Non-autonomous Dynamical Systems Research Articles

Embedding information onto a dynamical system

The celebrated Takens’ embedding theorem concerns embedding an attractor of a dynamical system in a Euclidean space of appropriate dimension through a generic delay-observation map. The embedding also establishes a topological conjugacy. In this paper, we show how an arbitrary sequence can be mapped into another space as an attractive solution of a nonautonomous dynamical system. Such mapping also entails a topological conjugacy and an embedding between the sequence and the attractive solution spaces. This result is not a generalisation of Takens embedding theorem but helps us understand what exactly is required by discrete-time state space models widely used in applications to embed an external stimulus onto its solution space. Our results settle another basic problem concerning the perturbation of an autonomous dynamical system. We describe what exactly happens to the dynamics when exogenous noise perturbs continuously a local irreducible attracting set (such as a stable fixed point) of a discrete-time autonomous dynamical system.

Influence of imperfections on the stability of a multi-disc friction system

This paper studies the impact of geometrical imperfections (gaps in the links, center distance, inclination angle, etc.) on the stability and the dynamic behavior of mechanical systems exhibiting friction-induced vibrations (braking systems, clutches, etc.). It comes further to two papers relating to improved modeling of friction and separation at the interface of a rotor-stator system. The latter is a phenomenological model which is composed of an embedded beam on which a disc is mounted, called the stator, in contact with another disc called the rotor. The phenomenological mechanical system used in this paper is a new version of the latter model taking into account some of the geometrical imperfections present in real mechanical systems. In the proposed model, the rotor and stator discs have the ability to move radially thanks to the gaps in the links. The inclination angle of the rotor disc and a center distance between the stator and the rotor discs are also introduced. The inclusion of these three geometrical imperfections induces additional complex phenomena that have a strong impact on the stability and dynamic behavior of the mechanical system, because the coupling of the imperfections with the rotor rotation gives rise to a non-autonomous dynamical system. Consequently, stability studies of fixed points must be replaced by stability studies of periodic orbits involving the construction of a monodromy matrix. In conclusion, significant differences in the values of the bifurcation point and the levels of the associated instabilities can be observed, unlike in a study that does not take geometrical imperfections into account.

Nonlinear and divergent responses of fluctuation-driven systems

Most network dynamical systems are out of equilibrium and externally driven by fluctuations. Linear response theory generically characterizes systems responses to such fluctuations for small driving amplitudes yet cannot capture response properties that are either due to strong driving or intrinsically nonlinear. For oscillation-driven systems, we here report average response offsets that scale quadratically with asymptotically small amplitudes. At some critical driving amplitude, responses cease to stay close to a given operating point and may diverge. Standard response theory fails to predict these amplitudes even at arbitrarily high orders. We propose an integral self-consistency condition that captures the full nonlinear system dynamics. We illustrate our approach for a minimal one-dimensional model and capture the nonlinear shift of voltages in the phase, frequency and voltage dynamics of AC power grid networks. Our approach may help to quantitatively predict intrinsically nonlinear response dynamics as well as bifurcations emerging at large driving amplitudes in non-autonomous dynamical systems.

Wong-Zakai approximations and long term behavior of second order non-autonomous stochastic lattice dynamical systems with additive noise

<abstract><p>In this article, we investigate the Wong-Zakai approximations of a class of second order non-autonomous stochastic lattice systems with additive white noise. We first prove the existence and uniqueness of tempered pullback random attractors for the original stochastic system and its Wong-Zakai approximation. Then, we establish the upper semicontinuity of these attractors for Wong-Zakai approximations as the step-length of the Wiener shift approaches zero.</p></abstract>

Multi-valued random dynamics of stochastic wave equations with infinite delays

<p style='text-indent:20px;'>This paper is devoted to the asymptotic behavior of solutions to a non-autonomous stochastic wave equation with infinite delays and additive white noise. The nonlinear terms of the equation are not expected to be Lipschitz continuous, but only satisfy continuity assumptions along with growth conditions, under which the uniqueness of the solutions may not hold. Using the theory of multi-valued non-autonomous random dynamical systems, we prove the existence and measurability of a compact global pullback attractor.</p>

An improved finite-time stabilization of discontinuous non-autonomous IT2 T-S fuzzy interconnected complex-valued systems: A fuzzy switching state-feedback control method

<abstract><p>Based on the type-2 Takagi-Sugeno (IT2 T-S) fuzzy theory, a non-autonomous fuzzy complex-valued dynamical system with discontinuous interconnection function is formulated. Under the framework of Filippov, the finite-time stabilization (FTS) problem is investigated by using an indefinite-derivative Lyapunov function method, where the derivative of the constructed Lyapunov function is allowed to be positive. By designing a fuzzy switching state feedback controller involving time-varying control gain parameters, several sufficient criteria are established to determine the considered system's stability in finite time. Correspondingly, due to the time-varying system parameters and the designed time-dependent control gain coefficients, a more flexible settling time (ST) is estimated. Finally, an example is presented to confirm the proposed methodology.</p></abstract>

