Bifurcation of Multiple Limit Cycles in an Epidemic Model on Adaptive Networks

Abstract

Very recently, Zhang et al. considered an epidemic model on adaptive networks [Zhang et al., 2019], in which Hopf bifurcation, homoclinic bifurcation and Bogdanov–Takens bifurcation are studied. Degenerate Hopf bifurcation is investigated via simulation and a numerical example is given to show the existence of two limit cycles. However, whether the codimension of the Hopf bifurcation is two is still open. In this paper, we will rigorously prove that the codimension of the Hopf bifurcation is two. That is, the maximal two limit cycles can bifurcate from the Hopf critical point. Moreover, the conditions for the existence of two limit cycles are derived.