Discretization, inflation and perturbation of attractors

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Grüne, Lars ; Kloeden, Peter E.:
Discretization, inflation and perturbation of attractors.
Bayreuth , 2001

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Abstract

The basic issues concerning the effect of discretization or perturbation on autonomous attractors are now quite well understood. For nonautonomous systems matters are, however, considerably more complicated as solutions now depend explicitly on both the initial and the current time, so limiting objects need not exist in current time or be invariant, the semigroup evolutionary property no longer holds, and the concept of an attractor for autonomous systems is generally too restrictive.Nonautonomous systems are ubiquitous. They are easily obtained by including time variation in the vector field of an autonomous differential equation and also arise naturally without an underlying autonomous model. Moreover, they cannot be entirely avoided when one is interested primarily in a particular autonomous system, since perturbations and noise terms are more realistically time dependent, while numerical schemes with variable step size are essentially nonautonomous difference equations even when the underlying differential equation is autonomous. This Chapter begins with a brief review of results for the autonomous case and more recent ideas on inflated autonomous attractors. The cocycle formalism for a nonautonomous system and the concepts of pullback convergence and pullback attractors in such systems are then outlined. Results on the existence of pullback attractors and of Lyapunov functions characterizing pullback attractors are presented, the formulation of a numerical scheme with variable time steps as a discrete time cocycle system is discussed and the comparison of numerical and original pullback attractors considered, at least in special cases, along with the inflation of pullback attractors. Finally, some open questions and desirable future developments are mentioned.