Stable invariant manifolds for parabolic dynamics

Abstract

We consider nonautonomous equations v ′ = A ( t ) v in a Banach space that exhibit stable and unstable behaviors with respect to arbitrary growth rates e c ρ ( t ) for some function ρ ( t ) . This corresponds to the existence of a “generalized” exponential dichotomy, which is known to be robust. When ρ ( t ) ≠ t this behavior can be described as a type of parabolic dynamics. We consider the general case of nonuniform exponential dichotomies, for which the Lyapunov stability is not uniform. We show that for any sufficiently small perturbation f of a “generalized” exponential dichotomy there is a stable invariant manifold for the perturbed equation v ′ = A ( t ) v + f ( t , v ) . We also consider the case of exponential contractions, which allow a simpler treatment, and we show that they persist under sufficiently small nonlinear perturbations.