On the integrability and the zero-Hopf bifurcation of a Chen–Wang differential system

Abstract

The first objective of this paper was to study the Darboux integrability of the polynomial differential system $$\begin{aligned} \dot{x} = y, \quad \dot{y} = z,\quad \dot{z} = -y - x^2 - x z + 3 y^2 + a, \end{aligned}$$ and the second one is to show that for $$a>0$$ sufficiently small this model exhibits two small amplitude periodic solutions that bifurcate from a zero-Hopf equilibrium point localized at the origin of coordinates when $$a=0$$ . We note that this polynomial differential system introduced by Chen and Wang (Nonlinear Dyn 71:429–436, 2013) is relevant in the sense that it is the first system in $$\mathbb {R}^3$$ exhibiting chaotic motion without having equilibria.