Non-autonomous Dynamical Systems Research Articles

Омниколесная реализация задачи Суслова с реономной связью: динамическая модель и управление

The classical Suslov problem of the motion of a rigid body with a fixed point is well known and has been studied in detail. In this paper, an omniwheel implementation of the Suslov problem is proposed. The controlled motion of a rigid body with a fixed point in the presence of scleronomic nonholonomic constraints and rheonomic artificial kinematic constraint is considered. The rigid body rotates around a fixed point, rolls around a spherical shell from the inside and contacts it by means of omniwheels with a differential actuator. We believe that the omniwheels are in contact with the spherical shell only at one point. In order to subordinate the motion of the rigid body to an artificial rheonomic constraint, a differential actuator creates control torques on omniwheels. Based on the d’Alembert–Lagrange principle, equations of motion of the mechanical system with indeterminate multipliers specifying constraint reactions are constructed. The problem is reduced to the study of a non-autonomous two-dimensional dynamical system. Using the generalized Poincar.e transformation, the study of a two-dimensional dynamical system is reduced to the study of the stability of a one-parameter family of fixed points for a system of differential equations with a degenerate linear part. We determine numerical parameters for which phase trajectories of the system are bounded and for which phase trajectories of the system are unbounded. The results of the study are illustrated graphically. Based on numerical integration, maps for the period (Poincar.e sections) and a map of dynamic regimes are constructed to confirm the Feigenbaum scenario of transition to chaotic dynamics.

The evolution of a non-autonomous chaotic system under non-periodic forcing: A climate change example.

In this article, we approach the problem of measuring and interpreting the mid-term climate of a non-autonomous chaotic dynamical system in the context of climate modeling. To do so, we use a low-dimensional, conceptual model for the Earth system with different timescales of variability and subjected to non-periodic external forcing. We introduce the concepts of an evolution set and its distribution, which are dependent on the starting state of the system, and explore their links to different types of initial condition uncertainty and the rate of external forcing. We define the convergence time as the time that it takes for the evolution distribution of one of the dependent variables to lose memory of its initial condition. We suspect a connection between convergence times and the classical concept of mixing times, but the precise nature of this connection needs to be explored. These results have implications for the design of influential climate and Earth system model ensembles and raise a number of issues of mathematical interest.

Entropies of nonautonomous dynamical systems

Entropies of nonautonomous dynamical systems

Pullback Exponential Attractors with Explicit Fractal Dimensions for Non-Autonomous Partial Functional Differential Equations

The aim of this paper is to propose a new method to construct pullback exponential attractors with explicit fractal dimensions for non-autonomous infinite-dimensional dynamical systems in Banach spaces. The approach is established by combining the squeezing properties and the covering of finite subspace of Banach spaces, which generalize the method established for autonomous systems in Hilbert spaces (Eden A, Foias C, Nicolaenko B, and Temam R Exponential attractors for dissipative evolution equations, Wiley, New York, 1994). The method is especially effective for non-autonomous partial functional differential equations for which phase space decomposition based on the exponential dichotomy of the linear part or variation techniques are available for proving squeezing property. The theoretical results are illustrated by applications to several specific non-autonomous partial functional differential equations, including a retarded reaction–diffusion equation, a retarded 2D Navier–Stokes equation and a retarded semilinear wave equation. The constructed exponential attractors possess explicit fractal dimensions which do not depend on the entropy number but only on some inner characteristics of the studied equations including the spectra of the linear part and the Lipschitz constants of the nonlinear terms and hence do not require the smooth embedding between two spaces in the previous work.

Chaotic Characteristics in Devaney’s Framework for Set-Valued Discrete Dynamical Systems

This paper focuses on the relationship between a non-autonomous discrete dynamical system (NDDS) (H,f1,∞) and its induced set-valued discrete dynamical systems (K(H),f¯1,∞). Specifically, it explores the chaotic properties of these systems. The main finding is that f1,∞ is Devaney chaotic if and only if f¯1,∞ is Devaney chaotic in we-topology. The paper also provides similar conclusions for weak mixing, mixing, mild mixing, chain-transitivity, and chain-mixing in non-autonomous set-valued discrete dynamical systems (NSDDSs). Additionally, the paper proves that weak mixing implies sensitivity in NSDDSs.

