Nonautonomous Dynamics Research Articles

Upper Semicontinuity of Strong Pullback Attractors for Delay Non-autonomous Dynamical Systems and Application to Double Time-delayed 2D MHD Equations

Upper Semicontinuity of Strong Pullback Attractors for Delay Non-autonomous Dynamical Systems and Application to Double Time-delayed 2D MHD Equations

Forward Attraction of Nonautonomous Dynamical Systems and Applications to Navier–Stokes Equations

Forward Attraction of Nonautonomous Dynamical Systems and Applications to Navier–Stokes Equations

Nonautonomous Dynamics: Classification, Invariants, and Implementation

Nonautonomous Dynamics: Classification, Invariants, and Implementation

Controllable nonautonomous localized waves and dynamics for a quasi-1D Gross-Pitaevskii equation in Bose-Einstein condensations with attractive interaction.

This paper investigates dynamical behaviors and controllability of some nonautonomous localized waves based on the Gross-Pitaevskii equation with attractive interatomic interactions. Our approach is a relation constructed between the Gross-Pitaevskii equation and the standard nonlinear Schrödinger equation through a new self-similarity transformation which is to convert the exact solutions of the latter to the former's. Subsequently, one can obtain the nonautonomous breather solutions and higher-order rogue wave solutions of the Gross-Pitaevskii equation. It has been shown that the nonautonomous localized waves can be controlled by the parameters within the self-similarity transformation, rather than relying solely on the nonlinear intensity, spectral parameters, and external potential. The control mechanism can induce an unusual number of loosely bound higher-order rogue waves. The asymptotic analysis of unusual loosely bound rogue waves shows that their essence is energy transfer among rogue waves. Numerical simulations test the dynamical stability of obtained localized wave solutions, which indicate that modifying the parameters in the self-similarity transformation can improve the stability of unstable localized waves and prolong their lifespan. We numerically confirm that the rogue wave controlled by the self-similarity transformation can be reproduced from a chaotic initial background field, hence anticipating the feasibility of its experimental observation, and propose an experimental method for observing these phenomena in Bose-Einstein condensates. The method presented in this paper can help to induce and observe new stable localized waves in some physical systems.

Intermittent phase dynamics of non-autonomous oscillators through time-varying phase

Oscillatory dynamics pervades the universe, appearing in systems of all scales. Whilst autonomous oscillatory dynamics has been extensively studied and is well understood, the very important problem of non-autonomous oscillatory dynamics is less well understood. Here, we provide a framework for non-autonomous oscillatory dynamics, within which we can define intermittent phenomena such as intermittent phase synchronisation. Moreover, we demonstrate this framework with a coupled pair of non-autonomous phase oscillators as well as a higher-dimensional system comprising of two interacting phase-oscillator networks.

A Note on Stronger Forms of Sensitivity for Non-Autonomous Dynamical Systems on Uniform Spaces.

This paper introduces the notion of multi-sensitivity with respect to a vector within the context of non-autonomous dynamical systems on uniform spaces and provides insightful results regarding N-sensitivity and strongly multi-sensitivity, along with their behaviors under various conditions. The main results established are as follows: (1) For a k-periodic nonautonomous dynamical system on a Hausdorff uniform space (S,U), the system (S,fk∘⋯∘f1) exhibits N-sensitivity (or strongly multi-sensitivity) if and only if the system (S,f1,∞) displays N-sensitivity (or strongly multi-sensitivity). (2) Consider a finitely generated family of surjective maps on uniform space (S,U). If the system (S,f1,∞) is N-sensitive, then the system (S,fk,∞) is also N-sensitive. When the family f1,∞ is feebly open, the converse statement holds true as well. (3) Within a finitely generated family on uniform space (S,U), N-sensitivity (and strongly multi-sensitivity) persists under iteration. (4) We present a sufficient condition under which an nonautonomous dynamical system on infinite Hausdorff uniform space demonstrates N-sensitivity.

Quenched limit theorems for expanding on average cocycles

We prove quenched versions of a central limit theorem, a large deviations principle as well as a local central limit theorem for expanding on average cocycles. This is achieved by building an appropriate modification of the spectral method for nonautonomous dynamics developed by [D. Dragičević, G. Froyland, C. Gonzàlez-Tokman and S. Vaienti, A spectral approach for quenched limit theorems for random expanding dynamical systems, Commun. Math. Phys. 360 (2018) 1121–1187], to deal with the case of random dynamics that exhibits nonuniform decay of correlations, which are ubiquitous in the context of the multiplicative ergodic theory.

