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.     

Taylor Approximation of Integral Manifolds

Integral manifolds generalize invariant manifolds to nonautonomous ordinary differential equations. In this paper, we develop a method to calculate their Taylor approximation with respect to the state space variables. This is of decisive importance, e.g., in nonautonomous bifurcation theory or for an application of the reduction principle in a time-dependent setting.     

Invariant Set and Attractor of Nonautonomous Functional Differential Systems: A Decomposition Approach

In this paper, the invariant set and attractor are addressed for the nonautonomous functional differential systems. An estimation of the existence range of the invariant set and attractor are given by using a decomposition approach and the properties of nonnegative matrices. In addition, an example is given to denote the application of the new results.     

Asymptotic pseudotrajectories and chain recurrent flows,with applications

We present a general framework to study compact limit sets of trajectories for a class of nonautonomous systems, including asymptotically autonomous differential equations, certain stochastic differential equations, stochastic approximation processes with decreasing gain, and fictitious plays in game theory. Such limit sets are shown to be internally chain recurrent, and conversely.     

Invariant manifolds of one class of systems of impulsive differential equations

We consider general problems related to the existence of invariant toroidal sets for linear and weakly nonlinear systems of impulsive differential equations defined in the direct product of an m-dimensional torus and an n-dimensional Euclidean space. We investigate classes of problems for which the conditions for the existence of invariant toroidal manifolds are satisfied.     

Multi-Multiplier Ambient-Space Formulations of Constrained Dynamical Systems, with an Application to Elastodynamics

Various formulations of the equations of motion for both finite- and infinite-dimensional constrained Lagrangian dynamical systems are studied. The different formulations correspond to different ways of enforcing constraints through multiplier fields. All the formulations considered are posed on ambient spaces whose members are unrestricted by the need to satisfy constraint equations, but each formulation is shown to possess an invariant set on which the constraint equations and physical balance laws are satisfied. The stability properties of the invariant set within its ambient space are shown to be different in each case. We use the specific model problem of linearized incompressible elastodynamics to compare properties of three different ambient-space formulations. We establish the well-posedness of one formulation in the particular case of a homogeneous, isotropic body subject to specified tractions on its boundary. Accepted October 11, 2000?Published online April 23, 2001     

Analysis of forced vibrations by nonlinear modes

The combination of Rausher method and nonlinear modes is suggested to analyze the forced vibrations of nonlinear discrete systems. The basis of the Rausher method is iterative procedure. In this case, the analysis of a nonautonomous dynamical system reduces to the multiple solutions of the autonomous ones. As an example, the forced vibrations of shallow arch close to equilibrium position are considered in this paper. The results of the analysis are shown on the frequency response.     

Nonexistence of invariant manifolds in fractional-order dynamical systems

Invariant manifolds are important sets arising in the stability theory of dynamical systems. In this article, we take a brief review of invariant sets. We provide some results regarding the existence of invariant lines and parabolas in planar polynomial systems. We provide the conditions for the invariance of linear subspaces in fractional-order systems. Further, we provide an important result showing the nonexistence of invariant manifolds (other than linear subspaces) in fractional-order systems.

     

Regularity of Invariant Manifolds for Nonuniformly Hyperbolic Dynamics

For any sufficiently small perturbation of a nonuniform exponential dichotomy, we show that there exist invariant stable manifolds as regular as the dynamics. We also consider the general case of a nonautonomous dynamics defined by the composition of a sequence of maps. The proof is based on a geometric argument that avoids any lengthy computations involving the higher order derivatives. In addition, we describe how the invariant manifolds vary with the dynamics.      

Maps of Convex Sets and Invariant Regions¶for Finite-Difference Systems¶of Conservation Laws

General results about maps of convex sets in ? ⁿ are proved. We outline their extensions to an infinite-dimensional context. Such extensions have applications in nonlinear analysis such as in the study of the invariance of convex sets under nonlinear maps. Here, we explore applications only in the finite-dimensional context. More specifically, we apply the general results to the problem of finding sufficient conditions for a region of the state space to be globally or locally invariant under finite-difference schemes applied to systems of conservation laws in several space variables. In particular, we establish a final characterization of the invariant regions under the Lax-Friedrichs scheme and also give sufficient conditions for the local invariance. Further, we give sufficient conditions for the global and local invariance of regions under flux-splitting finite-difference schemes. An example of the multi-dimensional Euler equations for non-isentropic gas dynamics is discussed.     

