The purpose of this article to provide an explicit example where continuation based on the homotopy method fails. The example is a one-parameter homotopy for periodic orbits between two well-known nonlinear systems, the normal form of the Hopf bifurcation and the van der Pol system. Our analysis shows that various types of obstructions can make approximation over the whole range of the homotopy parameter impossible. The Hopf-van der Pol system demonstrates that homotopy methods may fail even for seemingly innocent systems.
- Periodic orbits
- Global bifurcations
Meijer, H. G. E., & Kalmár-Nagy, T. (2012). The Hopf-van der Pol system: Failure of a homotopy method. Differential equations and dynamical systems, 20(3), 323-328. https://doi.org/10.1007/s12591-011-0091-5