Infinite-dimensional Dynamical Systems Research Articles

Existence of Random Attractors for a Stochastic Strongly Damped Plate Equations with Multiplicative Noise

In this article, we study the asymptotic dynamics of a stochastic strongly damped plate system with homogeneous Neumann boundary conditions and multiplicative noise. First, we investigate the existence and uniqueness of solutions in infinite-dimensional dynamical systems using the notion of mild solutions, and then we examine the presence of a bounded absorbing set. Finally, we investigate the asymptotic compactness by using the decomposition technique to prove the existence of a random attractor.

Dynamics of a three-molecule autocatalytic Schnakenberg model with cross-diffusion: Turing patterns of spatially homogeneous Hopf bifurcating periodic solutions

<abstract><p>In this paper, a three-molecule autocatalytic Schnakenberg model with cross-diffusion is established, the instability of bifurcating periodic solutions caused by diffusion is studied, that is, diffusion can destabilize the stable periodic solutions of the ordinary differential equation (ODE) system. First, utilizing the local Hopf bifurcation theory, the central manifold theory, the normal form method and the regular perturbation theory of the infinite dimensional dynamical system, the stability of periodic solutions for the ODE system is discussed. Second, for this model, according to the implicit function existence theorem and Floquet theory, the Turing instability of spatially homogeneous Hopf bifurcating periodic solutions is studied. It is proved that the otherwise stable Hopf bifurcating periodic solutions in the ODE system produces Turing instability in the Schnakenberg model with cross-diffusion. Finally, through numerical simulations, it is verified that Turing instability of periodic solutions is determined by cross-diffusion rates.</p></abstract>

On Evolving Natural Curvature for an Inextensible, Unshearable, Viscoelastic Rod

We formulate and consider the problem of an inextensible, unshearable, viscoelastic rod, with evolving natural configuration, moving on a plane. We prove that the dynamic equations describing quasistatic motion of an Eulerian strut, an infinite dimensional dynamical system, are globally well-posed. For every value of the terminal thrust, these equations contain a smooth embedded curve of static solutions (equilibrium points). We characterize the spectrum of the linearized equations about an arbitrary equilibrium point, and using this information and a convergence result for dynamical systems due to Brunovský and Polácik, we prove that every solution to the quasistatic equations of motion converges to an equilibrium point as time goes to infinity.

Asymptotic Properties of the Semigroup Generated by a Continuous Interval Map

The article's purpose is twofold. First, we wish to draw attention to the insufficiently known field of continuous-time difference equations. These equations are paradigmatic for modeling complexity and chaos. Even the simplest equation , easily leads to complex dynamics, its solutions are perfectly suited to simulate strong nonlinear phenomena such as large-to-small cascades of structures, intermixing, formation of fractals, etc. Second, in the main body of the article we present a small but very important part of the theory behind the above equation marked by . Just as the discrete-time analog of this equation induces the one-dimensional dynamical system on some interval , so the equation induces the infinite-dimensional dynamical system on the space of functions . In the latter case, not only are the long-term behaviours of solutions critically dependent on the limit behaviour of the sequence (as in the discrete case) but also on the internal structure of as . Assuming to be continuous, we consider the iterations of as the semigroup generated by on the space of continuous maps, and introduce the notion of a limit semigroup for in a wider map space in order to investigate asymptotic properties of . We construct a limit semigroup in the space of upper semicontinuous maps. This enables us to describe both of the aforementioned aspects of our interest around the iterations of .

Solving Cubic Matrix Equations Arising in Conservative Dynamics

In this paper we consider the spatial semi-discretization of conservative PDEs. Such finite dimensional approximations of infinite dimensional dynamical systems can be described as flows in suitable matrix spaces, which in turn leads to the need to solve polynomial matrix equations, a classical and important topic both in theoretical and in applied mathematics. Solving numerically these equations is challenging due to the presence of several conservation laws which our finite models incorporate and which must be retained while integrating the equations of motion. In the last thirty years, the theory of geometric integration has provided a variety of techniques to tackle this problem. These numerical methods require solving both direct and inverse problems in matrix spaces. We present three algorithms to solve a cubic matrix equation arising in the geometric integration of isospectral flows. This type of ODEs includes finite models of ideal hydrodynamics, plasma dynamics, and spin particles, which we use as test problems for our algorithms.

