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 $\mathscr{N}^\Omega$ of $T_\gamma M$ invariant under $\nabla X_t$. Under additional nonresonance conditions on $\Omega$, we prove, by an application of the parameterization method (Cabr\'e et al., 2003), the existence of an invariant topological submanifold $\mathscr{W}^\Omega \subset \mathbb{R} \times M$ with $C^r$ fibers $W_t^\Omega:= \mathscr{W}^\Omega \cap \{t\} \times M$ that is unique in a sufficiently smooth subclass of invariant manifolds uniformly tangent to $\mathscr{N}^\Omega$ (made precise later). We moreover show regularity of $\mathscr{W}^\Omega$ with respect to parameters, and characterize the attraction towards $\mathscr{W}^\Omega$ via a foliation in the case $\Omega$ is the slowest part of the spectrum.