Existence of heteroclinic and homoclinic orbits in two different chaotic dynamical systems

This paper presents the existence of Si'lnikov orbits in two different chaotic systems belong to the class of Lorenz systems, more exactly in the Lu system and in the Zhou's system. Both systems have exactly two heteroclinic orbits which are symmetrical with respect to the z-axis by using the undetermined coefficient method. The existence of the homoclinic orbit for the Zhou's system has been proven also by using the undetermined coefficient method. As a result, the Si'lnikov criterion along with some technical conditions guarantees that Lu and Zhou's systems have both Smale horseshoes and horseshoe type of chaos. Moreover, the geometric structures of attractors are determined by these heteroclinic orbits. (C) 2012 Elsevier Inc. All rights reserved.