Theory of Invariant Manifolds for Infinite-Dimensional Nonautonomous Dynamical Systems and Applications

Theory of Invariant Manifolds for Infinite-Dimensional Nonautonomous Dynamical Systems and Applications

Nonautonomous dynamics of local and nonlocal Fokas–Lenells models

We investigate the dynamics of solutions to the local and nonlocal nonautonomous Fokas–Lenells (FL) models. Solutions to the local and nonlocal reductions are presented using the Wronskian technique by imposing constraints on the double Wronskian solutions to the unreduced nonautonomous FL system. Classifications of the soliton solutions are shown in relation to the distribution of the eigenvalues. Moreover, dynamics of these solutions are presented, along with the impacts of the variable coefficient . Notably, the coefficient influences the top traces and velocities of the found solutions in both the local and nonlocal cases, enriching this study.

Rate and memory effects in bifurcation-induced tipping.

A variation in the environment of a system, such as the temperature, the concentration of a chemical solution, or the appearance of a magnetic field, may lead to a drift in one of the parameters. If the parameter crosses a bifurcation point, the system can tip from one attractor to another (bifurcation-induced tipping). Typically, this stability exchange occurs at a parameter value beyond the bifurcation value. This is what we call, here, the shifted stability exchange. We perform a systematic study on how the shift is affected by the initial parameter value and its change rate. To that end, we present numerical simulations and partly analytical results for different types of bifurcations and different paradigmatic systems. We show that the nonautonomous dynamics can be split into two regimes. Depending on whether we exceed the numerical or experimental precision or not, the system may enter the nondeterministic or the deterministic regime. This is determined solely by the conditions of the drift. Finally, we deduce the scaling laws governing this phenomenon and we observe very similar behavior for different systems and different bifurcations in both regimes.

The Variational Principle for the Packing Entropy of Nonautonomous Dynamical Systems

The Variational Principle for the Packing Entropy of Nonautonomous Dynamical Systems

A note on the differentiability of discrete Palmer's linearization

In the context of discrete nonautonomous dynamics, we prove that the homeomorphisms in the linearization theorem are $C^2$ diffeomorphisms. In contrast to other related works, our result does not involve non-resonance conditions or spectral gaps. Our approach is based on the interlacing of the properties of nonautonomous hyperbolicity of the linear part, and boundedness and Lipschitzness of the nonlinearities. Moreover, we propose a functional approach to find conditions for regularity of arbitrary degree.

A Note on Adiabatic Time Evolution and Quasi-Static Processes in Translation-Invariant Quantum Systems

We study the slowly varying, non-autonomous quantum dynamics of a translation-invariant spin or fermion system on the lattice $$\mathbb {Z}^d$$ . This system is assumed to be initially in thermal equilibrium, and we consider realizations of quasi-static processes in the adiabatic limit. By combining the Gibbs variational principle with the notion of quantum weak Gibbs states introduced in Jakšić et al. (Approach to equilibrium in translation-invariant quantum systems: some structural results, in the present issue of Ann. H. Poincaré, 2023), we establish a number of general structural results regarding such realizations. In particular, we show that such a quasi-static process is incompatible with the property of approach to equilibrium studied in this previous work.

Angular Values of Nonautonomous Linear Dynamical Systems: Part II – Reduction Theory and Algorithm

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. H\"uls, 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 spectral bundles. In particular, we prove a reduction theorem which shows that instead of maximizing over all subspaces, it suffices to maximize over so-called trace spaces which have their basis in the spectral fibers. The reduction leads to an algorithm for computing angular values of dimensions one and two. We apply the algorithm to several systems of dimension up to 4 and demonstrate its efficiency to detect the fastest rotating subspace even if it is not dominant under the forward dynamics.

