Infinite-dimensional Dynamical Systems Research Articles

Quasi-periodic solutions for a class of beam equation system

In this paper, we establish an abstract infinite dimensional KAM theorem. As an application, we use the theorem to study the higher dimensional beam equation system \begin{document}$ \left\{ \begin{array}{lll} u_{1tt}+ \Delta^2 u_1 +\sigma u_1 +u_1u_2^2 & = & 0 \\ &&\\ u_{2tt}+ \Delta^2 u_2 +\mu u_2 +u_1^2 u_2 & = & 0 \end{array} \right. $\end{document} under periodic boundary conditions, where $ 0<\sigma \in [ \sigma_1,\sigma_2 ], $ $ 0<\mu\in [ \mu_1,\mu_2 ] $ are real parameters. By establishing a block-diagonal normal form, we obtain the existence of a Whitney smooth family of small amplitude quasi-periodic solutions corresponding to finite dimensional invariant tori of an associated infinite dimensional dynamic system.

Inverse scattering transform for the complex reverse space-time nonlocal modified Korteweg-de Vries equation with nonzero boundary conditions and constant phase shift.

Recently, the complex reverse space-time (RST) nonlocal modified Korteweg-de Vries equation (mKdV) was introduced and shown to be an integrable infinite-dimensional dynamical system. The inverse scattering transform (IST) for zero boundary condition was studied by Ablowitz and Musslimani in 2016. In this paper, the IST for the complex RST nonlocal mKdV equation with nonzero boundary conditions at infinity is presented. The direct and inverse scattering problems are analyzed. The method to carry out the inverse problem employs two Riemann surfaces associated with square root branch points in the eigenfunctions/scattering data, which is more complicated than that for the zero boundary condition. Four cases are considered, special soliton solutions are discussed, and an explicit 1-soliton and two 2-soliton are found for three cases. In another case, there are no solitons.

Spatiotemporal Patterns Induced by Hopf Bifurcations in a Homogeneous Diffusive Predator-Prey System

In this paper, we consider a diffusive predator-prey system where the prey exhibits the herd behavior in terms of the square root of the prey population. The model is supposed to impose on homogeneous Neumann boundary conditions in the bounded spatial domain. By using the abstract Hopf bifurcation theory in infinite dimensional dynamical system, we are capable of proving the existence of both spatial homogeneous and nonhomogeneous periodic solutions driven by Hopf bifurcations bifurcating from the positive constant steady state solutions. Our results allow for the clearer understanding of the mechanism of the spatiotemporal pattern formations of the predator-prey interactions in ecology.

Long-Time Dynamics of Balakrishnan–Taylor Extensible Beams

This paper is concerned with well-posedness and long-time dynamics for a class extensible beams with nonlocal Balakrishnan–Taylor and frictional damping. The related model describes vibrations in nonlinear extensible beams arising in connection with models of oscillation in pipes and supersonic panel flutter. Our main results feature the study of the nonlinear dynamical system generated by the problem. The main novelty is to explore the global $$L^q$$ -regularity ( $$q\ge 2$$ ) in time of the nonlocal Balakrishnan–Taylor term and show how it generates a dissipative term that plays an important role in the asymptotic behavior of solutions, mainly in what concerns to achieve the useful property of quasi-stability in the theory of infinite-dimensional dynamical systems.

A Variation Evolving Method for Optimal Control Computation

A new method, which originates from the continuous-time dynamics stability theory in the control field, is proposed for the optimal control computation. By introducing a virtual dimension, the variation time, an infinite-dimensional dynamic system that describes the variation motion of variables is derived from the optimal control problem based on the Lyapunov principle. The optimal solution is its stable equilibrium point and will be obtained in an asymptotically evolving way. Through this method, the intractable optimal control problems are transformed to the initial-value problems and they may be solved with mature ordinary differential equation numerical integration methods. Especially, the deduced dynamic system is globally stable, so any initial value may evolve to an extremal solution ultimately.