Dynamical Symmetry Breaking of Infinite-Dimensional Stochastic System

The mapping relationship between the symmetry and the conserved quantity inspired researchers to seek the conserved quantity from the viewpoint of the symmetry for the dynamic systems. However, the symmetry breaking in the dynamic systems is more common than the symmetry in the engineering. Thus, as the supplement of our previous work on the symmetry breaking of infinite-dimensional deterministic dynamic systems, the dynamical symmetry breaking of infinite-dimensional stochastic systems is discussed in this paper. Following a brief review of the stochastic (multi-)symplectic for the dynamic system excited by stochastic white noise, two types of stochastic symmetry breaking factors, including the general stochastic excitation and the general stochastic parameters of the infinite-dimensional dynamic systems, are investigated in detail. We find that both the general stochastic excitation and the general stochastic parameters will not break the local multi-symplectic structure of the dynamic systems. However, the local energy conservation law will be broken by the general stochastic excitation, as well as by the stochastic parameters, which are given by the local energy dissipation in this paper. To illustrate the validity of the analytical results, the stochastic vibration of a clamped single-walled carbon nanotube is investigated and the critical condition of the appearance of chaos is obtained. The theoretical results obtained can be used to guide us to construct the structure-preserving method for the stochastic dynamic systems.

Hidden spatiotemporal symmetries and intermittency in turbulence

We consider general infinite-dimensional dynamical systems with the Galilean and spatiotemporal scaling symmetry groups. Introducing the equivalence relation with respect to temporal scalings and Galilean transformations, we define a representative set containing a single element within each equivalence class. Temporal scalings and Galilean transformations do not commute with the evolution operator (flow) and, hence, the equivalence relation is not invariant. Despite of that, we prove that a normalized flow with an invariant probability measure can be introduced on the representative set, such that symmetries are preserved in the statistical sense. We focus on hidden symmetries, which are broken in the original system but restored in the normalized system. The central motivation and application of this construction is the intermittency phenomenon in turbulence. We show that hidden symmetries yield power law scaling for structure functions, and derive formulas for their exponents in terms of normalized measures. The use of Galilean transformation in the equivalence relation leads to the quasi-Lagrangian description, making the developed theory applicable to the Euler and Navier–Stokes systems.

Second-Kind Equilibrium States of the Kuramoto–Sivashinsky Equation with Homogeneous Neumann Boundary Conditions

In this paper, we consider the boundary-value problem for the Kuramoto–Sivashinsky equation with homogeneous Neumann conditions. The problem on the existence and stability of second-kind equilibrium states was studied in two ways: by the Galerkin method and by methods of the modern theory of infinite-dimensional dynamical systems. Some differences in results obtained are indicated.

Dynamical systems with benign ghosts

We consider finite and infinite-dimensional ghost-ridden dynamical systems whose Hamiltonians involve nonpositive definite kinetic terms. We point out the existence of three classes of such systems where the ghosts are benign, i.e., systems whose evolution is unlimited in time: (i) systems obtained from the variation of bounded-motion systems; (ii) systems describing motions over certain Lorentzian manifolds and (iii) higher-derivative models related to certain modified Korteweg-de Vries equations.

Exponential attractors for two-dimensional nonlocal diffusion lattice systems with delay

<p style='text-indent:20px;'>In this paper, we study the long term dynamical behavior of a two-dimensional nonlocal diffusion lattice system with delay. First some sufficient conditions for the construction of an exponential attractor are presented for infinite dimensional autonomous dynamical systems with delay. Then, the existence of exponential attractors for the two-dimensional nonlocal diffusion delay lattice system is established by using the new method of tail-estimates of solutions and overcoming the difficulties caused by the nonlocal diffusion operator and the multi-dimensionality.</p>

Estimate of the attractive velocity of attractors for some dynamical systems

In this paper, we first prove an abstract theorem on the existence of polynomial attractors and the concrete estimate of their attractive velocity for infinite-dimensional dynamical systems, then apply this theorem to a class of wave equations with nonlocal weak damping and anti-damping in case that the nonlinear term~$f$~is of subcritical growth.

