Computer Assisted Proof for Normally Hyperbolic Invariant Manifolds

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.

33 pages, 16 figures

References in corpus (1)

• Random versus deterministic exponents in a rich family of diffeomorphisms

Cited by in corpus (2)

• Arnold diffusion in the planar elliptic restricted three-body problem: mechanism and numerical verification
• Geometric Proof of Strong Stable/Unstable Manifolds, with Application to the Restricted Three Body Problem