Non-autonomous Dynamical Systems Research Articles

Non-autonomous higher-order Moreau's sweeping process: Well-posedness, stability and Zeno trajectories

In this article, we study the higher-order Moreau's sweeping process introduced in [1], in the case where an exogenous time-varying functionu(·) is present in both the linear dynamics and in the unilateral constraints. First, we show that the well-posedness results (existence and uniqueness of solutions) obtained in [1] for the autonomous case, extend to the non-autonomous case whenu(·) is smooth and piece-wise analytic, after a suitable state transformation is done. Stability issues are discussed. The complexity of such non-smooth non-autonomous dynamical systems is illustrated in a particular case named the higher-order bouncing ball, where trajectories with accumulations of jumps are exhibited. Examples from mechanics and circuits illustrate some of the results. The link with complementarity dynamical systems and with switching differential-algebraic equations is made.

Relationships among some chaotic properties of non-autonomous discrete dynamical systems

This paper is concerned with relationships among some chaotic properties of non-autonomous discrete dynamical systems. Some relationships among weak mixing, topologically weak mixing, generic chaos, dense chaos, and sensitivity are investigated. In addition, some equivalent conditions of sensitivity are given and the relationships between sensitivity and Li–Yorke sensitivity are obtained. These results generalize some existing results of autonomous discrete systems, some of which relax the corresponding conditions.

Pathwise upper semi-continuity of random pullback attractors along the time axis

The pullback attractor of a non-autonomous random dynamical system is a time-indexed family of random sets, typically having the form {At(⋅)}t∈R with each At(⋅) a random set. This paper is concerned with the nature of such time-dependence. It is shown that the upper semi-continuity of the mapping t↦At(ω) for each ω fixed has an equivalence relationship with the uniform compactness of the local union ∪s∈IAs(ω), where I⊂R is compact. Applied to a semi-linear degenerate parabolic equation with additive noise and a wave equation with multiplicative noise we show that, in order to prove the above locally uniform compactness and upper semi-continuity, no additional conditions are required, in which sense the two properties appear to be general properties satisfied by a large number of real models.

Sensitive dependence for nonautonomous discrete dynamical systems

Given a nonautonomous discrete dynamical system (NDS) (X,f1,∞) we show that transitivity and density of periodic points do not imply sensitivity in general, i.e., in the definition of Devaney chaos there are no redundant conditions for NDS. In addition, we show that if we also assume uniform convergence of the sequence (fn) that induces the NDS, then sensitivity follows. Furthermore, in contrast to the autonomous case, we show that there exist minimal NDS which are neither equicontinuous nor sensitive.

On Points Focusing Entropy

In the paper, we consider local aspects of the entropy of nonautonomous dynamical systems. For this purpose, we introduce the notion of a (asymptotical) focal entropy point. The notion of entropy appeared as a result of practical needs concerning thermodynamics and the problem of information flow, and it is connected with the complexity of a system. The definition adopted in the paper specifies the notions that express the complexity of a system around certain points (the complexity of the system is the same as its complexity around these points), and moreover, the complexity of a system around such points does not depend on the behavior of the system in other parts of its domain. Any periodic system “acting” in the closed unit interval has an asymptotical focal entropy point, which justifies wide interest in these issues. In the paper, we examine the problems of the distortions of a system and the approximation of an autonomous system by a nonautonomous one, in the context of having a (asymptotical) focal entropy point. It is shown that even a slight modification of a system may lead to the arising of the respective focal entropy points.

Equi-attraction and backward compactness of pullback attractors for point-dissipative Ginzburg-Landau equations

A new concept of an equi-attractor is introduced, and defined by the minimal compact set that attracts bounded sets uniformly in the past, for a non-autonomous dynamical system. It is shown that the compact equi-attraction implies the backward compactness of a pullback attractor. Also, an eventually equi-continuous and strongly bounded process has an equi-attractor if and only if it is strongly point dissipative and strongly asymptotically compact. Those results primely strengthen the known existence result of a backward bounded pullback attractor in the literature. Finally, the theoretical criteria are applied to prove the existence of both equi-attractor and backward compact attractor for a Ginzburg-Landau equation with some varying coefficients and a backward tempered external force.

