Sil'nikov chaos of the Liu system

A new chaotic attractor is discovered for the Liu system. The homoclinic and heteroclinic orbits in the Liu system have been found by using the undetermined coefficient method. It analytically demonstrates that there exists one heteroclinic orbit of the Sil'nikov type that connects two nontrivial equilibrium points, and therefore Smale horseshoes and the horseshoe chaos occur for this system via the Sil'nikov criterion. In addition, there also exists one homoclinic orbit joined to the origin. The convergence of the series expansions of these two types of orbits is proved.