Bifurcation Analysis of Generalized Hopf Bifurcation in Ordinary and Delay Differential Equations

Abstract

The generalized Hopf (Bautin) bifurcation is a well-studied codimension two bifurcation characterized by an equilibrium with a pair of simple purely imaginary eigenvalues as the only critical eigenvalues and the vanishing first Lyapunov coefficient. This bifurcation arises in both ordinary differential equations (ODEs) and delay differential equations (DDEs). Generically, a codimension one bifurcation curve of nonhyperbolic (double) limit cycles ($LPC$ curve) emanates from a generalized Hopf point at the Hopf bifurcation curve $H$. By performing the parameter-dependent center manifold reduction near this point, the first-order predictors to initiate continuation of the $LPC$ curve have been derived in the literature. These predictors, however, do not distinguish the curves $H$ and $LPC$ in the parameter space. In this paper, we overcome this deficiency by deriving higher-order predictors for the $LPC$ curve in ODEs and DDEs for the first time. The new predictors, which require seventh-order derivatives of the system, have been implemented, and their effectiveness is demonstrated in several models.