Computer Assisted Proof of Drift Orbits Along Normally Hyperbolic Manifolds

TL;DR

This paper introduces a computer-assisted methodology leveraging normally hyperbolic invariant manifolds theory to rigorously prove diffusion in chaotic dynamical systems, demonstrated on the generalized standard map.

Contribution

It develops a finite-step, computer-implementable method for validating diffusion conditions in chaotic systems using rigorous interval arithmetic.

Findings

Validated diffusion in the generalized standard map

Method confirms diffusion over specific action ranges

Provides a practical computational approach for diffusion proofs

Abstract

Normally hyperbolic invariant manifolds theory provides an efficient tool for proving diffusion in dynamical systems. In this paper we develop a methodology for computer assisted proofs of diffusion in a-priori chaotic systems based on this approach. We devise a method, which allows us to validate the needed conditions in a finite number of steps, which can be performed by a computer by means of rigorous-interval-arithmetic computations. We apply our method to the generalized standard map, obtaining diffusion over an explicit range of actions.