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 'X PONTO' = y, 'Y PONTO' = z, 'Z PONTO' = −y − 'X POT.2' − xz + 3'Y POT.2' + a, 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 'R POT.3' exhibiting chaotic motion without having equilibria.