Non-autonomous Dynamical Systems Research Articles

Machine learning prediction of tipping in complex dynamical systems

Anticipating a tipping point, a transition from one stable steady state to another, is a problem of broad relevance due to the ubiquity of the phenomenon in diverse fields. The steady-state nature of the dynamics about a tipping point makes its prediction significantly more challenging than predicting other types of critical transitions from oscillatory or chaotic dynamics. Exploiting the benefits of noise, we develop a general data-driven and machine-learning approach to predicting potential future tipping in nonautonomous dynamical systems and validate the framework using examples from different fields. As an application, we address the problem of predicting the potential collapse of the Atlantic meridional overturning circulation, possibly driven by climate-induced changes in the freshwater input to the North Atlantic. Our predictions based on synthetic and currently available empirical data place a potential collapse window spanning from 2040 to 2065, in consistency with the results in the current literature. Published by the American Physical Society 2024

A note on non-autonomous discrete dynamical systems

In this paper, we define some qualitative properties of non-autonomous discrete dynamical systems such as orbit shift continuum-wise expansivity, orbit shift persistence and orbit shift α-persistence. Then we discuss the relation between these notions and give necessary examples. Moreover, we prove that every continuum-wise expansive non-autonomous discrete system on a compact metric space is orbit shift continuum-wise expansive.

Higher Cheeger ratios of features in Laplace-Beltrami eigenfunctions

This paper investigates links between the eigenvalues and eigenfunctions of the Laplace-Beltrami operator, and the higher Cheeger constants of smooth Riemannian manifolds, possibly weighted and/or with boundary. The higher Cheeger constants give a loose description of the major geometric features of a manifold. We give a constructive upper bound on the higher Cheeger constants, in terms of the eigenvalue of any eigenfunction with the corresponding number of nodal domains. Specifically, we show that for each such eigenfunction, a positive-measure collection of its superlevel sets have their Cheeger ratios bounded above in terms of the corresponding eigenvalue.Some manifolds have their major features entwined across several eigenfunctions, and no single eigenfunction contains all the major features. In this case, there may exist carefully chosen linear combinations of the eigenfunctions, each with large values on a single feature, and small values elsewhere. We can then apply a soft-thresholding operator to these linear combinations to obtain new functions, each supported on a single feature. We show that the Cheeger ratios of the level sets of these functions also give an upper bound on the Laplace-Beltrami eigenvalues. We extend these level set results to nonautonomous dynamical systems, and show that the dynamic Laplacian eigenfunctions reveal sets with small dynamic Cheeger ratios.

Pullback and uniform exponential attractors for non-autonomous Oregonator systems

Abstract We consider the long-time global dynamics of non-autonomous Oregonator systems. This system is a coupled system of three reaction-diffusion equations, that arises from the Belousov-Zhabotinskii reaction. We first present some sufficient conditions for the existence of pullback and uniform exponential attractors for non-autonomous dynamical system. Then, we apply abstract results to prove the existence of a pullback exponential attractor for Oregonator systems affected by time-dependent forces and a uniform exponential attractor for Oregonator systems driven by quasi-periodic external forces.

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

Time-dependent uniform upper semicontinuity of pullback attractors for non-autonomous delay dynamical systems: Theoretical results and applications

In this paper we provide general results on the uniform upper semicontinuity of pullback attractors with respect to the time parameter for non-autonomous delay dynamical systems. Namely, we establish a criteria in terms of the multi-index convergence of solutions for the delay system to the non-delay one, locally pointwise convergence and local controllability of pullback attractors. As an application, we prove the upper semicontinuity of pullback attractors for a non-autonomous delay reaction-diffusion equation to the corresponding nondelay one over any bounded time interval as the delay parameter tends to zero.

Accurate deep learning-based filtering for chaotic dynamics by identifying instabilities without an ensemble.