Simplified conditions of initial observability for infinite-dimensional second-order damped dynamical systems

In the paper, the initial observability of abstract second-order infinite-dimensional dynamical systems is considered. Two cases of observation operator are explored. First, the observation operator is assumed to be bounded. Then results are extended to more general class of observations related to the observation operators that are assumed to be bounded with respect to square root of elasticity operator which covers the case of some unboundedness in observation. Using the frequency-domain method it is proved, that initial observability of second-order system under discussion can be verified by the initial observability conditions for the corresponding simplified first-order system. General results are then applied for initial observability investigation of infinite dimensional systems described by partial differential equations modelling flexible mechanical structures. Some special cases are also discussed and practical comments and remarks are given. The paper extends earlier results on initial observability for a class of infinite-dimensional second-order abstract dynamical systems.

The generalized Solow model

A modification of the known mathematical macroeconomics model of macroeconomics are studied in which a delay factor is presumed. This led to the replacement of the ordinary differential equation, which cannot exhibit periodic cycles on the equations with a deviating argument (functional-differential equations). It was possible to show the existence of periodic solutions that can and are intended to describe the periodic cycles in the market economy.The mathematical portion is based on the application of the modern theory of dynamical systems with an infinite-dimensional space of initial conditions. This will allow us to apply the Andronov-Hopf Theorem for equations with a deviating argument in such a form that the parameters of the cycles are located. We will also apply the well-known Krylov-Bogolyubov algorithm that is extended to infinite-dimensional dynamical systems that is used and reduces the problem to the analysis of the finite-dimensional system of ordinary differential equations-the normal Poincare-Dulac form.

PERIODIC CYCLES IN THE SOLOW MODEL WITH A DELAY EFFECT

The three natural modifications of the known mathematical macroeconomics model of macroeconomics are studied in which a delay factor is presumed. This led to the replacement of the ordinary differential equation, which cannot exhibit periodic cycles on the equations with a deviating argument (functional-differential equations). It was possible to show the existence of periodic solutions that can and are intended to describe the periodic cycles in the market economy in two of the three variants of such changes in the classical form of the model. The mathematical portion is based on the application of the modern theory of dynamical systems with an infinite-dimensional space of initial conditions. This will allow us to apply the Andronov-Hopf Theorem for equations with a deviating argument in such a form that the parameters of the cycles are located. We will also apply the well-known Krylov-Bogolyubov algorithm that is extended to infinite-dimensional dynamical systems that is used and reduces the problem to the analysis of the finite-dimensional system of ordinary differential equations-the normal Poincare-Dulac form.

Numerical dynamics of integrodifference equations: Basics and discretization errors in a C0-setting

Besides being interesting infinite-dimensional dynamical systems in discrete time, integrodifference equations successfully model growth and dispersal of populations with nonoverlapping generations, and are often illustrated by simulations. This paper points towards and initiates a mathematical foundation of such simulations using generic methods to numerically discretize (and solve) integral equations. We tackle basic properties of a flexible class of integrodifference equations, as well as of their collocation and degenerate kernel semi-discretizations on the state space of continuous functions over a compact domain. Moreover, various estimates for the global discretization error are provided. Numerical simulations illustrate and confirm our theoretical results.

Nontrivial Attractors of the Perturbed Nonlinear Schrödinger Equation: Applications to Associative Memory and Pattern Recognition

AbstractComputers are dynamical systems that carry out information processing through their change of state with time. For instance, neural networks such as Hopfield's associative memory are dissipative dynamical systems in a finite dimensional configuration space with attractors that represent stored patterns. In particular, dissipative dynamical systems with an infinite dimensional configuration space are of broad interest and have the possibility to store and restore complex and strongly distorted data structures. In this work, the nonlinear Schrödinger equation (NLSE) with a dissipative perturbation which creates a frictional force acting on soliton is considered. It is shown that the control of the perturbative term allows one to decrease the velocity of soliton to zero and conserve a positive value of its amplitude. The existence of such a frictional force would suggest that the perturbation makes the zero‐velocity solitonic solution of non‐zero amplitude into an attractor for all evolution trajectories whose initial conditions are moving solitons. This allows a way to store information in the infinite dimensional dynamical system governed by NLSE using principles which are completely analogous to that of Hopfield's associative memory. This approach can be realized with solitons in Bose–Einstein condensates and nonlinear optical systems.

