The dynamics of a self-exciting homopolar dynamo system

Abstract

The periodic orbits of the Hide’s dynamo system, which is simplified from a self-exciting homopolar dynamo equation introduced by Hide et al. (1996), were studied by Swinnerton-Dyer and Wagenknecht (2008). In this paper, we study the Hopf bifurcation and Bogdanov–Takens bifurcation of the Hide’s dynamo system. Although Swinnerton-Dyer and Wagenknecht have already shown that the Hide’s dynamo system can undergo supercritical Hopf bifurcation and subcritical Hopf bifurcation, they provided only a brief review and did not give a detailed derivation. We give a rigorous derivation of the Hopf bifurcation, and show that the Hide’s dynamo system undergoes degenerate Hopf bifurcation of codimension two. For the Bogdanov–Takens bifurcation, it is shown that the Hide’s dynamo system admits Bogdanov–Takens bifurcation of codimension two and degenerate Bogdanov–Takens bifurcation of codimension three. These bifurcation phenomena give rise to several dynamical behaviors, including double limit cycles and homoclinic orbits.