Smoothness of invariant manifolds for nonautonomous equations. (English) Zbl 1098.34041
The authors investigate the smoothness of invariant manifolds for nonlinear equations of the form
\[ v'=A(t)v+f(t,v) \]
with zero solution under the weakest assumption on the linear part \(v' =A(t)v\), namely the existence of a nonuniform exponential dichotomy, e.g., for stable manifolds it is enough to assume that there exist negative Lyapunov exponents. The paper builds on [J. Differ. Equations 221, No. 1, 58–90 (2006; Zbl 1098.34036)] by the authors and proves \(C^1\) smoothness of the invariant manifolds of the trivial solution (cf. also with the authors [Nonlinearity 18, No. 5, 2373–2390 (2005; Zbl 1089.34040)] for higher regularity).
\[ v'=A(t)v+f(t,v) \]
with zero solution under the weakest assumption on the linear part \(v' =A(t)v\), namely the existence of a nonuniform exponential dichotomy, e.g., for stable manifolds it is enough to assume that there exist negative Lyapunov exponents. The paper builds on [J. Differ. Equations 221, No. 1, 58–90 (2006; Zbl 1098.34036)] by the authors and proves \(C^1\) smoothness of the invariant manifolds of the trivial solution (cf. also with the authors [Nonlinearity 18, No. 5, 2373–2390 (2005; Zbl 1089.34040)] for higher regularity).
Reviewer: Stefan Siegmund (Frankfurt)
MSC:
| 34D09 | Dichotomy, trichotomy of solutions to ordinary differential equations |
| 34C30 | Manifolds of solutions of ODE (MSC2000) |
| 37C10 | Dynamics induced by flows and semiflows |
| 37D10 | Invariant manifold theory for dynamical systems |
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