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Generalized Wright stability for distributed fractional-order nonlinear dynamical systems and their synchronization. (English) Zbl 1430.34007
Summary: In this article, we present a generalization of stability theorems for Caputo fractional derivative to the distributed fractional-order (DFO) case by using the Laplace transform and the asymptotical expansion of the generalized Mittag-Leffler function. We propose the definition of the generalized Wright stability to study the stability of DFO nonlinear dynamical system using Lyapunov direct method. The linear feedback control is used to stabilize a class of chaotic DFO nonlinear dynamical systems. Using Lyapunov direct method, we study the synchronization between two identical chaotic systems and between two other different in the linear terms. The chaotic DFO Lorenz system is given as an example to achieve the linear feedback control technique. Another two examples which are chaotic DFO complex Chen and Lü systems are used to show the validity and feasibility of our proposed synchronization scheme. Numerical simulations are implemented to verify the results of these investigations.
MSC:
34A08 Fractional ordinary differential equations and fractional differential inclusions
34D06 Synchronization of solutions to ordinary differential equations
26A33 Fractional derivatives and integrals
34C28 Complex behavior and chaotic systems of ordinary differential equations
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