Perturbation Theory for Infinite Dimensional Dynamical Systems

Abstract

When considering the effect of perturbations on initial value problems over long time intervals it is not possible, in general, to uniformly approximate individual trajectories. This is because well-posed initial value problems allow exponential divergence of trajectories and this fact is reflected in the error bound relating trajectories of the perturbed and unperturbed problems. In order to interpret data obtained from numerical simulations over long time intervals, and from other forms of perturbations, it is hence often necessary to ask different questions concerning the behavior as the approximation is refined. One possibility, which we concentrate on in this review, is to study the effect of perturbation on sets which are invariant under the evolution equation. Such sets include equilibria, periodic solutions, stable and unstable manifolds, phase portraits, inertial manifolds and attractors; they are crucial to the understanding of long-time dynamics. An abstract semilinear evolution equation in a Hilbert space X is considered, yielding a semigroup S(t) actlng on a subspace V of X. A general class of perturped semigroups S^h(t) are considered which are C^1 close to S(t) uniformly on bounded subsets of V and time intervals [t_1, t_2] with 0 < t_1 < t_2 < ∞. A variety of perturbed problems are shown to satisfy these approximation properties. Examples include a Galerkin method based on the eigenfunctions of the linear part of the abstract sectorial evolution equation, a backward Euler approximation of the same equation and a singular perturbation of the Cahn-Hilliard equation arising from the phase-field model of phase transitions. The invariant sets of S(t) and S^h(t) are compared and convergence properties established.