From Hopf to Neimark-Sacker bifurcation: a computational algorithm

TL;DR

This paper develops a Fourier spectral algorithm to approximate invariant tori at Neimark-Sacker bifurcation points, extending methods used for Hopf bifurcations to higher-dimensional invariant structures.

Contribution

It introduces a novel parametrisation approach for invariant tori, enabling low- and high-order approximations for both autonomous and forced systems.

Findings

Algorithm successfully approximates invariant tori at bifurcation points

Applicable to both autonomous and periodically-forced systems

Provides a systematic method for higher-order approximations

Abstract

We construct an algorithm for approximating the invariant tori created at a Neimark-Sacker bifurcation point. It is based on the same philosophy as many algorithms for approximating the periodic orbits created at a Hopf bifurcation point, i.e. a Fourier spectral method. For Neimark-Sacker bifurcation, however, we use a simple parametrisation of the tori in order to determine low-order approximations, and then utilise the information contained therein to develop a more general parametrisation suitable for computing higher-order approximations. Different algorithms, applicable to either autonomous or periodically-forced systems of differential equations, are obtained.