Dynamics for the complex Ginzburg-Landau equation on non-cylindrical domains II: The monotone case

In this article, we continue the study of the dynamics of the following complex Ginzburg-Landau equation ∂tu − (λ + iα)Δu + (κ + iβ)|u|p−2u − γu = f(t) on non-cylindrical domains. We assume that the spatial domains are bounded and increase with time, which is different from the diffeomorphism case presented in Zhou and Sun [Discrete Contin. Dyn. Syst., Ser. B 21, 3767–3792 (2016)]. We develop a new penalty function to establish the existence and uniqueness of a variational solution satisfying energy equality as well as some energy inequalities and prove the existence of a D-pullback attractor for the non-autonomous dynamical system generated by this class of solutions.

A Spectral Approach for Quenched Limit Theorems for Random Expanding Dynamical Systems

We prove quenched versions of (i) a large deviations principle (LDP), (ii) a central limit theorem (CLT), and (iii) a local central limit theorem (LCLT) for non-autonomous dynamical systems. A key advance is the extension of the spectral method, commonly used in limit laws for deterministic maps, to the general random setting. We achieve this via multiplicative ergodic theory and the development of a general framework to control the regularity of Lyapunov exponents of \emph{twisted transfer operator cocycles} with respect to a twist parameter. While some versions of the LDP and CLT have previously been proved with other techniques, the local central limit theorem is, to our knowledge, a completely new result, and one that demonstrates the strength of our method. Applications include non-autonomous (piecewise) expanding maps, defined by random compositions of the form $T_{\sigma^{n-1}\omega}\circ\cdots\circ T_{\sigma\omega}\circ T_\omega$. An important aspect of our results is that we only assume ergodicity and invertibility of the random driving $\sigma:\Omega\to\Omega$; in particular no expansivity or mixing properties are

Is uniform persistence a robust property in almost periodic models? A well-behaved family: almost-periodic Nicholson systems

Using techniques of non-autonomous dynamical systems, we completely characterize the persistence properties of an almost periodic Nicholson system in terms of some numerically computable exponents. Although similar results hold for a class of cooperative and sublinear models, in the general non-autonomous setting one has to consider persistence as a collective property of the family of systems over the hull: the reason is that uniform persistence is not a robust property in models given by almost periodic differential equations.

Relationship between Li-Yorke chaos and positive topological sequence entropy in nonautonomous dynamical systems

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.

Parrondo's dynamic paradox for the stability of non-hyperbolic fixed points

We show that for periodic non-autonomous discrete dynamical systems, even when a common fixed point for each of the autonomous associated dynamical systems is repeller, this fixed point can became a local attractor for the whole system, giving rise to a Parrondo's dynamic type paradox.

Koopman Operator Family Spectrum for Nonautonomous Systems

For any nonautonomous dynamical system, the family of Koopman operators, as well as related Koopman eigenvalues and eigenfunctions, is parameterized by a time pair. Therefore, a logical approach in the data-driven algorithms for the nonautonomous Koopman mode decomposition is the application of a dynamic mode decomposition (DMD) method on the moving stencils of snapshots in order to capture the time dependency. In this paper, we investigate the issues that arise in such an approach. These issues do not appear if we use the moving stencil approach as the model fitting method; they appear as significant errors in the computed nonautonomous Koopman operator eigenvalues. The first issue manifests itself in the hybrid dynamical systems when the moving stencil passes over a nonautonomous switching point. We show that such stencils can be detected through the Krylov subspace projection error and propose an algorithm that computes correct eigenvalues by avoiding such stencils. The second issue appears in the continuous-in-time nonautonomous systems. Even if we apply techniques of finding good observables that solve all issues in the autonomous case, the nonautonomous Koopman eigenvalues will still be computed with a significant error. In the presented theorems, we reveal the nature of this error and propose a second algorithm that is based on the reduction of the stencil size. The application of the two new data-driven algorithms on various nonautonomous systems shows complete recovery from the errors otherwise present in computation of the nonautonomous Koopman operator eigenvalues.

Criteria on the existence and stability of pullback exponential attractors and their application to non-autonomous kirchhoff wave models

In this paper, we are concerned with the existence and stability of pullback exponential attractors for a non-autonomous dynamical system. (ⅰ) We propose two new criteria for the discrete dynamical system and continuous one, respectively. (ⅱ) By applying the criteria to the non-autonomous Kirchhoff wave models with structural damping and supercritical nonlinearity we construct a family of pullback exponential attractors which are stable with respect to perturbations.

