Zero-Hopf bifurcation in a 3-D jerk system

Abstract

We consider the 3-D system defined by the jerk equation $\stackrel{...}{x}=-a\ddot{x}+x{\dot{x}}^2-x^3-bx+c\dot{x}$, with $a,b,c\in \mathbb{R}$. When $a=b=0$ and $c<0$ the equilibrium point localized at the origin is a zero-Hopf equilibrium. We analyse the zero-Hopf Bifurcation that occur at this point when we persuade a quadratic perturbation of the coefficients, and prove that one, two or three periodic orbits can born when the parameter of the perturbation goes to $0$.