Nonautonomous dynamics: classification, invariants, and implementation

The work is a brief review of the results obtained in nonautonomous dynamics based on the concept of uniform equivalence of nonautonomous systems. This approach to the study of nonautonomous systems was proposed in [10] and further developed in the works of the second author, and recently - jointly by both authors. Such an approach seems to be fruitful and promising, since it allows one to develop a nonautonomous analogue of the theory of dynamical systems for the indicated classes of systems and give a classi cation of some natural classes of nonautonomous systems using combinatorial type invariants. We show this for classes of nonautonomous gradient-like vector elds on closed manifolds of dimensions one, two, and three. In the latter case, a new equivalence invariant appears, the wild embedding type for stable and unstable manifolds [14,17], as shown in a recent paper by the authors [5].

Evolution maps for delay-difference equations

To any one-sided nonautonomous dynamics obtained from a delay-difference equation, we associate a lifted autonomous delay-difference equation that can be used, together with its solutions and its perturbations, to characterize the hyperbolicity in three different ways: in terms of the hyperbolicity of the corresponding evolution map, which describes the solutions of the lifted autonomous equation; in terms of an admissibility property for this evolutions map, which corresponds to the invertibility of a certain linear operator; and in terms of an admissibility property for the perturbations of the lifted delay-difference equation. We emphasize that the evolution map is necessarily noninvertible in our setting, which led us to introduce a new notion of exponential dichotomy for this map whose existence is equivalent to the existence of an exponential dichotomy for the original nonautonomous dynamics.

Sufficient Conditions for the Existence and Uniqueness of Maximal Attractors for Autonomous and Nonautonomous Dynamical Systems

Sufficient Conditions for the Existence and Uniqueness of Maximal Attractors for Autonomous and Nonautonomous Dynamical Systems

Topological string amplitudes and Seiberg-Witten prepotentials from the counting of dimers in transverse flux

Important illustration to the principle “partition functions in string theory are τ-functions of integrable equations” is the fact that the (dual) partition functions of 4d mathcal{N} = 2 gauge theories solve Painlevé equations. In this paper we show a road to self-consistent proof of the recently suggested generalization of this correspondence: partition functions of topological string on local Calabi-Yau manifolds solve q-difference equations of non-autonomous dynamics of the “cluster-algebraic”integrable systems.We explain in details the “solutions” side of the proposal. In the simplest non-trivial example we show how 3d box-counting of topological string partition function appears from the counting of dimers on bipartite graph with the discrete gauge field of “flux” q. This is a new form of topological string/spectral theory type correspondence, since the partition function of dimers can be computed as determinant of the linear q-difference Kasteleyn operator. Using WKB method in the “melting” q → 1 limit we get a closed integral formula for Seiberg-Witten prepotential of the corresponding 5d gauge theory. The “equations” side of the correspondence remains the intriguing topic for the further studies.

Measure and Statistical Attractors for Nonautonomous Dynamical Systems

Various inequivalent notions of attraction for autonomous dynamical systems have been proposed, each of them useful to understand specific aspects of attraction. Milnor’s notion of a measure attractor considers invariant sets with positive measure basin of attraction, while Ilyashenko’s weaker notion of a statistical attractor considers positive measure points that approach the invariant set in terms of averages. In this paper we propose generalisations of these notions to nonautonomous evolution processes in continuous time. We demonstrate that pullback/forward measure/statistical attractors can be defined in an analogous manner and relate these to the respective autonomous notions when an autonomous system is considered as nonautonomous. There are some subtleties even in this special case–we illustrate an example of a two-dimensional flow with a one-dimensional measure attractor containing a single point statistical attractor. We show that the single point can be a pullback measure attractor for this system. Finally, for the particular case of an asymptotically autonomous system (where there are autonomous future and past limit systems) we relate pullback (respectively, forward) attractors to the past (respectively, future) limit systems.

Perturbations with nonpositive Lyapunov exponents

The notion of uniform hyperbolicity is equivalent to various admissibility properties. For example, one such property is expressed in terms of the existence of bounded solutions for any bounded perturbation of the dynamics. Our main objective is to describe a weaker hyperbolicity property for a nonautonomous dynamics with discrete time that is equivalent to a modification of the latter admissibility property, namely the existence of solutions with nonpositive Lyapunov exponents for any perturbation with nonpositive Lyapunov exponent. Our characterization is expressed in terms of appropriate exponents that somehow mix the Bohl exponents and the Lyapunov exponents.