Computer-Assisted Proofs in Dynamical Systems: A Case Study of a Heteroclinic Orbit in the Shimizu--Morioka System

TL;DR

This paper demonstrates a computer-assisted proof of a heteroclinic orbit in the Shimizu--Morioka system using the radii polynomial approach, validating key dynamical features with a structured four-step method.

Contribution

It introduces a unified four-step validation procedure for heteroclinic orbits, equilibria, and invariant manifolds, with implementation in Julia using the RadiiPolynomial library.

Findings

Validated the existence of a transverse heteroclinic orbit in the system.

Confirmed the equilibria and eigenpairs with rigorous bounds.

Provided computational code snippets in Julia for the validation process.

Abstract

The radii polynomial approach is an a posteriori validation method based on the contraction of a quasi-Newton operator. We apply this strategy to give a computer-assisted proof of a transverse heteroclinic orbit in the Shimizu--Morioka system, validating the equilibria and eigenpairs, the local invariant manifolds via the parameterization method, and the connecting orbit via a boundary-value problem. For each subproblem we present a four-step procedure: $(i)$ zero-finding formulation, $(ii)$ approximate zero, $(iii)$ approximate inverse, and $(iv)$ bound estimates. This highlights the unifying structure behind the a posteriori validation method. Alongside the analysis, we include code snippets implemented in Julia using the RadiiPolynomial library.