Synchronization of the Chaotic Pandey-Baghel-Singh Systems of Fractional Order

Abstract

Abstract. The paper considers the modified Pandey-Baghel-Singh system of fractional order. Chaotic behavior of the system is analyzed and the problem of synchronization of two modified Pandey-Baghel-Singh systems via masterslave configuration with linear coupling is considered. A simple sufficient condition for synchronization using the Lyapunov and Gershgorin stability theory is proposed. The considerations are illustrated by numerical simulations.Keywords: chaos, fractional, Pandey-Baghel-Singh system, synchronization. IntroductionDynamical systems described by fractional order differential or difference equations have been investigated in several areas such as viscoelasticity, electrochemistry, diffusion processes, control theory, electrical engineering, etc. The problems of analysis and synthesis of dynamic systems described by fractional order differential (or difference) equations recently have considerable attention, see monographs [8,13,15,[17][18][19], for example. Many non-linear dynamical systems have behavior known as chaos. Chaos is a very interesting non-linear phenomenon. Chaotic systems are deterministic and highly sensitive on initial condition and system parameters. There are many chaotic systems, e.g. in paper [16] was introduced Pandey-Baghel-Singh chaotic system of integer order. Synchronization of chaos is a very interesting problem, enjoying a wide interest, for example, in control technology, cryptography, communications [9,20], etc. The problem of synchronization of chaos recently has been intensively studied in many papers, see for example [3,4,7,10,11,25] for systems of integer order and [2,6,12,14,17,21,22] for system of fractional order. In the paper [2] the method of the synchronization of two coupled general chaotic fractional order Van der PolDuffing systems with a unidirectional linear error feedback coupling is presented.In this paper we consider the modified Pandey-Baghel-Singh oscillator of fractional order. Chaotic behavior of this system will be analyzed for different values of fractional orders. Simple sufficient condition for synchronization of two such systems with non-commensurate fractional orders via master/slave configuration with linear coupling will be given. The proposed condition is obtained in a similar way as in the paper [2] using the Lyapunov and Gershgorin stability theory.