Hamiltonian systems under small nonautonomous perturbations

Hamiltonian systems under small nonautonomous and aperiodic perturbations are considered. Sufficient conditions under which the first integrals of the unperturbed system vary slightly along the solution to the perturbed system are formulated. Some mechanical systems are considered as examples.     

Variational Principles for Entropies of Nonautonomous Dynamical Systems

In this paper, we first introduce the measure-theoretic entropy for arbitrary Borel probability measure in nonautonomous case. Then we show that there is certain variational relation for nonautonomous dynamical systems.     

Dependence of Topological Conjugacies on Parameters

For impulsive differential equations, we establish the existence of invariant stable manifolds under sufficiently small perturbations of a linear equation. We consider the general case of nonautonomous equations for which the linear part has a nonuniform exponential dichotomy. One of the main advantages of our work is that our results are optimal, in the sense that for vector fields of class C ¹ outside the jumping times, we show that the invariant manifolds are also of class C ¹ outside these times. The novelty of our proof is the use of the fiber contraction principle to establish the smoothness of the invariant manifolds. In addition, using the same approach we can also consider linear perturbations.     

Dissipation and Compact Attractors

We present an approach to the study of the qualitative theory of infinite dimensional dynamical systems. In finite dimensions, most of the success has been with the discussion of dynamics on sets which are invariant and compact. In the infinite dimensional case, the appropriate setting is to consider the dynamics on the maximal compact invariant set. In dissipative systems, this corresponds to the compact global attractor. Most of the time is devoted to necessary and sufficient conditons for the existence of the compact global attractor. Several important applications are given as well as important results on the qualitative properties of the flow on the attractor.     

Stable Manifolds for Impulsive Equations Under Nonuniform Hyperbolicity

For impulsive differential equations, we establish the existence of invariant stable manifolds under sufficiently small perturbations of a linear equation. We consider the general case of nonautonomous equations for which the linear part has a nonuniform exponential dichotomy. One of the main advantages of our work is that our results are optimal, in the sense that for vector fields of class C ¹ outside the jumping times, we show that the invariant manifolds are also of class C ¹ outside these times. The novelty of our proof is the use of the fiber contraction principle to establish the smoothness of the invariant manifolds. In addition, using the same approach we can also consider linear perturbations.     

Singularly Perturbed Ordinary Differential Equations with Nonautonomous Fast Dynamics

Embedding of nonautonomous dynamics in a skew-product flow is employed in the analysis of singularly perturbed equations, where the fast dynamics is time-varying. Uniform convergence of the slow dynamics and statistical convergence of the fast dynamics are established. The limits are characterized in terms of projections of invariant probability measures of the skew-product flow in which the fast dynamics is embedded. These invariant measures are generated by the limiting equations of the original time-dependent process.     

Spatial Independence of Invariant Sets of Parabolic Systems

In this paper we study the -limit sets of semiflows generated by systems of autonomous parabolic systems under Neumann boundary conditions. Under weaker assumptions than previous works, we show the -limit sets consist of space independent solutions. We also prove similar results for shadow systems and some systems with periodic time dependence and calculate the Conley indices of invariant sets.     

Multi-pulse orbits and chaotic dynamics in a nonlinear forced dynamics of suspended cables

The global bifurcations in mode of a nonlinear forced dynamics of suspended cables are investigated with the case of the 1:1 internal resonance. After determining the equations of motion in a suitable form, the energy phase method proposed by Haller and Wiggins is employed to show the existence of the Silnikov-type multi-pulse orbits homoclinic to certain invariant sets for the two cases of Hamiltonian and dissipative perturbation. Furthermore, some complex chaos behaviors are revealed for this class of systems.     

Detecting invariant manifolds, attractors, and generalized KAM tori in aperiodically forced mechanical systems

We show how the recently developed theory of geodesic transport barriers for fluid flows can be used to uncover key invariant manifolds in externally forced, one-degree-of-freedom mechanical systems. Specifically, invariant sets in such systems turn out to be shadowed by least-stretching geodesics of the Cauchy–Green strain tensor computed from the flow map of the forced mechanical system. This approach enables the finite-time visualization of generalized stable and unstable manifolds, attractors and generalized KAM curves under arbitrary forcing, when Poincaré maps are not available. We illustrate these results by detailed visualizations of the key finite-time invariant sets of conservatively and dissipatively forced Duffing oscillators.     

On Invariant Tori for a Stochastic Ito System

By using the Green function, we obtain conditions implying existence of invariant sets for Ito systems that are extensions of dynamical systems on a torus.