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.
International Journal of Computing Science and Mathematics, 2008 Vol.2 No.1/2, pp.132 - 180
Published online: 25 Jul 2008 *Full-text access for editors Access for subscribers Purchase this article Comment on this article