A Study on Some Properties for Weak Stability of Non-Autonomous Discrete Dynamical Systems

The interest for non-autonomous discrete dynamical systems has been increasing in recent years, because they are adequate to tell of real activities. For examples, when the mapping is disturbed in each iteration because of external factors, or model of some phenomena in physics, biology and economy, in specific, the population of human and weather forecasting, plus to solve problems generated in mathematics. In mathematics, stability theory addresses the stability of solutions of trajectories of dynamical systems under small perturbations of initial conditions. Besides that, the topological dynamics is the main method used in this paper, since we studied the non-autonomous discrete dynamical systems on a topological space. Next, we present a conception of weak stability (stableness) of non-autonomous discrete dynamical systems (NADDS). We evaluate a collection of weak stable points, and survey the connection among weak stableness and shadowing property. We also consider the connection among weak stableness of a non-autonomous discrete dynamical system and its induced system. In conclusion, the results of this paper will provide some helps in the modelling of a general dynamical system.

Stability of regular attractors for non-autonomous random dynamical systems and applications to stochastic Newton–Boussinesq equations with delays

In this paper, we establish theoretical results on the stability of random regular attractors. First, we introduce a backward regular attractor, which is a new type of attractor defined by a minimal backward pullback attracting set. We then establish an existence theorem for such an attractor, and prove it is long time stable. Eventually, we prove the long time stability of regular pullback random attractors. As an application, we consider stochastic non-autonomous Newton–Boussinesq equations with variable and distributed delays. Since solutions of the equations have no higher regularity, we prove their regular asymptotic compactness via the spectrum decomposition technique.

A Design Method to Eliminate Sub Synchronous Response in Automotive Turbochargers utilizing Nonlinear Dynamics of Time Periodic Systems

The present work introduces a methodology for the dynamic design evaluation of high-speed rotors, applying nonlinear dynamics of time periodic systems. The paper focuses on the definition of design variables so as the system to avoid sub synchronous components in the response. The response of an unbalanced turbocharger rotor mounted on oil bearings with wire mesh dampers (WMDs) is evaluated with an efficient collocation method coupled to the pseudo arc length continuation to evaluate response limit cycles in the entire range of the rotating speed, which acts as bifurcation parameter. The specific system is used to demonstrate the method without loss of generality. The analytical model of the WMD component is implemented directly from the literature, while rigid rotor model and short bearing approximations are used for the rest components. Floquet theory is applied in the non-autonomous dynamic system and the stability characteristics of the response limit cycles are evaluated through Floquet multipliers, the magnitude of which act in the algorithm as stability index. A sensitivity analysis is performed at first to evaluate the correlation between WMD key design parameters and the time response of the system.A multi objective optimization method is embedded to set Floquet multipliers at specific values; in this way the response limit cycles retain stability and periodicity, avoiding any type of bifurcation which would render sub-synchronous component. Acceptable sets for design variables of journal bearings and WMDs are defined by multi-objective optimization and these are verified in efficiency by evaluating the response of the system by time integration, as a virtual experiment, for several operating conditions. The method requires severely lower computational time and capacity to render an acceptable design compared to the conventional design of experiment DoE procedure, and it is not limited on the analytical models of rotor, bearing, or other component.

A new result on the fractal dimension estimates of random attractor for non-autonomous random 2D stochastic dynamical type systems

We prove some conditions for bounding the fractal dimension of random invariant sets of non-autonomous random dynamical systems on separable Banach spaces. Then we apply these conditions to prove the finiteness of fractal dimension of random attractor for stochastic 2D hydrodynamical type equations with linear additive white noise in bounded domains or unbounded domains satisfying the Poincaré inequality.

Global Asymptotic Stability of Generalized Homogeneous Dynamical Systems

The goal of the paper is to study the relationship between asymptotic stability and exponential stability of the solutions of generalized homogeneous nonautonomous dynamical systems. This problem is studied and solved within the framework of general non-autonomous (cocycle) dynamical system. The application of our general results for differential and difference equations is given.

