Zero-Hopf bifurcation in a 3D jerk system

Abstract

Let the three-dimensional differential system defined by the jerk equation x⃛=−aẍ+xẋ2−x3−bx+cẋ, with a,b,c∈R. When a=b=0 and c<0 the equilibrium point localized at the origin of coordinates is a zero-Hopf equilibrium. We analyse the zero-Hopf bifurcation occurring at this singular point after persuading a quadratic perturbation of the coefficients. Particularly, by using averaging theory of second order, we prove that up to three periodic orbits born as the parameter of the perturbation tends to zero.