Nonautonomous equations, generalized dichotomies and stable manifolds

Details Nonautonomous equations, generalized dichotomies and stable manifolds Nonautonomous equations, generalized dichotomies and stable manifolds

2009-05-29

arxiv.org/abs/0905.4935v1

Mathematics

Dynamical Systems

16 pages

Scientific paper

Assuming the existence of a general nonuniform dichotomy for the evolution operator of a non-autonomous ordinary linear differential equation in a Banach space, we establish the existence of invariant stable manifolds for the semiflow generated by sufficiently small nonlinear perturbations of the linear equation. The family of dichotomies considered satisfies a general growth rate given by some increasing differentiable function, allows situations for which the classical Lyapunov exponents are zero, and contains the nonuniform exponential dichotomies as a very particular case. In addition we also give explicit examples of linear equations that admit all the possible considered dichotomies.

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