Renormalization group based implicit function approach to connecting orbits

TL;DR

This paper introduces a renormalization group-based implicit function method for efficiently approximating connecting orbits in nonlinear systems, simplifying computations and accurately identifying homoclinic and heteroclinic connections.

Contribution

It presents a novel implicit function scheme leveraging renormalization group techniques to represent connecting orbits, with simplified linear algebraic equations for coefficients and a method to determine unknown parameters.

Findings

Successfully finds homoclinic and heteroclinic connections in five examples

Reduces computational load using symmetry considerations

Provides effective uniform approximation of connecting orbits

Abstract

Connecting orbits are important invariant structures in the state space of nonlinear systems and various techniques are designed for their computation. However, a uniform analytic approximation of the whole orbit seems rare. Here, based on renormalization group, an implicit function scheme is designed to effectively represent connections of disparate types, where coefficients of the defining function satisfy a set of linear algebraic equations, which greatly simplifies their computation. Unknown system parameters are conveniently determined by minimizing an error function. Symmetry may be profitably utilized to reduce the computation load. Homoclinic or heteroclinic connections are found in five popular examples approximately or exactly, demonstrating the effectiveness of the new scheme.