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

Applied Mathematics and Computation 218 (2012) 11859–11870 Contents lists available at SciVerse ScienceDirect Applied Mathematics and Computation journal homepage: /locate/amc Existence of heteroclinic and homoclinic orbits in two different chaotic dynamical systems M.M. El-Dessoky a,b,⇑, M.T. Yassen b, E. Saleh b, E.S. Aly b a Mathematics Department, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia b Mathematics Department, Faculty of Science, Mansoura University, Mansoura 35516, Egypt a r t i c l e i n f o a b s t r a c t Keywords: This paper presents the existence of Ši’lnikov orbits in two different chaotic systems belong Ši’lnikov criterion to the class of Lorenz systems, more exactly in the Lü system and in the Zhou’s system. Lü system Both systems have exactly two heteroclinic orbits which are symmetrical with respect to Zhou’s system the z-axis by using the undetermined coefficient method. The existence of the homoclinic Heteroclinic orbits orbit for the Zhou’s system has been proven also by using the undetermined coefficient Homoclinic orbits Smale horseshoes method. As a result, the Ši’lnikov criterion along with some technical conditions guarantees Undetermined coefficients method 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 heteroclin