Title: From Hopf to Neimark–Sacker bifurcation: a computational algorithm

Authors: Gerald Moore

Addresses: Department of Mathematics, Imperial College of Science, Technology and Medicine, 180 Queen's Gate, SW7 2AZ, London, UK

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.

Keywords: Neimark–Sacker bifurcation; Hopf bifurcation; Fourier spectral method; normal form; Floquet theory; invariant tori; differential equations.

DOI: 10.1504/IJCSM.2008.019726

International Journal of Computing Science and Mathematics, 2008 Vol.2 No.1/2, pp.132 - 180

Published online: 25 Jul 2008 *

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