^{1, a)} Rizgar H. Salih and Bashdar M. Mohammed ^{2, b)}

^1,2 Department of Mathematics, College of Basic Education,

University of Raparin, Raniyah, Kurdistan Region – Iraq.

^a)rizgar.salih@uor.edu.krd

^b)bashdar.mahmoodmuhamad@uor.edu.krd

This paper is devoted to studying the stability of the unique equilibrium point and the occurrence of the Hopf bifurcation as well as limit cycles of a three-dimensional chaotic system. We characterize the parameters for which a Hopf equilibrium point takes place at the equilibrium point. In addition, the system has only one equilibrium point which is E_0=(0,0,0). It was proved that E_0 is asymptotically stable and unstable when α<(-13)/7 and α>(-13)/7, respectively. Moreover, for studying the cyclicity of the system, two techniques are used which are dynamics on the center manifold and Liapunov quantities. It was shown that at most two limit cycles can be bifurcated from the origin. All the results presented in this paper have been verified by a program via Maple software.