Pullback attractors via quasi-stability for non-autonomous lattice dynamical systems

<p style='text-indent:20px;'>In this paper we study long-time behavior of first-order non-autono-mous lattice dynamical systems in square summable space of double-sided sequences using the cooperation between the discretized diffusion operator and the discretized reaction term. We obtain existence of a pullback global attractor and construct pullback exponential attractor applying the introduced notion of quasi-stability of the corresponding evolution process.</p>

Controller and asymptotic autonomy of random attractors for stochastic p-Laplace lattice equations

<p style='text-indent:20px;'>A non-autonomous random dynamical system is called to be controllable if there is a pullback random attractor (PRA) such that each fibre of the PRA converges upper semi-continuously to a nonempty compact set (called a controller) as the time-parameter goes to minus infinity, while the PRA is called to be asymptotically autonomous if there is a random attractor for another (autonomous) random dynamical system as a controller. We establish the criteria for ensuring the existence of the minimal controller and the asymptotic autonomy of a PRA respectively. The abstract results are illustrated in possibly non-autonomous stochastic p-Laplace lattice equations with tempered convergent external forces.</p>

Asymptotic behavior of plate equations with memory driven by colored noise on unbounded domains

<abstract><p>The paper investigates mainly the asymptotic behavior of the non-autonomous random dynamical systems generated by the plate equations with memory driven by colored noise defined on $ \mathbb{R}^n $. Firstly, we prove the well-posedness of the equation in the natural energy space. Secondly, we define a continuous cocycle associated with the solution operator. Finally, we establish the existence and uniqueness of random attractors of the equation by the uniform tail-ends estimates methods and the splitting technique.</p></abstract>

A nonautonomous chemostat model for the growth of gut microbiome with varying nutrient

<p style='text-indent:20px;'>A mathematical model describing the growth of gut microbiome inside and on the wall of the gut is developed based on the chemostat model with wall growth. Both the concentration and flow rate of the nutrient input are time-dependent, which results in a system of non-autonomous differential equations. First the stability of each meaningful equilibrium is studied for the autonomous counterpart. Then the existence of pullback attractors and its detailed structures for the nonautonomous system are investigated using theory and techniques of nonautonomous dynamical systems. In particular, sufficient conditions under which the microbiome vanishes or persists are constructed. Numerical simulations are provided to illustrate the theoretical results.</p>

On the global uniform stability analysis of non-autonomous dynamical systems: A survey

In this survey, we introduce the notion of stability of time varying nonlinear systems. In particular we investigate the notion of global practical exponential stability for non-autonomous systems. The proposed approach for stability analysis is based on the determination of the bounds of perturbations that characterize the asymptotic convergence of the solutions to a closed ball centered at the origin.

Asymptotics of singularly perturbed damped wave equations with super-cubic exponent

<p style='text-indent:20px;'>This work is devoted to studying the relations between the asymptotic behavior for a class of hyperbolic equations with super-cubic nonlinearity and a class of heat equations, where the problem is considered in a smooth bounded three dimensional domain. Based on the extension of the Strichartz estimates to the case of bounded domain, we show the regularity of the pullback, uniform, and cocycle attractors for the non-autonomous dynamical system given by hyperbolic equation. Then we prove that all types of non-autonomous attractors converge, upper semicontiously, to the natural extension global attractor of the limit parabolic equations.</p>

DYNAMICAL ANALYSIS OF NONAUTONOMOUS <i>RLC</i> CIRCUIT WITH THE ABSENCE AND PRESENCE OF ATANGANA-BALEANU FRACTIONAL DERIVATIVE

<abstract> The Fractional-order derivative (FOD) has a rich history in mathematical science. In comparison, the half derivative has not been extensively used in applied science and engineering. On the other hand, it is widely applicable in many branches of engineering modeling, especially control fractional-order proportional integral derivative(PID), semi-infinite lossy transmission, and many more. Thus, this work investigates the classical RLC circuit with the sense of the Atangana-Baleanu FOD. Concurrently, the Lyapunov spectral analysis is applied to determine whether or not stability and instability appear in the RLC circuit. In the first look, the RLC likes a straightforward dynamical system when the external driving force is zero. But, the dynamical system of the RLC evolves more complicated, and the wandering phase spaces do not illustrate the stability or instability when the time-dependent driving force is not zero. As a result, the Lyapunov exponents play a significant role to analyze the trait of the trajectories of the phase state. It is found from the bifurcation plots and Lyapunov exponent's spectral analysis that the non-autonomous dynamical systems lead to instability and the chaotic state with the manifestation of the local dynamical system, which switches to a stable spiral node with the presence Atangana-Baleanu FOD. </abstract>

