Chaos Synchronization in a Circular Restricted Three Body Problem Under the Effect of Radiation
In this paper, we have investigated the synchronization behaviour of two identical nonlinear dynamical systems of a circular restricted three body problem (CRTBP) under the effect of radiation evolving from different initial conditions using the active co
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Abstract In this paper, we have investigated the synchronization behaviour of two identical nonlinear dynamical systems of a circular restricted three body problem (CRTBP) under the effect of radiation evolving from different initial conditions using the active control technique based on the Lyapunov stability theory and the Routh-Hurwitz criteria. The designed controller, with our own choice of the coefficient matrix of the error dynamics, are found to be effective in the stabilization of the error states at the origin, thereby, achieving synchronization between the states variables of two dynamical systems under consideration. Numerical simulations are presented to illustrate the effectiveness of the proposed control techniques using mathematica.
1 Introduction The idea of synchronization of two identical chaotic systems that start from different initial conditions consists of linking the trajectory of one system to the same values in the other so that they remain in step with each other, through the transmission of a signal. After the pioneering work on chaos control and synchronization of chaotic systems, it has received a broad attention [1–7] and has become a very active topic in nonlinear sciences since last couple of years. Since then in this direction, various effective methods have been proposed and utilized to achieve the control and stabilization of various nonlinear dynamical systems [8–14].
A. Khan Department of Mathematics, Zakir Husain College, University of Delhi, New Delhi 110002, India e-mail: [email protected] M. Shahzad () Department of General Requirements, College of Applied Sciences, Nizwa, Oman e-mail: [email protected] S.G. Stavrinides et al. (eds.), Chaos and Complex Systems, DOI 10.1007/978-3-642-33914-1 8, © Springer-Verlag Berlin Heidelberg 2013
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A. Khan and M. Shahzad
So far several attempts have been made for the different chaotic systems based on different controllers to realize synchronization [15–22]. In particular, backstepping design and active control have been recognised as two powerful design methods to synchronize. It has been reported that backstepping design can guarantee global stability, tracking and transient performance for a broad class of strict-feedback nonlinear systems [23–25]. In recent time, it has been employed for controlling, tracking and synchronizing many chaotic systems [26–32]. Some of the advantages in the method include applicability to a variety of chaotic systems whether they contain external excitation or not; needs only one controller to realize synchronization between chaotic systems and finally there are no derivatives in the controller and also the controller is singularity free from the nonlinear term of quadratic type, gives flexibility to construct a control law which can be extended to higher dimensional hyperchaotic system and the closed-loop system is globally stable [25, 26]. Keeping in mind the above studies, we have written this article to study the synchronization behavior of the two identical circular restricted thr
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