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.

Appendix: Roots of a cubic polynomial

We recall that the discriminant $\Delta$ of the polynomial $ax^3+bx^2+cx+d$ is

$$\begin{aligned} \Delta = 18abcd -4b^3d + b^2c^2 - 4ac^3 - 27a^2d^2. \end{aligned}$$

It is known that

• If $\Delta > 0$, then the equation has three distinct real roots.

• If $\Delta = 0$, then the equation has a root of multiplicity $2$ and all its roots are real.

• If $\Delta < 0$, then the equation has one real root and two non–real complex conjugate roots.

For more details see [1].