%0 Journal Article
%T Harmonic Balance Method for Chaotic Dynamics in Fractional-Order R£¿ssler Toroidal System
%A Huijian Zhu
%J Discrete Dynamics in Nature and Society
%D 2013
%I Hindawi Publishing Corporation
%R 10.1155/2013/593856
%X This paper deals with the problem of determining the conditions under which fractional order R£¿ssler toroidal system can give rise to chaotic behavior. Based on the harmonic balance method, four detailed steps are presented for predicting the existence and the location of chaotic motions. Numerical simulations are performed to verify the theoretical analysis by straightforward computations. 1. Introduction The concept of fractional calculus has been known since the development of the regular calculus, with the first reference probably being associated with Leibniz and L¡¯H£¿spital in 1695. In the past three decades or so, fractional calculus gained considerable popularity and importance, due mainly to its demonstrated applications in numerous seemingly diverse and widespread fields of science and engineering [1¨C3]. In particular, by utilizing fractional calculus technique, many investigations were devoted to the chaotic behaviors and chaotic control of dynamical systems involving the fractional derivative, called fractional-order chaotic system [4¨C8]. For example, it has been shown that Chua circuit of the order as low as 2.7 can behave in chaotic manner [4]. In [5], the nonautonomous Duffing systems of the order less than 2 can still produce a chaotic attractor. In [6], chaotic behavior of the fractional-order Lorenz system was further studied. In [7], chaos and hyperchaos in the fractional-order R£¿ssler equations were also studied, in which the authors showed that chaos can exist in the fractional-order R£¿ssler equation with the order as low as 2.4, and hyperchaos can exist in the fractional-order R£¿ssler hyperchaos equation with the order as low as 3.8. Later on, the chaotic behavior and its control in the fractional-order Chen system were investigated in [8]. And recently, more dynamic behaviors of fractional order chaotic systems were analyzed by using different approaches; we refer the readers to [9¨C12]. However, to our knowledge, the conditions for chaos existence in dynamical systems (including integer-order system and fractional-order system) are still incomplete. For a given dynamical system, can we decide (without invoking numerical simulations) whether and in what parameter ranges, chaotic behavior might exist? It is still an open problem [13]. Even though some theorems such as Melnikov¡¯s criteria [14] and Shil¡¯nikov¡¯s theorem [15] may be helpful in some special cases, it seems that a powerful tool is not generally available to determine the accurate parameter ranges for chaos existence in a given dynamical system. Moreover, the fact that
%U http://www.hindawi.com/journals/ddns/2013/593856/