Pullback Attracting Inertial Manifolds for Nonautonomous Dynamical Systems

In this paper we present an abstract approach to inertial manifolds for nonautonomous dynamical systems. Our result on the existence of inertial manifolds requires only two geometrical assumptions, called cone invariance and squeezing property, and two additional technical assumptions, called boundedness and coercivity property. Moreover we give conditions which ensure that the global pullback attractor is contained in the inertial manifold. In the second part of the paper we consider special nonautonomous dynamical systems, namely processes (or two-parameter semi-flows). As a first application of our abstract approach and for reason of comparison with known results we verify the assumptions for semilinear nonautonomous evolution equations whose linear part satisfies an exponential dichotomy condition and whose nonlinear part is globally bounded and globally Lipschitz.     

Convergent families of approximate inertial manifolds for nonautonomous evolution equations

I.IntroduCtionAtpresent.autollomousinfillitedimensionaldynamicalsystemshavebeenthoroughlystLldicdilltllcory.andwidelyappliedinpracticel'-'1.Forthe11onautonomouscase,[5--91havesttldicd1ilocxistenceanddimensionestimateofattractorsofnonautonomouscase;[12].hasconsideredtileexislellceofinertialmanifolds.Theoretically,inertialmanifoldisaveryusefulInethodtodiscussthelongtimebehaviorofthesolutionstononautonomousinfinitedimcnsiollaldynamicalsystems.Butitcannotbeexpressedexplicitly.Soitisnotconvenient…     

INERTIAL MANIFOLDS FOR NONAUTONOMOUS INFINITE DIMENSIONAL DYNAMICAL SYSTEMS

I.IntroductionInthelasttwodecades,thetheoriesofan'qnomousinfinitedimensionaldynamicalsystemshavebeenthoroughlystudiedandsystematicallyimprovedl'--'l.Comparatively,thestudiesofnonautonomousonesincreaseslowly.Themaindifficultyliesinthatthesemiflowsgeneratedbythesolutionstoautonomouscasesatisfythesemigroupproperty.whilethoseofnonautonomousonesdonot.Sothemethodsusedtostudytheautonomouscasecan'tbeappropriatefornonautonomouscase.Anditrequiresustoestablisll11on'theoriesandmethods.[5--91havediscussed…     

Inertial manifolds for nonautomous infinite dimensional dynamical systems

In this paper, the long time behavior of nonautonomous infinite dimensional dynamical systems is discussed. Under the spectral gap condition, It is proved that there exist inertial manifolds for a class of nonautonomous evolution equations. Project supported by the National Natural Science Foundation of China     

关于近似惯性流形及其数值方法的研究

本文简述了近年来在无穷维动力系统研究中一些数学理论的进展状况,主要目的是结合二维Ｎａｖｉｅｒ－Ｓｔｏｋｅｓ方程讨论近似惯性流形及其数值方法的构造和意义．      

Smooth Stable and Unstable Manifolds for Stochastic Evolutionary Equations

Invariant manifolds are fundamental tools for describing and understanding nonlinear dynamics. In this paper, we present a theory of stable and unstable manifolds for infinite dimensional random dynamical systems generated by a class of stochastic partial differential equations. We first show the existence of Lipschitz continuous stable and unstable manifolds by the Lyapunov–Perrons method. Then, we prove the smoothness of these invariant manifolds.Dedicated to Professor Shui-Nee Chow on the occasion of his 60th birthday.     

Invariant Manifolds for Random Dynamical Systems with Slow and Fast Variables

We consider random dynamical systems with slow and fast variables driven by two independent metric dynamical systems modeling stochastic noise. We establish the existence of a random inertial manifold eliminating the fast variables. If the scaling parameter tends to zero, the inertial manifold tends to another manifold which is called the slow manifold. We achieve our results by means of a fixed point technique based on a random graph transform. To apply this technique we need an asymptotic gap condition.      

