Si´lnikov Chaos of a 3-D quadratic autonomous system with a four-wing chaotic attractor

The stability and chaotic motions of a 3-D quadratic autonomous system with a four-wing chaotic attractor are investigated in this paper. Base on the linearization analysis, the stability of the equilibrium points is studied. By using the undetermined coefficient method, the homoclinic and heteroclinic orbits are found and the series expansions of these two types of orbits is given. It analytically demonstrates that there exist homoclinic orbits of Silnikov type that join the equilibrium points to themselves and heteroclinic orbits of Silnikov type connecting the equilibrium points. Therefore, Smale horseshoes and the horseshoe chaos occur for this system via the Silnikov criterion.