On the coexistence of periodic or quasi-periodic oscillations near a Hopf-pitchfork bifurcation in NFDE

Normal form method is first employed to study the Hopf-pitchfork bifurcation in neutral functional differential equation (NFDE), and is proved to be an efficient approach to show the rich dynamics (periodic and quasi-periodic oscillations) around the bifurcation point. We give an algorithm for calculating the third-order normal form in NFDE models, which naturally arise in the method of extended time delay autosynchronization (ETDAS). The existence of Hopf-pitchfork bifurcation in a van der Pol’s equation with extended delay feedback is given and the unfoldings near this critical point is obtained by applying our algorithm. Some interesting phenomena, such as the coexistence of several stable periodic oscillations (or quasi-periodic oscillations) and the existence of saddle connection bifurcation on a torus, are found by analyzing the bifurcation diagram and are illustrated by numerical method.