We investigate the ability to discover data assimilation (DA) schemes meant for chaotic dynamics with deep learning. The focus is on learning the analysis step of sequential DA, from state trajectories and their observations, using a simple residual convolutional neural network, while assuming the dynamics to be known. Experiments are performed with the Lorenz 96 dynamics, which display spatiotemporal chaos and for which solid benchmarks for DA performance exist. The accuracy of the states obtained from the learned analysis approaches that of the best possibly tuned ensemble Kalman filter and is far better than that of variational DA alternatives. Critically, this can be achieved while propagating even just a single state in the forecast step. We investigate the reason for achieving ensemble filtering accuracy without an ensemble. We diagnose that the analysis scheme actually identifies key dynamical perturbations, mildly aligned with the unstable subspace, from the forecast state alone, without any ensemble-based covariances representation. This reveals that the analysis scheme has learned some multiplicative ergodic theorem associated to the DA process seen as a non-autonomous random dynamical system.

Transitions in echo index and dependence on input repetitions

The echo index counts the number of simultaneously stable asymptotic responses of a nonautonomous (i.e. input-driven) dynamical system. It generalises the well-known echo state property for recurrent neural networks — this corresponds to the echo index being equal to one. In this paper, we investigate how the echo index depends on parameters that govern typical responses to a finite-state ergodic external input that forces the dynamics. We consider the echo index for a nonautonomous system that switches between a finite set of maps, where we assume that each map possesses a finite set of hyperbolic equilibrium attractors. We find the minimum and maximum repetitions of each map are crucial for the resulting echo index. Casting our theoretical findings in the RNN computing framework, we obtain that for small amplitude forcing the echo index corresponds to the number of attractors for the input-free system, while for large amplitude forcing, the echo index reduces to one. The intermediate regime is the most interesting; in this region the echo index depends not just on the amplitude of forcing but also on more subtle properties of the input.

Predicting transient dynamics in a model of reed musical instrument with slowly time-varying control parameter.

When playing a self-sustained reed instrument (such as the clarinet), initial acoustical transients (at the beginning of a note) are known to be of crucial importance. Nevertheless, they have been mostly overlooked in the literature on musical instruments. We investigate here the dynamic behavior of a simple model of reed instrument with a time-varying blowing pressure accounting for attack transients performed by the musician. In practice, this means studying a one-dimensional non-autonomous dynamical system obtained by slowly varying in time the bifurcation parameter (the blowing pressure) of the corresponding autonomous systems, i.e., whose bifurcation parameter is constant. In this context, the study focuses on the case for which the time-varying blowing pressure crosses the bistability domain (with the coexistence of a periodic solution and an equilibrium) of the corresponding autonomous model. Considering the time-varying blowing pressure as a new (slow) state variable, the considered non-autonomous one-dimensional system becomes an autonomous two-dimensional fast-slow system. In the bistability domain, the latter has attracting manifolds associated with two stable branches of the bifurcation diagram of the system with constant parameter. In the framework of the geometric singular perturbation theory, we show that a single solution of the two-dimensional fast-slow system can be used to describe the global system behavior. Indeed, this allows us to determine, depending on the initial conditions and rate of change of the blowing pressure, which manifold is approached when the bistability domain is crossed and to predict whether a sound is produced during transient as a function of the musician's control.

Boundary dynamics for holomorphic sequences, non-autonomous dynamical systems and wandering domains

There are many classical results, related to the Denjoy–Wolff theorem, concerning the relationship between orbits of interior points and orbits of boundary points under iterates of holomorphic self-maps of the unit disc. Here, we address such questions in the very general setting of sequences (Fn) of holomorphic maps between simply connected domains. We show that, while some classical results can be generalised, with an interesting dependence on the geometry of the domains, a much richer variety of behaviours is possible. One of our main results is new even in the classical setting.Our methods apply in particular to non-autonomous dynamical systems, when (Fn) are forward compositions of holomorphic maps, and to the study of wandering domains in holomorphic dynamics.The proofs use techniques from geometric function theory, measure theory and ergodic theory, and the construction of examples involves a ‘weak independence’ version of the second Borel–Cantelli lemma and the concept from ergodic theory of ‘shrinking targets’.

