From Hopf to Neimark-Sacker bifurcation: a computational algorithm
This provides a computational tool for analyzing Neimark-Sacker bifurcations, which is relevant for researchers studying dynamical systems and bifurcation theory.
The authors developed an algorithm to approximate invariant tori arising at Neimark-Sacker bifurcations, extending Fourier spectral methods used for Hopf bifurcations. The method uses a parametrization approach to compute low- and high-order approximations for both autonomous and periodically-forced systems.
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.