Cone Conditions and Covering Relations for Topologically Normally Hyperbolic Invariant Manifolds

TL;DR

This paper introduces a topological method to prove the existence of invariant manifolds in dynamical systems with hyperbolic-like properties, using conditions verifiable through computer assistance, demonstrated on the rotating Hénon map.

Contribution

It provides a novel topological proof framework that does not rely on perturbation assumptions, enabling rigorous computer-assisted verification of invariant manifolds.

Findings

Validated invariant manifold existence for the rotating Hénon map within specific parameters.

Developed conditions applicable to a broad class of maps with hyperbolic-like dynamics.

Enabled computer-assisted proofs for invariant structures in dynamical systems.

Abstract

We present a topological proof of the existence of invariant manifolds for maps with normally hyperbolic-like properties. The proof is conducted in the phase space of the system. In our approach we do not require that the map is a perturbation of some other map for which we already have an invariant manifold. We provide conditions which imply the existence of the manifold within an investigated region of the phase space. The required assumptions are formulated in a way which allows for rigorous computer assisted verification. We apply our method to obtain an invariant manifold within an explicit range of parameters for the rotating H\'enon map.