An application of the parameterization method to evolution flows associated to nonautonomous systems

Abstract

Let $M$ be a finite-dimensional complete Riemannian manifold of bounded geometry. We consider a time-dependent vector field $X:\mathbb{R}\times M\to TM$ with $X_t:=X(t,\cdot )\in {\mathfrak{X}}^r(M)$ for all $t\in \mathbb{R}$ that satisfies some further uniform continuity assumptions. We suppose that $X$ admits a globally defined integral curve $\gamma :\mathbb{R}\to M$. To any isolated subset $\Omega$ of the Sacker-Sell spectrum of $t\mapsto \nabla X_t(\gamma (t))$, Sacker-Sell theory associates a subbundle ${\mathcal{N}}^{\Omega }$ of $T_{\gamma }M$ invariant under $\nabla X_t$. Under additional nonresonance conditions on $\Omega$, we prove, by an application of the parameterization method (Cabré et al., 2003), the existence of an invariant topological submanifold ${\mathcal{W}}^{\Omega }\subset \mathbb{R}\times M$ with $C^r$ fibers $W_t^{\Omega }:={\mathcal{W}}^{\Omega }\cap \{t\}\times M$ that is unique in a sufficiently smooth subclass of invariant manifolds uniformly tangent to ${\mathcal{N}}^{\Omega }$ (made precise later). We moreover show regularity of ${\mathcal{W}}^{\Omega }$ with respect to parameters, and characterize the attraction towards ${\mathcal{W}}^{\Omega }$ via a foliation in the case $\Omega$ is the slowest part of the spectrum.

arXiv comment63 pages, 9 figures. Comments welcome!