Periodic approximation of exceptional Lyapunov exponents for semi-invertible operator cocycles

We prove that for semi-invertible and H\"older continuous linear cocycles $A$ acting on an arbitrary Banach space and defined over a base space that satisfies the Anosov Closing Property, all exceptional Lyapunov exponents of $A$ with respect to an ergodic invariant measure for base dynamics can be approximated with Lyapunov exponents of $A$ with respect to ergodic measures supported on periodic orbits. Our result is applicable to a wide class of infinite-dimensional dynamical systems.

The Numerical Computation of Unstable Manifolds for Infinite Dimensional Dynamical Systems by Embedding Techniques

In this work we extend the novel framework developed by Dellnitz, Hessel-von Molo and Ziessler to the computation of finite dimensional unstable manifolds of infinite dimensional dynamical systems. To this end, we adapt a set-oriented continuation technique developed by Dellnitz and Hohmann for the computation of such objects of finite dimensional systems with the results obtained in the work of Dellnitz, Hessel-von Molo and Ziessler. We show how to implement this approach for the analysis of partial differential equations and illustrate its feasibility by computing unstable manifolds of the one-dimensional Kuramoto-Sivashinsky equation as well as for the Mackey-Glass delay differential equation.

Numerical Dynamics of Integrodifference Equations: Global Attractivity in a $C^0$-Setting

Integrodifference equations are successful and popular models in theoretical ecology to describe spatial dispersal and temporal growth of populations with nonoverlapping generations. In relevant situations, such infinite-dimensional discrete dynamical systems have a globally attractive periodic solution. We show that this property persists under sufficiently accurate spatial (semi-) discretizations of collocation and degenerate kernel-type using linear splines. Moreover, convergence preserving the order of the method is established. This justifies theoretically that simulations capture the behavior of the original problem. Several numerical illustrations confirm our results.

Critical Points of Strichartz Functional

We study a pair of infinite dimensional dynamical systems naturally associated with the study of minimizing/maximizing functions for the Strichartz inequalities for the Schrödinger equation. One system is of gradient type and the other one is a Hamiltonian system. For both systems, the corresponding sets of critical points, their stability, and the relation between the two are investigated. By a combination of numerical and analytical methods we argue that the Gaussian is a maximizer in a class of Strichartz inequalities for dimensions one, two, and three. The argument reduces to verification of an apparently new combinatorial inequality involving binomial coefficients.

Bifurcation and Stability Analysis of Pulsating Solitons

Pulsating soliton solutions bifurcation analysis of the two-dimensional (2D) Complex Swift-Hohenberg equation (CSHE) is presented. The approach is based on a reduction from an infinite-dimensional dynamical dissipative system to a finite-dimensional model. Thanks to the collective variable approach, we investigated the influence of the nonlinear gain and the saturation of the Kerr nonlinearity on the pulsations of the solitons. Research has shown that the transformation between pulsating soliton and fronts can be realized through a series of period-doubling bifurcations. The complete bifurcation diagrams of the total energy have been obtained for a definite range of the nonlinear gain and the saturation of the Kerr nonlinearity values. The detailed analysis reveals that when the saturation of the Kerr nonlinearity increases one-period pulsating solution bifurcates to double-period pulsations. While the increase of the nonlinear gain leads the double-period pulsations to return into one-period pulsation before transforming into a stationary pulsating solitons.

