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

This paper presents the existence of Ši’lnikov orbits in two different chaotic systems belong to the class of Lorenz systems, more exactly in the Lü 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 Ši’lnikov criterion along with some technical conditions guarantees that Lü 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.