Multi-Objective Physics-Guided Recurrent Neural Networks for Identifying Non-Autonomous Dynamical Systems*

While trade-offs between modeling effort and model accuracy remain a major concern with system identification, resorting to data-driven methods often leads to a complete disregard for physical plausibility. To address this issue, we propose a physics-guided hybrid approach for modeling non-autonomous systems under control. Starting from a traditional physics-based model, this is extended by a recurrent neural network and trained using a sophisticated multi-objective strategy yielding physically plausible models. While purely data-driven methods fail to produce satisfying results, experiments conducted on real data reveal substantial accuracy improvements by our approach compared to a physics-based model.

Dynamics for the 3D incompressible Navier-Stokes equations with double time delays and damping

&lt;p style='text-indent:20px;'&gt;This paper is concerned with the tempered pullback attractors for 3D incompressible Navier-Stokes model with a double time-delays and a damping term. The delays are in the convective term and external force, which originate from the control in engineer and application. Based on the existence of weak and strong solutions for three dimensional hydrodynamical model with subcritical nonlinearity, we proved the existence of minimal family for pullback attractors with respect to tempered universes for the non-autonomous dynamical systems.&lt;/p&gt;

Polynomial entropy of nonautonomous dynamical systems for noncompact sets

This paper is devoted to the study of the Bowen polynomial entropy for nonautonomous dynamical systems, which is an extension of the classical definition of Bowen topological entropy. We establish a variational principle for polynomial entropy on compact subsets in the context of nonautonomous dynamical systems. Also, we show that the Bowen topological entropy can be determined by the local entropies of measures for nonautonomous dynamical systems, which extends Ma and Wen's result [33].

Controlling coexisting attractors of a class of non-autonomous dynamical systems

This paper studies a control method for switching stable coexisting attractors of a class of non-autonomous dynamical systems. The central idea is to introduce a continuous path for the system’s trajectory to transition from its original undesired stable attractor to a desired one by varying one of the system parameters according to the information of the desired attractor. The behaviour of the control is studied numerically for both non-smooth and smooth dynamical systems, using a soft-impact and a Duffing oscillators as examples. Special attention is given to identify the parametric regions where the proposed control strategy is applicable by using path-following methods implemented via the continuation platform COCO. It is shown that the proposed control concept can be implemented through either using an external control input or varying a system parameter. Finally, extensive numerical results are presented to validate the proposed control methods.

Finite-Dimensionality of Tempered Random Uniform Attractors

Finite-dimensional attractors play an important role in finite-dimensional reduction of PDEs in mathematical modelization and numerical simulations. For non-autonomous random dynamical systems, Cui and Langa (J Differ Equ, 263:1225–1268, 2017) developed a random uniform attractor as a minimal compact random set which provides a certain description of the forward dynamics of the underlying system by forward attraction in probability. In this paper, we study the conditions that ensure a random uniform attractor to have finite fractal dimension. Two main criteria are given, one by a smoothing property and the other by a squeezing property of the system, and neither of the two implies the other. The upper bound of the fractal dimension consists of two parts: the fractal dimension of the symbol space plus a number arising from the smoothing/squeezing property. As an illustrative application, the random uniform attractor of a stochastic reaction–diffusion equation with scalar additive noise is studied, for which the finite-dimensionality in L^2 is established by the squeezing approach and that in H_0^1 by the smoothing framework. In addition, a random absorbing set that absorbs itself after a deterministic period of time is also constructed.

Nonautonomous attractors and Young measures

Motivated by the problem of identifying a mathematical framework for the formal definition of concepts such as weather, climate and connections between them, we discuss a question of convergence of short-time time averages for random nonautonomous dynamical systems depending on a parameter. The problem is formulated by means of Young measures. Using the notion of pull-back attractor, we prove a general theorem giving a sufficient condition for the tightness of the law of the approximating problems. In a specific example, we show that the theorem applies and we characterize the unique limit point.

Random attractor and exponentially stability for the second order nonautonomous retarded lattice systems with multiplicative white noise

In this paper, we first present some conditions for the existence of a random attractor of continuous non-autonomous random dynamical systems (cocycle) and locally coupled stochastic retarded lattice dynamical systems in a space of continuous functions from finite closed interval into a product space of finite weighted spaces of infinite sequences. Then we consider the existence, singleton and exponentially stability of the random attractor for the second order non-autonomous retarded lattice systems with multiplicative white noise.