Projective synchronization in fractional order chaotic systems and its control

The chaotic dynamics of fractional (non-integer) order systems have begun to attract much attention in recent years. In this paper, we study the projective synchronization in two coupled fractional order chaotic oscillators. It is shown that projective synchronization can also exist in coupled fractional order chaotic systems. A simple feedback control method for controlling the scaling factor onto a desired value is also presented. PACS. 05.45.XtSynchronization; coupled oscillators. Although fractional calculus has a mathematical history nearly as long as that of the integer-order calculus, the applications of it to physics and engineering are just a recent focus of interest [1, 2]. Many systems are known to display fractional order dynamics, such as viscoelastic systems [3-5], dielectric polarization [6], electrode-electrolyte polarization [7] and electromagnetic waves [8], so it is important to study the properties of fractional order systems. The dynamics of fractional order systems has not yet been fully studied, and it is by no mean trivial. The definitions of fractional order calculus are geometrically and physically less intuitive than the integer-order ones, and can’t be simulated directly in time-domain. It is still unclear whether the dynamics of fractional order systems is similar to the integer-order ones. More recently, many authors have begun to investigate the chaotic dynamics of fractional order dynamical systems [9-17]. In [9], it was shown that the fractional order Chua’s system of order as low as 2.7 can produce a chaotic attractor. In [10], it was shown that nonautonomous Duffing systems of order less than 2 can still behave in a chaotic manner. In [11], chaotic behaviors of the fractional order “jerk” model was studied, in which chaotic attractor was obtained with system orders as low as 2.1, and in [12] the control of this fractional order chaotic system was reported. In [13], chaotic behavior of the fractional order Lorenz system was studied, but unfortunately, the results presented in this paper are not correct. In [14] and [15], bifurcation and chaotic dynamics of the fractional order cellular neural networks were studied. In [16], chaos and hyperchaos in the fractional order Rössler equations were studied, in which we showed that chaos can exist in the fractional order Rössler equation with Progress of Theoretical Physics, Vol. 115, No. 3, March 2006, pp.661-666. Email: [email protected]