts of \textit{nonuniform almost reducibility} and \textit{nonuniform contractibility} which are generalization of this notions originally defined in a uniform context.
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 or…
In this paper we generalize the method of Lagrangian descriptors to two dimensional, area preserving, autonomous and nonautonomous discrete time dynamical systems. We consider four generic model problems--a hyperbolic saddle point for a linear, area-preserving autonomous map, a hyperbolic saddle point for a nonlinear, area-preserving autonomous map, a hyperbolic saddle point for linear, area-preserving nonautonomous map, and a hyperbolic saddle point for nonlinear, area-preserving nonautonomous map. The discrete time setting allows us to evaluate the expression for the Lagrangian descriptors e…
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 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 discuss the nonlinear phenomena of irreversible tipping for non-autonomous systems where time-varying inputs correspond to a smooth "parameter shift" from one asymptotic value to another. We express tipping in terms of pullback attraction and present some results on how nontrivial dynamics for non-autonomous systems can be deduced from analysis of the bifurcation diagram for an associated autonomous system where parameters are fixed. In particular, we show that there is a unique local pullback point attractor associated with each linearly stable equilibrium for the past limit. If there is a…
We show that an $R^d$-topological dynamical system equipped with an invariant ergodic measure has discrete spectrum if and only it is $μ$-mean equicontinuous (proven for $Z^d$ before). In order to do this we introduce mean equicontinuity and mean sensitivity with respect to a function. We study this notion in the topological and measure theoretic setting. In the measure theoretic case we characterize almost periodic functions and in the topological case we show that weakly almost periodic functions are mean equicontinuous (the converse does not hold).
For a general one-sided nonautonomous dynamics defined by a sequence of linear operators, we consider the notion of a polynomial dichotomy with respect to a sequence of norms and we characterize it completely in terms of the admissibility of bounded solutions. As a nontrivial application, we establish the robustness of the notion of a nonuniform polynomial dichotomy.
The present work is concerned with the eqiucontinuity and sensitivity of iterated function systems (IFSs). Here, we consider more general case of IFSs, i.e. the IFSs generated by a family of relations. We generalize the concepts of transitivity, sensitivity and equicontinuity to these kinds of systems. This note investigates the relationships between these concepts. Then, several sufficient conditions for sensitivity of IFSs are presented. We introduce the notion of weak topologically exact for IFSs generated by a family of relations. It is proved that non-minimal weak topologically exact IFSs…
In this paper, we investigate the dynamics on the hyperspace induced by a non-autonomous dynamical system $(X,\mathbb{F})$, where the non-autonomous system is generated by a sequence $(f_n)$ of continuous self maps on $X$. We relate the dynamical behavior of the induced system on the hyperspace with the dynamical behavior of the original system $(X,\mathbb{F})$. We derive conditions under which the dynamical behavior of the non-autonomous system extends to its induced counterpart(and vice-versa). In the process, we discuss properties like transitivity, weak mixing, topological mixing, topologi…