The effect of self-limiting on the prevention and control of the diffuse COVID-19 epidemic with delayed and temporal-spatial heterogeneous

BackgroundThe global spread of the novel coronavirus pneumonia is still continuing, and a new round of more serious outbreaks has even begun in some countries. In this context, this paper studies the dynamics of a type of delayed reaction-diffusion novel coronavirus pneumonia model with relapse and self-limiting treatment in a temporal-spatial heterogeneous environment.MethodsFirst, focus on the self-limiting characteristics of COVID-19, incorporate the relapse and self-limiting treatment factors into the diffusion model, and study the influence of self-limiting treatment on the diffusion of the epidemic. Second, because the traditional Lyapunov stability method is difficult to determine the spread of the epidemic with relapse and self-limiting treatment, we introduce a completely different method, relying on the existence conditions of the exponential attractor of our newly established in the infinite-dimensional dynamic system to determine the diffusion of novel coronavirus pneumonia. Third, relapse and self-limiting treatment have led to a change in the structure of the delayed diffusion COVID-19 model, and the traditional basic reproduction number R_0 no longer has threshold characteristics. With the help of the Krein-Rutman theorem and the eigenvalue method, we studied the threshold characteristics of the principal eigenvalue and found that it can be used as a new threshold to describe the diffusion of the epidemic.ResultsOur results prove that the principal eigenvalue uplambda ^{*} of the delayed reaction-diffusion COVID-19 system with relapse and self-limiting treatment can replace the basic reproduction number R_0 to describe the threshold effect of disease transmission. Combine with the latest official data and the prevention and control strategies, some numerical simulations on the stability and global exponential attractiveness of the diffusion of the COVID-19 epidemic in China and the USA are given.ConclusionsThrough the comparison of numerical simulations, we find that self-limiting treatment can significantly promote the prevention and control of the epidemic. And if the free activities of asymptomatic infected persons are not restricted, it will seriously hinder the progress of epidemic prevention and control.

Chaotic vibrations of 3D linear hyperbolic PDEs with linear perturbations of superlinear boundary conditions

This article studies the chaotic vibrations of the infinite-dimensional dynamical systems governed by linear hyperbolic partial differential equations (PDEs) in three-dimensional (3D) space, where the boundary conditions include two linear perturbations of superlinear types. A rigorous mathematical theorem that guarantees the occurrence of chaos of such systems is obtained. The main theme of this article is the advancement of existing chaos to 3D hyperbolic PDEs. As applications, two examples are provided for showing the effectiveness of the theoretical chaotic results.

Attracting sets for one class of asymptotically compact systems with pulsed perturbation

The authors consider the pulsed dynamical systems generated by evolutionary processes. The trajectories of these processes undergo the pulsed perturbation when the energy functional reaches some fixed limit value. The generalization of the classical theory of global attractors of infinite dimensional dynamical systems in case of systems with impulse actions is carried out. It is established that for the dissipative pulsed dynamical system generated by the asymptotically compact semigroup, there exists a uniform attractor, i.e., a compact uniformly attracting set, minimal among all such sets in the phase space of the system. The result is applied to the weakly nonlinear wave equation with dissipation, the trajectories of which are subjected to impulsive perturbations upon attainment of a certain fixed subset in the phase space, so called the impulse set.

A Reduced Order Model Approach to Inverse Scattering in Lossy Layered Media

We introduce a reduced order model (ROM) methodology for inverse electromagnetic wave scattering in layered lossy media, using data gathered by an antenna which generates a probing wave and measures the time resolved reflected wave. We recast the wave propagation problem as a passive infinite-dimensional dynamical system, whose transfer function is expressed in terms of the measurements at the antenna. The ROM is a low-dimensional dynamical system that approximates this transfer function. While there are many possible ROM realizations, we are interested in one that preserves passivity and in addition is: (1) data driven (i.e., is constructed only from the measurements) and (2) it consists of a matrix with special sparse algebraic structure, whose entries contain spatially localized information about the unknown dielectric permittivity and electrical conductivity of the layered medium. Localized means in the intervals of a special finite difference grid. The main result of the paper is to show with analysis and numerical simulations that these unknowns can be extracted efficiently from the ROM.

