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

TL;DR

This paper develops higher-order predictors for bifurcation curves emanating from generalized Hopf points in ODEs and DDEs, improving the accuracy of continuation methods by utilizing seventh-order derivatives.

Contribution

It introduces the first higher-order predictors for the LPC curve at generalized Hopf bifurcations in both ODEs and DDEs, enhancing bifurcation analysis techniques.

Findings

New predictors require seventh-order derivatives.

Predictors effectively distinguish H and LPC curves.

Implementation demonstrates improved bifurcation continuation.

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.