The Kuramoto–Sivashinsky Equation. A Local Attractor Filled with Unstable Periodic Solutions

A periodic boundary value problem is considered for one of the first versions of the Kuramoto–Sivashinsky equation, which is widely known in mathematical physics. Local bifurcations in a neighborhood of spatially homogeneous equilibrium points are studied in the case when they change stability. It is shown that the loss of stability of homogeneous equilibrium points leads to the occurrence of a two-dimensional local attractor on which all solutions are periodic functions of time, except for one spatially inhomogeneous state. The spectrum of frequencies of this family of periodic solutions fills the entire number line, and all of them are unstable in the sense of Lyapunov’s definition in the metric of the phase space (the space of initial conditions) of the corresponding initial boundary value problem. As the phase space, a Sobolev functional space natural for this boundary value problem is chosen. Asymptotic formulas are given for periodic solutions filling the two-dimensional attractor. To analyze the bifurcation problem, analysis methods for an infinite-dimensional dynamical system are used: the integral (invariant) manifold method combined with the methods of the Poincare normal form theory and asymptotic methods. Analyzing the bifurcations for the periodic boundary value problem is reduced to analyzing the structure of the neighborhood of the zero solution to the homogeneous Dirichlet boundary value problem for the equation under consideration.

Bounds on the Hausdorff dimension of random attractors for infinite-dimensional random dynamical systems on fractals

<p style='text-indent:20px;'>We consider a stochastic nonlinear evolution equation where the domain is given by a fractal set. The linear part of the equation is given by a Laplacian defined on the fractal. This equation generates a random dynamical system. The long time behavior is given by an attractor which has a finite Hausdorff dimension. We would like to reveal the connections between upper and lower estimates of this Hausdorff dimension and the geometry of the fractal. In particular, the parameter which determines these bounds is the spectral exponent of the fractal. Especially for the lower estimate we construct a local unstable random Lipschitz manifold.

Centre manifolds for infinite dimensional random dynamical systems

ABSTRACTStochastic centre manifolds theory are crucial in modelling the dynamical behaviour of complex systems under stochastic influences. The existence of stochastic centre manifolds for infinite dimensional random dynamical systems is shown under the assumption of exponential trichotomy. The theory provides a support for the discretisations of nonlinear stochastic partial differential equations with space–time white noise.

Inverse Scattering Transform for the Nonlocal Reverse Space–Time Nonlinear Schrödinger Equation

Nonlocal reverse space–time equations of the nonlinear Schrodinger (NLS) type were recently introduced. They were shown to be integrable infinite-dimensional dynamical systems, and the inverse scattering transform (IST) for rapidly decaying initial conditions was constructed. Here, we present the IST for the reverse space–time NLS equation with nonzero boundary conditions (NZBCs) at infinity. The NZBC problem is more complicated because the branching structure of the associated linear eigenfunctions is complicated. We analyze two cases, which correspond to two different values of the phase at infinity. We discuss special soliton solutions and find explicit one-soliton and two-soliton solutions. We also consider spatially dependent boundary conditions.

Analysis of Integrodifference Equations with a Separable Dispersal Kernel

Integrodifference equations are a class of infinite-dimensional dynamical systems in discrete time that have recently received great attention as mathematical models of population dynamics in spatial ecology. The dispersal of individuals between generations is described by a ‘dispersal kernel’, a probability density function for the distance that an individual moves within a season. Previous authors recognized that the dynamics are reduced to a finite-dimensional problem when the dispersal kernel is separable. We prove some open questions from their work on the dynamics of a single population and then extend the idea to investigate the dynamics of two spatially distributed species in (i) a competitive relation, and (ii) a predator-prey relation. In all cases, we discuss how the dynamics of the population(s) depend on the amount of suitable space that is available to them. We find a number of bifurcations, such as period-doubling sequences and Naimark-Sacker bifurcations, which we illustrate through simulations.