From heteroclinic loops to homoclinic snaking in reversible systems: rigorous forcing through computer-assisted proofs

TL;DR

This paper employs computer-assisted proofs to rigorously establish the existence of homoclinic snaking phenomena in reversible systems, specifically in the Swift-Hohenberg and Gray-Scott models, advancing understanding of pattern formation in nonlinear dynamics.

Contribution

It introduces a novel computer-assisted approach to prove the existence of heteroclinic and homoclinic connections in non-perturbative regimes of reversible systems.

Findings

Proof of homoclinic snaking in Swift-Hohenberg and Gray-Scott models

Development of computer-assisted methods for connecting orbits

Enhanced understanding of global dynamics dependence on parameters

Abstract

Homoclinic snaking is a widespread phenomenon observed in many pattern-forming systems. Demonstrating its occurrence in non-perturbative regimes has proven difficult, although a forcing theory has been developed based on the identification of patterned front solutions. These heteroclinic solutions are themselves challenging to analyze due to the nonlinear nature of the problem. In this paper, we use computer-assisted proofs to find parameterized loops of heteroclinic connections between equilibria and periodic orbits in time reversible systems. This leads to a proof of homoclinic snaking in both the Swift-Hohenberg and Gray-Scott problems. Our results demonstrate that computer-assisted proofs of continuous families of connecting orbits in nonlinear dynamical systems are a powerful tool for understanding global dynamics and their dependence on parameters.