An Improved Numerical Algorithm for the Fractional Differential Equations and Its Application in the Fractional-Order Nonlinear Systems

In this paper, an improved numerical algorithm for the fractional differential equations is proposed based on the variational iteration method. Using the improved numerical scheme, we investigate a new fractional-order hyperchaotic system, and find that hyperchaos does exist in the new fractional-order four-dimensional system with order less than 4. The lowest order we find to yield hyperchaos is 3.46 in this new fractional-order system. The existence of two positive Lyapunov exponents further verifies our results. Numerical results show that the proposed method has a faster speed and more accurate comparing with the traditional predictor-corrector algorithm. Based on the stability theory of the fractional-order system, a nonlinear controller is constructed to achieve synchronization fora class of nonlinear fractional-order systems using the pole placement technique. The nonlinear control method can synchronize two different fractional-order hyperchaotic systems. This synchronization scheme which is simple and theoretically rigorous enables synchronization of fractional-order hyperchaotic systems to be achieved in a systematic way and does not need to compute the conditional Lyapunov exponents. Simulation results are given to validate the effectiveness of the proposed synchronization method.