Review article: Dynamical systems, algebraic topology and the climate sciences

Abstract. The definition of climate itself cannot be given without a proper understanding of the key ideas of long-term behavior of a system, as provided by dynamical systems theory. Hence, it is not surprising that concepts and methods of this theory have percolated into the climate sciences as early as the 1960s. The major increase in public awareness of the socio-economic threats and opportunities of climate change has led more recently to two major developments in the climate sciences: (i) the Intergovernmental Panel on Climate Change's successive Assessment Reports and (ii) an increasing understanding of the interplay between natural climate variability and anthropogenically driven climate change. Both of these developments have benefited from remarkable technological advances in computing resources, relating throughput as well as storage, and in observational capabilities, regarding both platforms and instruments. Starting with the early contributions of nonlinear dynamics to the climate sciences, we review here the more recent contributions of (a) the theory of non-autonomous and random dynamical systems to an understanding of the interplay between natural variability and anthropogenic climate change and (b) the role of algebraic topology in shedding additional light on this interplay. The review is thus a trip leading from the applications of classical bifurcation theory to multiple possible climates to the tipping points associated with transitions from one type of climatic behavior to another in the presence of time-dependent forcing, deterministic as well as stochastic.

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

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

Stochastic paleoclimatology: Modeling the EPICA ice core climate records.

We analyze and model the stochastic behavior of paleoclimate time series and assess the implications for the coupling of climate variables during the Pleistocene glacial cycles. We examine 800 kiloyears of carbon dioxide, methane, nitrous oxide, and temperature proxy data from the European Project for Ice Coring in Antarctica (EPICA) Dome-C ice core, which are characterized by 100 ky glacial cycles overlain by fluctuations across a wide range of timescales. We quantify this behavior through multifractal time-weighted detrended fluctuation analysis, which distinguishes near-red-noise and white-noise behavior below and above the 100 ky glacial cycle, respectively, in all records. This allows us to model each time series as a one-dimensional periodic nonautonomous stochastic dynamical system, and assess the stability of physical processes and the fidelity of model-simulated time series. We extend this approach to a four-variable model with intervariable coupling terms, which we interpret in terms of possible interrelationships among the four time series. Within the framework of our coupling coefficients, we find that carbon dioxide and temperature act to stabilize each other and methane and nitrous oxide, whereas the latter two destabilize each other and carbon dioxide and temperature. We also compute the response function for each pair of variables to assess the model performance by comparison to the data and confirm the model predictions regarding stability amongst variables. Taken together, our results are consistent with glacial pacing dominated by carbon dioxide and temperature that is modulated by terrestrial biosphere feedbacks associated with methane and nitrous oxide emissions.

The w-Expansive With The Shadowing Property in g-Nonautonomous Discrete Dynamical Systems

In this study, we define the w-expansive property of generic non-autonomous discrete dynamical systems. We also show how the w-expansive feature is preserved under conjugate, inverse limits, and iterate. In addition, we discuss the relationship between the w-expansive and shadowing properties in generic non-autonomous discrete dynamical systems.

Modeling the Spread of COVID-19 Using Nonautonomous Dynamical System with Simplex Algorithm-Based Optimization for Time-Varying Parameters

The S I R D V (Susceptible, Infected, Recovered, Death, Vaccinated) compartmental model along with time-varying parameters is used to model the spread of COVID-19 in the United States. Time-varying parameters account for changes in transmission rates, people’s behaviors, safety precautions, government regulations, the rate of vaccinations, and also the probabilities of recovery and death. By using a parameter estimation based on the simplex algorithm, the system of differential equations is able to match real COVID-19 data for infections, deaths, and vaccinations in the United States of America with relatively high precision. Autoregression is used to forecast parameters in order to forecast solutions. Van den Driessche’s next-generation approach for basic reproduction number agrees well across the entire time period. Analyses on sensitivity and elasticity are performed on the reproduction number with respect to transmission, exit, and natural death rates in order to observe the changes from a small change in parameter values. Model validation through the Akaike Information Criterion ensures that the model is suitable and optimal for modeling the spread of COVID-19.