Approximate inertial manifolds for reaction-diffusion equations in high space dimension

The concept of approximate inertial manifolds was introduced by Foiaset al. (1987) in the case of the two-dimensional Navier-Stokes equations. These manifolds are finite dimensional smooth manifolds such that the orbits enter a very thin neighborhood of the manifold after a transient time; this concept replaces the one of inertial manifold when either an inertial manifold does not exist or its existence is not known. Our aim in this paper is to prove that approximate inertial manifolds exist for reaction-diffusion equations in high space dimension by opposition with exact inertial manifolds whose existence has only been proved in low dimension and for which nonexistence results have been obtained in space dimensionn=4.     

Small Delay Inertial Manifolds Under Numerics: A Numerical Structural Stability Result

In this paper we formulate a numerical structural stability result for delay equations with small delay under Euler discretization. The main ingredients of our approach are the existence and smoothness of small delay inertial manifolds, the C ¹-closeness of the small delay inertial manifolds and their numerical approximation and M.-C. Li's recent result on numerical structural stability of ordinary differential equations under the Euler method.     

The research of longtime dynamic behavior in weakly damped forced KdV equation

I.IntroductionTheKorteweg-deVries(KdV)equationul uu, u.;:=o(l.1)wasinitiallyder1vedasamodelforonedirectionallongwaterwavesofsmal1amplitudepropagatinginachannel.SincetheworkofKorteweganddeVries,ithasbeenshownthatthisequationoccursinalargevarietyofphysicals…     

A Smoothness Theorem for Invariant Fiber Bundles

Invariant fiber bundles are the generalization of invariant manifolds from discrete dynamical systems (mappings) to non-autonomous difference equations. In this paper we present a self-contained proof of their existence and smoothness. Our main result generalizes the so-called Hadamard–Perron-Theorem for time-dependent families of pseudo-hyperbolic mappings from the finite-dimensional invertible to the infinite-dimensional non-invertible case.     

Stability for manifolds of equilibrium states of fractional generalized Hamiltonian systems

For a fractional generalized Hamiltonian system, in terms of Riesz derivatives, stability theory for the manifolds of equilibrium states is presented. The gradient representation and second order gradient representation of a fractional generalized Hamiltonian system are studied, and the conditions under which the system can be considered as a gradient system and a second order gradient system are given, respectively. Then, equilibrium equations, disturbance equations, and first approximate equations of a fractional generalized Hamiltonian system are obtained. A theorem for the stability of the manifolds of equilibrium states of the general autonomous system is used to a fractional generalized Hamiltonian system, and three propositions on the stability of the manifolds of equilibrium states of the system are investigated. As the special cases of this article, the conditions which a fractional generalized Hamiltonian system can be reduced to a generalized Hamiltonian system, a fractional Hamiltonian system and a Hamiltonian system are given, respectively, and the stability theory for the manifolds of equilibrium states of these systems are obtained. Further, a fractional dynamical system and a fractional Volterra model of the three species groups are given to illustrate the method and results of the application. Finally, by using the method in this paper, we construct a new kind of fractional dynamical model, i.e. the fractional Hénon–Heiles model, and we study its stability of the manifolds of equilibrium states.     

Dynamical Spectrum in Random Dynamical Systems

Dynamical spectrum is a concept in terms of exponential dichotomy. The theory of dynamical spectrum, due to Sacker and Sell, plays important roles in many fields of dynamical systems and differential equations. Noticing its significance and importance, we study in this paper the theory of dynamical spectrum for some general random dynamical systems. More precisely, after introducing a random version of the concept of exponential dichotomy, by using some methods and techniques from dynamical systems and ergodic theory, under some general integrability conditions, we establish the dynamical spectral decomposition theorem in framework of random dynamical systems, which can be regarded as a random version of the deterministic dynamical systems due to Sacker and Sell. In our result, the dynamical spectral intervals and the corresponding spectral subbundles will be given.     

Principal Lyapunov Exponents and Principal Floquet Spaces of Positive Random Dynamical Systems. III. Parabolic Equations and Delay Systems

This is the third part in a series of papers concerned with principal Lyapunov exponents and principal Floquet subspaces of positive random dynamical systems in ordered Banach spaces. The current part focuses on applications of general theory, developed in the authors’ paper Mierczyński and Shen (Trans Am Math Soc 365(10):5329–5365, 2013), to positive continuous-time random dynamical systems on infinite dimensional ordered Banach spaces arising from random parabolic equations and random delay systems. It is shown under some quite general assumptions that measurable linear skew-product semidynamical systems generated by random parabolic equations and by cooperative systems of linear delay differential equations admit measurable families of generalized principal Floquet subspaces, and generalized principal Lyapunov exponents.     

