Geometric proof for normally hyperbolic invariant manifolds

TL;DR

This paper introduces a non-perturbative, geometric proof for the existence and smoothness of normally hyperbolic invariant manifolds for maps, utilizing bounds on the map and its derivative, suitable for computer-assisted validation.

Contribution

It provides a new geometric, non-perturbative proof for normally hyperbolic manifolds that does not require invertibility, enabling rigorous computer-assisted validation.

Findings

Existence of normally hyperbolic manifolds established within given neighborhoods.

Manifolds' smoothness depends on the map's smoothness.

Method applicable for rigorous computer-assisted validation.

Abstract

We present a new proof of the existence of normally hyperbolic manifolds and their whiskers for maps. Our result is not perturbative. Based on the bounds on the map and its derivative, we establish the existence of the manifold within a given neighbourhood. Our proof follows from a graph transform type method and is performed in the state space of the system. We do not require the map to be invertible. From our method follows also the smoothness of the established manifolds, which depends on the smoothness of the map, as well as rate conditions, which follow from bounds on the derivative of the map. Our method is tailor made for rigorous, interval arithmetic based, computer assisted validation of the needed assumptions.