Global dynamical aspects of a generalized Chen–Wang differential system

Abstract

We study theoretically the global chaotic behavior of the generalized Chen–Wang differential system

$$\begin{aligned} \dot{x} = y, \quad \dot{y} = z,\quad \dot{z} = -y - b x^2 - x z + 3 y^2 + a, \end{aligned}$$

where $a,b \in {\mathbb {R}}$ are parameters and $b\ne 0$. This polynomial differential system is relevant because is the first polynomial differential system in ${\mathbb {R}}^3$ with two parameters exhibiting chaotic motion without having equilibria. We first show that for $a>0$ sufficiently small it can exhibit up to three small amplitude periodic solutions that bifurcate from a zero-Hopf equilibrium point located at the origin of coordinates when $a=0$. We also show that the system exhibits two limit cycles emerging from two classical Hopf bifurcations at the equilibrium points $({\pm }\sqrt{2a}, 0,0)$, for $a>0, b=1/2$. We also give a complete description of its dynamics on the Poincaré sphere at infinity by using the Poincaré compactification of a polynomial vector field in ${\mathbb {R}}^3$, and we show that it has no first integrals neither in the class of analytic functions nor in the class of Darboux functions.