Squeezing and finite dimensionality of cocycle attractors for 2D stochastic Navier-Stokes equation with non-autonomous forcing

In this paper, we study the squeezing property and finite dimensionality of cocycle attractors for non-autonomous dynamical systems (NRDS). We show that the generalized random cocycle squeezing property (RCSP) is a sufficient condition to prove a determining modes result and the finite dimensionality of invariant non-autonomous random sets, where the upper bound of the dimension is uniform for all components of the invariant set. We also prove that the RCSP can imply the pullback flattening property in uniformly convex Banach space so that could also contribute to establish the asymptotic compactness of the system. The cocycle attractor for 2D Navier-Stokes equation with additive white noise and translation bounded non-autonomous forcing is studied as an application.

Dynamical system modeling fermionic limit

The existence of multiple radial solutions to the elliptic equation modeling fermionic cloud of interacting particles is proved for the limiting Planck constant and intermediate value of mass parameters. It is achieved by considering the related nonautonomous dynamical system for which the passage to the limit can be established due to the continuity of the solutions with respect to the parameter going to zero.

Act-and-wait time-delayed feedback control of autonomous systems

Recently an act-and-wait modification of time-delayed feedback control has been proposed for the stabilization of unstable periodic orbits in nonautonomous dynamical systems (Pyragas and Pyragas, 2016 [30]). The modification implies a periodic switching of the feedback gain and makes the closed-loop system finite-dimensional. Here we extend this modification to autonomous systems. In order to keep constant the phase difference between the controlled orbit and the act-and-wait switching function an additional small-amplitude periodic perturbation is introduced. The algorithm can stabilize periodic orbits with an odd number of real unstable Floquet exponents using a simple single-input single-output constraint control.

Continuity of pullback and uniform attractors

We study the continuity of pullback and uniform attractors for non-autonomous dynamical systems with respect to perturbations of a parameter. Consider a family of dynamical systems parameterized by λ∈Λ, where Λ is a complete metric space, such that for each λ∈Λ there exists a unique pullback attractor Aλ(t). Using the theory of Baire category we show under natural conditions that there exists a residual set Λ⁎⊆Λ such that for every t∈R the function λ↦Aλ(t) is continuous at each λ∈Λ⁎ with respect to the Hausdorff metric. Similarly, given a family of uniform attractors Aλ, there is a residual set at which the map λ↦Aλ is continuous. We also introduce notions of equi-attraction suitable for pullback and uniform attractors and then show when Λ is compact that the continuity of pullback attractors and uniform attractors with respect to λ is equivalent to pullback equi-attraction and, respectively, uniform equi-attraction. These abstract results are then illustrated in the context of the Lorenz equations and the two-dimensional Navier–Stokes equations.

On disruptions of nonautonomous discrete dynamical systems in the context of their local properties

Abstract The aim of the paper is to examine topological properties of disruptions of nonautonomous discrete dynamical system by other “close” systems. Our considerations will be connected with entropy of a system at a point and with stable points of a system.

Dynamics of a rolling robot

Equations describing the rolling of a spherical ball on a horizontal surface are obtained, the motion being activated by an internal rotor driven by a battery mechanism. The rotor is modeled as a point mass mounted inside a spherical shell and caused to move in a prescribed circular orbit relative to the shell. The system is described in terms of four independent dimensionless parameters. The equations governing the angular momentum of the ball relative to the point of contact with the plane constitute a six-dimensional, nonholonomic, nonautonomous dynamical system with cubic nonlinearity. This system is decoupled from a subsidiary system that describes the trajectories of the center of the ball. Numerical integration of these equations for prescribed values of the parameters and initial conditions reveals a tendency toward chaotic behavior as the radius of the circular orbit of the point mass increases (other parameters being held constant). It is further shown that there is a range of values of the initial angular velocity of the shell for which chaotic trajectories are realized while contact between the shell and the plane is maintained. The predicted behavior has been observed in our experiments.

Weakly nonlinear theory for oscillating wave surge converters in a channel

We present a weakly nonlinear theory on the natural modes’ resonance of an array of oscillating wave surge converters (OWSCs) in a channel. We first derive the evolution equation of the Stuart–Landau type for the gate oscillations in uniform and modulated incident waves and then evaluate the nonlinear effects on the energy conversion performance of the array. We show that the gates are unstable to side-band perturbations so that a Benjamin–Feir instability similar to the case of Stokes’ waves is possible. The non-autonomous dynamical system presents period doubling bifurcations and strange attractors. We also analyse the competition of two natural modes excited by one incident wave. For weak damping and power take-off coefficient, the dynamical effects on the generated power of the OWSCs are investigated. We show that the occurrence of subharmonic resonance significantly increases energy production.