Abstract
In this paper the synchronization of fractional-order chaotic systems and a new property of fractional derivatives are studied. Then we propose a new fractional-order extension of Lyapunov direct method to control the fractional-order chaotic systems. A new synchronization method and a linear feedback controller are given to achieve the synchronization of fractional-order chaotic systems based on a simple Lyapunov candidate function. The proposed synchronization method can be applied to the synchronization of an arbitrary fractional-order chaotic system. This method is universal, simple, and theoretically rigorous. Numerical simulations of three fractional-order chaotic systems to verify the effectiveness and the universality of the proposed method.
Highlights
Fractional calculus is a topic of more than years old
The fractional-order nonlinear dynamic systems have many dynamic behaviors which are similar to the integer-order systems, such as chaos, bifurcation, and attractor [ – ]
5 Conclusion In this letter we proposed a new synchronization method for fractional-order chaotic systems based on a simple Lyapunov function
Summary
Fractional calculus is a topic of more than years old. The idea of fractional calculus has been known since the regular calculus, with the first reference probably being associated with Leibniz and L’Hospital in where a half-order derivative was mentioned. In [ ] the authors proposed the Lyapunov direct method to prove the stability of fractional-order nonlinear system with time delay. Some sufficient conditions of synchronization for the fractional-order chaotic systems are proposed based on a simple Lyapunov function. We adopt the improved version of Adams-Bashforth-Moulton algorithm [ , ] to numerically solve the fractional differential equations, which is proposed based on the predictor-correctors scheme. For explaining this method, the following differential equation is considered: Dαt y(t) = f t, y(t) , ≤ t ≤ T, ( ).
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