Modified projective synchronization of stochastic fractional order chaotic systems with uncertain parameters
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O R I G I N A L PA P E R
Modified projective synchronization of stochastic fractional order chaotic systems with uncertain parameters Shao-Juan Ma · Qiong Shen · Jing Hou
Received: 15 June 2012 / Accepted: 8 January 2013 / Published online: 30 January 2013 © Springer Science+Business Media Dordrecht 2013
Abstract In this paper, the synchronization of fractional order chaotic systems with random and uncertain parameters is analyzed. Firstly, based on the orthogonal polynomial expansion, the fractional order Lü and Lorenz systems with random and uncertain parameters are reduced into the equivalent deterministic systems. Secondly, modified projective synchronization of equivalent deterministic Lü and Lorenz systems is explored. Lastly, the theoretical results are verified by the numerical simulations. Keywords Modified projective synchronization · Stochastic system · Orthogonal polynomial expansion · Uncertain parameters 1 Introduction Fractional calculus is a 300-year-old mathematical topic, but it has gained a lot of attentions in variety of research fields until the recent 10 years [1, 2], such as physics, engineering, and secure communications. There are essential differences between integer order differential equation and fractional order differential equation, especially in dynamical systems and control. S.-J. Ma () · Q. Shen · J. Hou School of Information and Computation Science, Beifang University of Nationalities, Yinchuan 750021, China e-mail: [email protected] Q. Shen e-mail: [email protected]
Gu and Xu [3] has analyzed chaos in a fractional order dynamical model of love and its control. Gao and Zhang have presented an impulsive multidelayed feedback control method for stabilizing discrete chaotic systems [4]. In [5], nonlinear dynamics of Duffing system with fractional-order damping is studied. Liu and Xie have researched the dynamical characteristics of the fractional-order Fitzhugh–Nagumo model neuron and its synchronization [6]. In [7], chaos in diffusionless Lorenz system with fractional order and its control is researched by Xu. It is important to note that more and more explorers begin to research stochastic fractional-order systems. [8] has considered noise analysis of single stage fractional-order low-pass filter using stochastic and fractional calculus. The stochastic stability for nonlinear systems driven by Lévy noise is studied in [9]. Control of an uncertain fractional-order Liu system via fuzzy fractional-order sliding mode control is researched by Mohammad Reza Faieghi [10]. The research about the synchronization of deterministic fractional-order system has a lot of references [11–21]. However, the study about synchronization of stochastic fractional-order system is not many, and it is few that the research about synchronization of fractional-order systems with random and uncertain parameters. So the synchronization of fractional-order chaotic systems with random and uncertain parameters has became a remarkable research field. In the present paper, the main innovative point is using
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