Chaos Synchronization of the Modified Autonomous Van der Pol-Duffing Circuits via Active Control
In this work, we study the dynamics and synchronization of a chaotic system describes the Modified Autonomous Van der Pol-Duffing (MAVPD) circuit. The detailed bifurcation diagrams are given to show the rich dynamics of the proposed system. Lyapunov expon
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Chaos Synchronization of the Modified Autonomous Van der Pol-Duffing Circuits via Active Control Ahmed Sadek Hegazi and Ahmed Ezzat Matouk
Abstract In this work, we study the dynamics and synchronization of a chaotic system describes the Modified Autonomous Van der Pol-Duffing (MAVPD) circuit. The detailed bifurcation diagrams are given to show the rich dynamics of the proposed system. Lyapunov exponents are calculated to verify the existence of chaos in this system. Chaos synchronization of MAVPD system is obtained using active control method. According to the qualitative theory of fractional differential equations, the existence and uniqueness of solutions for a class of commensurate fractional-order MAVPD systems are investigated. Furthermore, based on the stability theory of fractional-order systems, the conditions of local stability of linear fractional-order system are discussed. Moreover, the existence of chaotic behaviors in the fractional-order MAVPD system is shown. A necessary condition for this system to remain chaotic is obtained. It is found that chaos exists in this system with order less than three. Phase synchronization of the fractional-order MAVPD system is also achieved using an active control technique. Numerical simulations show the effectiveness of the proposed synchronization schemes.
7.1 Introduction Chaos is an important dynamical phenomenon which has been extensively studied and developed by scientists since the work of Lorenz [1]. Lorenz chaotic system consists of three-dimensional autonomous integer-order differential equations. A.S. Hegazi Faculty of Science, Mathematics Department, Mansoura University, Mansoura 35516, Egypt e-mail: [email protected] A.E. Matouk () Faculty of Science, Mathematics Department, Mansoura University, Mansoura 35516, Egypt College of Prep Year, Mathematics Department, Hail University, Hail 2440, Saudi Arabia e-mail: [email protected] S. Banerjee and L. Rondoni (eds.), Applications of Chaos and Nonlinear Dynamics in Science and Engineering 3, Understanding Complex Systems, DOI 10.1007/978-3-642-34017-8 7, © Springer-Verlag Berlin Heidelberg 2013
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A.S. Hegazi and A.E. Matouk
A chaotic system has complex dynamical behaviors such as the unpredictability of the long-term future behavior and irregularity. Chaos has great potential applications in many disciplines such as, fluid mixing, chaotic heating of plasma for a nuclear fusion reactor and secure communications. Fractional calculus is a 300 years old topic [2]. Recently, it has been found that differential equations has many applications in many fields of science like engineering [2], physics [3], finance [4], social sciences [5], mathematical biology [6,7] and game theory [8]. Hence, fractional differential equations have been utilized to study dynamical systems in general and applications of chaos in particular. There are many definitions of the fractional derivative; one of the most common definitions is the Caputo definition of fractional derivatives [9]: d˛ f .t/ D D ˛ f .t/ D I m˛ f .
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