Stable invariant manifolds for parabolic dynamics |
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Authors: | Luis Barreira Claudia Valls |
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Affiliation: | Departamento de Matemática, Instituto Superior Técnico, 1049-001 Lisboa, Portugal |
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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 ecρ(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. |
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Keywords: | Invariant manifolds Parabolic dynamics Stability theory |
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