Zero-Hopf bifurcation of a three dimensional quadratic oscillator system

In this paper, sufficient conditions for the existence of the zero-Hopf singular point of the three-dimensional chaotic system which is proposed by D. Veeman are found. The system has a line of singularity, it is located on the x3-axis and the origin is the only zero-Hopf. For studying the zero-Hopf bifurcation at the origin, averaging theory of first order is used. It is shown that for this kind of constructed 3D chaotic system, the first equation without nonlinear term and the third equation without linear term, the averaged method does not give any knowledge about bifurcated limit cycles from the zero-Hopf singular point.