Parameter dependence of stable manifolds under nonuniform hyperbolicity |
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Authors: | Luis Barreira Claudia Valls |
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Institution: | Departamento de Matemática, Instituto Superior Técnico, 1049-001 Lisboa, Portugal |
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Abstract: | For a nonautonomous linear equation v′=A(t)v in a Banach space with a nonuniform exponential dichotomy, we show that the nonlinear equation v′=A(t)v+f(t,v,λ) has stable invariant manifolds Vλ which are Lipschitz in the parameter λ provided that f is a sufficiently small Lipschitz perturbation. Since any linear equation with nonzero Lyapunov exponents has a nonuniform exponential dichotomy, the above assumption is very general. We emphasize that passing from a classical uniform exponential dichotomy to a general nonuniform exponential dichotomy requires a substantially new approach. |
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Keywords: | Nonuniform hyperbolicity Parameters Stable manifolds |
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