The entropy formula of shrinking target problem in nonautonomous dynamical systems

This paper is devoted to studying the topological entropy of shrinking target problem in nonautonomous dynamical systems (NDSs). After providing the shrinking target problem of the topological version in NDSs, we obtain a general formula for computing the topological entropy of it. More precisely, we show that the entropy distribution principle still holds in NDSs and apply it to estimate the lower bound of the formula. In addition to the specification property, the concept of control non-decreasing property about the Bowen metric proposed by us is taken as a supplementary condition. Furthermore, we get the connection about the topological entropy between the shrinking target problem and the whole system. This extends the result of Ma et al. [Discrete Contin. Dyn. Syst. 39(5) (2019), pp. 2455–2471], corresponding to the case when the maps are the same as the more general case of different maps.

Nonlinear model reduction to temporally aperiodic spectral submanifolds.

We extend the theory of spectral submanifolds (SSMs) to general non-autonomous dynamical systems that are either weakly forced or slowly varying. Examples of such systems arise in structural dynamics, fluid-structure interactions, and control problems. The time-dependent SSMs we construct under these assumptions are normally hyperbolic and hence will persist for larger forcing and faster time dependence that are beyond the reach of our precise existence theory. For this reason, we also derive formal asymptotic expansions that, under explicitly verifiable nonresonance conditions, approximate SSMs and their aperiodic anchor trajectories accurately for stronger, faster, or even temporally discontinuous forcing. Reducing the dynamical system to these persisting SSMs provides a mathematically justified model- reduction technique for non-autonomous physical systems whose time dependence is moderate either in magnitude or speed. We illustrate the existence, persistence, and computation of temporally aperiodic SSMs in mechanical examples under chaotic forcing.

Data-driven models of nonautonomous systems

Nonautonomous dynamical systems are characterized by time-dependent inputs, which complicates the discovery of predictive models describing the spatiotemporal evolution of the state variables of quantities of interest from their temporal snapshots. When dynamic mode decomposition (DMD) is used to infer a linear model, this difficulty manifests itself in the need to approximate the time-dependent Koopman operators. Our approach is to approximate the original nonautonomous system with a modified system derived via a local parameterization of the time-dependent inputs. The modified system comprises a sequence of local parametric systems, which are subsequently approximated by a parametric surrogate model using the DRIPS (dimension reduction and interpolation in parameter space) framework. The offline step of DRIPS relies on DMD to build a linear surrogate model, endowed with reduced-order bases for the observables mapped from training data. The online step interpolates on suitable manifolds to construct a sequence of iterative parametric surrogate models; the target/test parameter points on these manifolds are specified by a local parameterization of the test time-dependent inputs. We use numerical experimentation to demonstrate the robustness of our method and compare its performance with that of deep neural networks.

Fusion dynamical systems with machine learning in imitation learning: A comprehensive overview

Imitation Learning (IL), also referred to as Learning from Demonstration (LfD), holds significant promise for capturing expert motor skills through efficient imitation, facilitating adept navigation of complex scenarios. A persistent challenge in IL lies in extending generalization from historical demonstrations, enabling the acquisition of new skills without re-teaching. Dynamical system-based IL (DSIL) emerges as a significant subset of IL methodologies, offering the ability to learn trajectories via movement primitives and policy learning based on experiential abstraction. This paper emphasizes the fusion of theoretical paradigms, integrating control theory principles inherent in dynamical systems into IL. This integration notably enhances robustness, adaptability, and convergence in the face of novel scenarios. This survey aims to present a comprehensive overview of DSIL methods, spanning from classical approaches to recent advanced approaches. We categorize DSIL into autonomous dynamical systems and non-autonomous dynamical systems, surveying traditional IL methods with low-dimensional input and advanced deep IL methods with high-dimensional input. Additionally, we present and analyze three main stability methods for IL: Lyapunov stability, contraction theory, and diffeomorphism mapping. Our exploration also extends to popular policy improvement methods for DSIL, encompassing reinforcement learning, deep reinforcement learning, and evolutionary strategies. The primary objective is to expedite readers’ comprehension of dynamical systems’ foundational aspects and capabilities, helping identify practical scenarios and charting potential future directions. By offering insights into the strengths and limitations of dynamical system methods, we aim to foster a deeper understanding among readers. Furthermore, we outline potential extensions and enhancements within the realm of dynamical systems, outlining avenues for further exploration.