Introducing some classes of stable systems without any smooth Lyapunov functions

This paper proves that certain classes of stable nonlinear systems do not admit any smooth Lyapunov functions. In fact, the stability analysis of two different classes of nonlinear dynamical systems is provided, and it is demonstrated that their stability properties cannot be established by any convex Lyapunov functions or smooth Lyapunov functions. In this regard, we begin by studying our first class of autonomous dynamical systems and prove that, despite stability, there are no convex Lyapunov functions in the form of V(x) or V(t,x) to establish the stability properties. For the second class, we consider a much more general form of nonlinear autonomous dynamical systems with a stable origin, and prove that this class, too, does not admit any smooth Lyapunov functions in the form of V(x) or V(t,x). Furthermore, another general class of stable non-autonomous dynamical systems is presented, and it is demonstrated that there is no smooth Lyapunov function in the form of V(x) to establish the stability properties. Finally, for the second class of more general stable non-autonomous systems, it is proved that the class does not admit even a continuous Lyapunov function in the form of V(x) or V(t,x).

Necessary and sufficient criteria for existence, regularity, and asymptotic stability of enhanced pullback attractors with applications to 3D primitive equations

We introduce several new concepts called enhanced pullback attractors for nonautonomous dynamical systems by improving the compactness and attraction of the usual pullback attractors in strong topology spaces uniformly over some infinite time intervals. Then we establish several necessary and sufficient criteria for the existence, regularity and asymptotic stability of these enhanced pullback attractors under general theoretical frameworks which are expected to be applied to many kinds of nonautonomous PDEs. These abstract results are applied to the famous 3D primitive equations modeling the large-scale ocean and atmosphere dynamics. We establish the existence, regularity and asymptotic stability of the enhanced pullback attractors for the nonautonomous primitive equations in [Formula: see text] and [Formula: see text] when the forces satisfy some mild conditions. The uniform pullback asymptotic compactness of the solution operators in [Formula: see text] and [Formula: see text] are established by deriving the uniform “flattening effects” of the solutions in [Formula: see text] and the uniform [Formula: see text]-Hölder continuity of the solutions from [Formula: see text] to [Formula: see text] with respect to the initial data, respectively. Our approaches are applicable to prove the existence of global attractors for the autonomous primitive equations in [Formula: see text] provided [Formula: see text].

One-dimensional monotone nonautonomous dynamical systems

One-dimensional monotone nonautonomous dynamical systems

Learning the dynamical response of nonlinear non-autonomous dynamical systems with deep operator neural networks

We propose using operator learning to approximate the dynamical response of non-autonomous systems, such as nonlinear control systems. Unlike classical function learning, operator learning maps between two function spaces, does not require discretization of the output function, and provides flexibility in data preparation and solution prediction. Particularly, we apply and redesign the Deep Operator Neural Network (DeepONet) to recursively learn the solution trajectories of the dynamical systems. Our approach involves constructing and training a DeepONet that approximates the system’s local solution operator. We then develop a numerical scheme that recursively simulates the system’s long/medium-term dynamic response for given inputs and initial conditions, using the trained DeepONet. We accompany the proposed scheme with an estimate for the error bound of the associated cumulative error. Moreover, we propose a data-driven Runge–Kutta (RK) explicit scheme that leverages the DeepONet’s forward pass and automatic differentiation to better approximate the system’s response when the numerical scheme’s step size is small. Numerical experiments on the predator–prey, pendulum, and cart pole systems demonstrate that our proposed DeepONet framework effectively learns to approximate the dynamical response of non-autonomous systems with time-dependent inputs.

Parametric Topological Entropy for Multivalued Maps and Differential Inclusions with Nonautonomous Impulses

The main purpose of this paper is to investigate a parametric topological entropy for impulsive differential inclusions on tori. In this way, besides other matters, we would like to extend our recent results concerning impulsive differential equations as well as those on “nonparametric” topological entropy to impulsive differential inclusions. Parametric topological entropy, which is usually called a topological entropy for nonautonomous dynamical systems, is considered here via the compositions of associated multivalued Poincaré translation operators with the single-valued time-dependent impulsive maps. On compact polyhedra and, in particular on tori, parametric topological entropy for families of admissible multivalued maps can be estimated from below by means Ivanov-type inequality in terms of the asymptotic Nielsen and Lefschetz numbers which are, unlike the topological entropy, homotopy invariants. In the scalar case, an effective criterion for a positive parametric topological entropy can be given by topological degree arguments for equi-continuous impulsive maps. In a single-valued nonparametric case, a positive topological entropy usually signifies topological chaos. Some simple illustrative examples are provided.