Zero-Hopf bifurcations of 3D quadratic jerk system

This paper is devoted to local bifurcations of three-dimensional (3D) quadratic jerk system. First, we start by analysing the saddle-node bifurcation. Then we introduce the concept of canonical system. Next, we study the transcritial bifurcation of canonical system. Finally we study the zero-Hopf bifurcations of canonical system, which constitutes the core contributions of this paper. By averaging theory of first order, we prove that, at most, one limit cycle bifurcates from the zero-Hopf equilibrium. By averaging theory of second order, third order, and fourth order, we show that, at most, two limit cycles bifurcate from the equilibrium. Overall, this paper can help to increase our understanding of local behaviour in the jerk dynamical system with quadratic non-linearity.