On the smoothness of principal angles between subspaces and their application to angular values of dynamical systems

In this work we provide detailed estimates of maximal principal angles between subspaces and we analyse their smoothness for smoothly varying subspaces. This leads to a new definition of angular values for linear dynamical systems in continuous time. We derive some of their properties complementary to the theory of angular values developed in Beyn, et al. [Angular values of nonautonomous and random linear dynamical systems: Part I – fundamentals, SIAM J. Appl. Dyn. Syst. 21(2) (2022), pp. 1245–1286; Angular values of nonautonomous linear dynamical systems: Part II – reduction theory and algorithm, SIAM J. Appl. Dyn. Syst. 22(1) (2023), pp. 162–198] for discrete time systems. The estimates are further employed to establish upper semicontinuity of angular values for some parametric model examples of discrete and continuous type.

Nonparametric estimation of nonautonomous uncertain differential equations with application to temperature models

In recent years, there has been a great development in parameter estimation methods for uncertain differential equations (UDEs). However, the observations we can obtain in real life are limited, in which case the form of function in a UDE is unknown. When dealing with such UDEs, we may use observational data to make nonparametric estimates. There are many nonautonomous systems in real life, and nonautonomous UDEs can simulate some uncertain nonautonomous dynamical systems well. In this paper, a nonparametric estimation method based on the nonautonomous UDEs of the binary Legendre polynomial is proposed. Then, three numerical examples are given to verify the reliability of nonparametric estimation. As an application, a real data example of global average monthly temperatures is used to illustrate the effectiveness of our method.

Asymptotically autonomous stability of kernel sections for lattice plate equations with nonlinear damping

We develop the theory of kernel sections of non-autonomous dynamical systems. Under some sufficient conditions, we establish some abstract results on the asymptotically autonomous stability of kernel sections, which suggests that the time-section of a kernel section is asymptotically stable to a global attractor of the autonomous dynamical system. A lattice plate equation with nonlinear damping and non-autonomous forcing is considered as an application. It is shown that the time-section of kernel sections of the non-autonomous lattice plate equations are nonempty, uniformly compact, pullback attracting, and asymptotically stable to a global attractor of the autonomous lattice plate equations.

Multivalued dynamics of non-autonomous reaction–diffusion equation with nonlinear advection term

In this paper, we investigate a reaction–diffusion population model with a nonlinear advection term and a time-dependent force given by the equation ut−Δu+α→⋅∇up=f(u)+h(t)in(τ,∞)×Ω,subject to the boundary condition u=0 on (τ,∞)×∂Ω. Here, Ω⊂RN with N≥1 is a bounded domain with smooth boundary, τ∈R, α→=(α1,…,αN) is a given advective direction and p>1. The presence of the nonlinear advection term α→⋅∇up introduces technical difficulties in the analysis, leading to a scenario where the uniqueness of weak solutions cannot be guaranteed. Consequently, the equation generates a multi-valued nonautonomous dynamical system. In this context, we establish the existence of minimal pullback attractors, considering universes of bounded and tempered sets. Moreover, we explore the relationships between these pullback attractors. Finally, we prove the upper semicontinuity of pullback attractors with respect to the advective vector α→.

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.