Coupling Dynamic Behavior of Space Flexible Hollow Beam

The coupling between the plane motion and transverse vibration is a challenge for the dynamic analysis of space structures. The coupling dynamic behavior of the on-orbit space flexible hollow beam is studied in this paper. As a common configuration of spacecraft and their components, the space structure with a large slenderness ratio and hollow structure is considered. With weak damping and variable structural parameters, the dynamic model is derived for the space flexible hollow beam by the Hamilton least-action principle. Then, a complex structure-preserving method that has been proved to be feasible for the infinite-dimensional dynamic system is constructed. Subsequently, effects of the hollow structure, weak damping and variable structural parameters on the orbit-attitude-vibration coupling dynamics are discussed comprehensively. Meanwhile, the evolutions of energy of the coupling dynamic system with/without damping are reproduced. The results show that the effects of the hollow structure, damping and variable structural parameters are all notable for the dynamic behavior of the space coupling dynamic systems. The constructed complex structure-preserving method can effectively characterize the coupling dynamic behavior and energy evolution of the space system. The numerical examples may serve as practical references for the design of space flexible hollow structures.

On the Morse theory of attractors: A functional approach

This paper aims at developing a functional approach to investigate the Morse structures of attractors for infinite-dimensional dynamical systems generated by PDEs. Let S(t) be a semiflow on a complete metric space X, and A an attractor of S(t) with a Morse decomposition M. We first construct some good Morse–Lyapunov functions for M which seem to be of independent interest in their own right. Then we establish some fundamental deformation lemmas. Based on these results, we finally introduce the concept of critical groups for Morse sets and prove Morse inequalities and equations. The problem of how to calculate Morse equations of attractors by using a natural Morse–Lyapunov function which may be defined only on a dense subspace of X is also addressed, and an illustrating example of parabolic equation is presented.

Local stable and unstable sets for positive entropy C1 dynamical systems

For any C1 diffeomorphism on a smooth compact Riemannian manifold that admits an ergodic measure with positive entropy, a lower bound of the Hausdorff dimension for the local stable and unstable sets is given in terms of the measure-theoretic entropy and the maximal Lyapunov exponent. The main line of our approach to this result is under the setting of topological dynamical systems, which is also applicable to infinite-dimensional C1 dynamical systems.

The Cosmological Semiclassical Einstein Equation as an Infinite-Dimensional Dynamical System

We develop a comprehensive framework in which the existence of solutions to the semiclassical Einstein equation (SCE) in cosmological spacetimes is shown. Different from previous work on this subject, we do not restrict to the conformally coupled scalar field and we admit the full renormalization freedom. Based on a regularization procedure, which utilizes homogeneous distributions and is equivalent to Hadamard point splitting, we obtain a reformulation of the evolution of the quantum state as an infinite-dimensional dynamical system with mathematical features that are distinct from the standard theory of infinite-dimensional dynamical systems (e.g., unbounded evolution operators). Nevertheless, applying methods closely related to Ovsyannikov’s method, we show existence of maximal/global solutions to the SCE for vacuum-like states and of local solutions for thermal-like states. Our equations do not show the instability of the Minkowski solution described by other authors.

Two-component symplectic universal characters and integrable hierarchies

In this paper, we first construct a symplectic Schur function solution to a newly defined two-component symplectic Kadomtsev–Petviashvili hierarchy. As a generalization of a two-component symplectic Schur function, we construct two-component symplectic universal characters which satisfy quadratic equations in an infinite-dimensional integrable dynamic system called a two-component symplectic universal character hierarchy. Then, we define a modified symplectic universal character hierarchy whose tau function can be represented by free fermions in Clifford algebras.