%0 Journal Article
%T The Proposed Modified Liu System with Fractional Order
%A Alireza K. Golmankhaneh
%A Roohiyeh Arefi
%A Dumitru Baleanu
%J Advances in Mathematical Physics
%D 2013
%I Hindawi Publishing Corporation
%R 10.1155/2013/186037
%X The chaos in a new system with order 3 is studied. We have shown that this chaotic system again will be chaotic when the order of system is less than 3. Generalized Adams-Bashforth algorithm has been used for investigating in stability of fixed points and existence of chaos. 1. Introduction It is well known that the nonlinear equations of dynamical systems with special condition have chaotic behavior [1]. Subsequently, additional studies were performed on the chaos and chaotic systems. Therefore solutions of different systems display their chaotic behavior such as Chen's system, Chua's dynamical system, the motion of double pendulum, and Rossler system amongst others. At first, it was thought that the chaos exists only when the order of system of differential equation is exactly 3. When the system of differential equation is composed of three first order differential equations, the order of the system is the sum of orders. But later on, a very interesting thing was realized; that is, it is also possible to observe chaotic behavior in a fractional order system. The system is composed of differential equations with fractional order derivatives [1¨C28]. For example, Sheu et al. reviewed the chaotic behavior of the Newton-Leipnik system with fractional order [10]. The important thing in the study of fractional-order systems is the minimum effective dimension of the system for which the system remains chaotic. This quantity has been numerically calculated for different systems including fractional order Lorenz system [11], fractional order Chua's system [12], and fractional order R£¿ssler system [13]. Recently, the chaos has been studied in fractional ordered Liu system, where the numerical investigations on the dynamics of this system have been carried out, and properties of the system have been analyzed by means of Lyapunov exponent [14]. In this paper we study the chaotic behavior of a generalization of the Liu system with fractional order. The framework of the paper is as follows. In Section 2, we study the behavior of a new fractional order system (modification of Liu system), and we study commensurate and incommensurate ordered systems and find lowest order at which chaos exists by numerical experiments. We have investigated the instability of fixed points and used Lyapunov exponent for the existence of chaos. In Section 3, we state the main conclusions. 2. The Proposed Modified Liu System In this section, we review the condition for asymptotic stability of the commensurate and incommensurate fractional ordered systems. We suggest the readers to see
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