Zero‐Hopf bifurcation of a cubic jerk system via the third order averaging method

This paper is devoted to analyze the zero‐Hopf bifurcation of a generalized three‐dimensional (3D) jerk system, the jerk function of this system has all quadratic and cubic terms. Due to the averaging method of second order, we show that at most three periodic orbits bifurcate form the zero‐Hopf equilibrium point of this jerk system, and this upper bound is sharp. Furthermore, by using the averaging method of third order, we show that three is also the maximal number of periodic orbits bifurcate from this zero‐Hopf equilibrium point. Finally, the numerical method is used to justify the theoretical analysis.