Constrained controllability of semilinear systems with delays

In the paper, finite-dimensional dynamical control systems described by semilinear ordinary differential state equations with multiple point time-variable delays in control are considered. Using a generalized open mapping theorem, sufficient conditions for constrained local relative controllability are formulated and proved. It is generally assumed that the values of admissible controls are in a convex and closed cone with the vertex at zero. The special case of constant multiple point delays is also discussed. Moreover, some remarks and comments on the existing results for controllability of nonlinear and semilinear dynamical systems are also presented.     

A sharp condition for existence of an inertial manifold

It is shown that a perturbation argument that guarantees persistence of inertial (invariant and exponentially attracting) manifolds for linear perturbations of linear evolution equations applies also when the perturbation is nonlinear. This gives a simple but sharp condition for existence of inertial manifolds for semi-linear parabolic as well as for some nonlinear hyperbolic equations. Fourier transform of the explicitly given equation for the tracking solution together with the Plancherel's theorem for Banach valued functions are used.     

Optimal bounded control for nonlinear stochastic smart structure systems based on extended Kalman filter

An optimal bounded control strategy for smart structure systems as controlled Hamiltonian systems with random excitations and noised observations is proposed. The basic dynamic equations for a smart structure system with smart sensors and actuators are firstly given. The nonlinear stochastic control system with noised observations is then obtained from the simplified smart structure system, and the system is expressed by generalized Hamiltonian equations with control, random excitation and dissipative forces. The optimal control problem for nonlinear stochastic systems with noised observations includes two parts: optimal state estimation and optimal response control based on estimated states, which are coupled each other. The probability density of optimally estimated systems has generally infinite dimensions based on the separation theorem. The proposed optimal control strategy gives an approximate separate solution. First, the optimally estimated system state is determined by the observations based on the extended Kalman filter, and the estimated nonlinear system with controls and stochastic excitations is obtained which has finite-dimensional probability density. Second, the dynamical programming equation for the estimated system is determined based on the stochastic dynamical programming principle. The control boundedness due to actuator saturation is considered, and the optimal bounded control law is obtained by the programming equation with the bounded control constraint. The optimal control depends on the estimated system state which is determined by noised observations. The proposed optimal bounded control strategy is finally applied to a single-degree-of-freedom nonlinear stochastic system with control and noised observation. The remarkable vibration control effectiveness is illustrated with numerical results. Thus the proposed optimal bounded control strategy is promising for application to nonlinear stochastic smart structure systems with noised observations.     

Invariant Manifolds for Parabolic Partial Differential Equations on Unbounded Domains

In this paper finite-dimensional invariant manifolds for nonlinear parabolic partial differential equations of the form

Continuity of Dynamical Structures for Nonautonomous Evolution Equations Under Singular Perturbations

In this paper we study the continuity of invariant sets for nonautonomous infinite-dimensional dynamical systems under singular perturbations. We extend the existing results on lower-semicontinuity of attractors of autonomous and nonautonomous dynamical systems. This is accomplished through a detailed analysis of the structure of the invariant sets and its behavior under perturbation. We prove that a bounded hyperbolic global solutions persists under singular perturbations and that their nonlinear unstable manifold behave continuously. To accomplish this, we need to establish results on roughness of exponential dichotomies under these singular perturbations. Our results imply that, if the limiting pullback attractor of a nonautonomous dynamical system is the closure of a countable union of unstable manifolds of global bounded hyperbolic solutions, then it behaves continuously (upper and lower) under singular perturbations.     

Differentiability of Parabolic Semi-Flows in Lp-Spaces and Inertial Manifolds

We consider the semi-flow defined by semi-linear parabolic equations in L ^p-spaces and study the differentiable dependence on the initial data. Together with a spectral gap condition, this implies the existence of inertial manifolds in arbitrary space dimensions. The spectral gap condition is satisfied by the Laplace–Beltrami operator for a class of manifolds. The simplest examples are products of spheres.