Validated Computation of Heteroclinic Sets

TL;DR

This paper introduces a rigorous computational method for enclosures of heteroclinic sets in nonlinear maps, combining high-order Taylor approximations, interval Newton methods, and error control techniques.

Contribution

It develops a novel, validated approach for computing heteroclinic manifolds with rigorous error bounds using advanced numerical and interval analysis methods.

Findings

Successfully computes heteroclinic sets for a volume-preserving map in 3D.

Provides computer-assisted proofs of heteroclinic invariant sets.

Demonstrates the method's effectiveness in nonlinear dynamical systems.

Abstract

In this work we develop a method for computing mathematically rigorous enclosures of some one dimensional manifolds of heteroclinic orbits for nonlinear maps. Our method exploits a rigorous curve following argument build on high order Taylor approximation of the local stable/unstable manifolds. The curve following argument is a uniform interval Newton method applied on short line segments. The definition of the heteroclinic sets involve compositions of the map and we use a Lohner-type representation to overcome the accumulation of roundoff errors. Our argument requires precise control over the local unstable and stable manifolds so that we must first obtain validated a-posteriori error bounds on the truncation errors associated with the manifold approximations. We illustrate the utility of our method by proving some computer assisted theorems about heteroclinic invariant sets for a volume preserving map of $\mathbb{R}^3$.