Existence of heteroclinic and homoclinic orbits in two different chaotic dynamical systems
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- 作者：M.M. El-Dessoky ; M.T. Yassen ; E. Saleh ; E.S. Aly
- 关键词：&Scaron ; i&rsquo ; lnikov criterion ; Lü ; system ; Zhou&rsquo ; s system ; Heteroclinic orbits ; Homoclinic orbits ; Smale horseshoes ; Undetermined coefficients method
- 刊名：Applied Mathematics and Computation
- 出版年：2012
- 出版时间：15 August, 2012
- 年：2012
- 卷：218
- 期：24
- 页码：11859-11870
- 全文大小：453 K
文摘
This paper presents the existence of ?i¡¯lnikov orbits in two different chaotic systems belong to the class of Lorenz systems, more exactly in the L¨¹ 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 ?i¡¯lnikov criterion along with some technical conditions guarantees that L¨¹ 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.