Computer Assisted Proof for Normally Hyperbolic Invariant Manifolds

TL;DR

This paper introduces a computer-assisted topological method to rigorously prove the existence of normally hyperbolic invariant manifolds in maps, validated through application to a driven logistic map.

Contribution

It provides a novel topological proof framework that does not require the map to be a perturbation of a known invariant manifold, enabling rigorous verification from approximate guesses.

Findings

Proved the existence of a normally hyperbolic invariant curve in a driven logistic map.

Validated the chaotic attractor as a normally hyperbolic invariant manifold, contradicting initial numerical simulations.

Established a method applicable to other maps for rigorous computer-assisted proofs.

Abstract

We present a topological proof of the existence of a normally hyperbolic invariant manifold for maps. 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. But a non-rigorous, good enough, guess is necessary. The required assumptions are formulated in a way which allows for rigorous computer assisted verification. We apply our method for a driven logistic map, for which non-rigorous numerical simulation in plain double precision suggests the existence of a chaotic attractor. We prove that this numerical evidence is false and that the attractor is a normally hyperbolic invariant curve.