Study On The Stability Of Four Types Of Fractional-order Nonlinear System With Time-Delay

In recent years, with the leap forward development of computer science and crossing discipline, fractional calculus has got dramatic advances in both theory and application. Fractional calculus is the extension of integer-order calculus in order, with the no local property, fractional calculus has been used to describe the basic nature of almost all sciences and engineering fields, such as fluid flow in porous materials, anomalous diffusion transport, acoustic wave propagation in viscoelastic materials, dynamics in self-similar structures, signal processing, financial theory, security communication, electric conductance of biological systems and so on. And fractional calculus has been the hot and difficult topic. And in this thesis, efforts are mainly paid on the study of an modified predictor-corrector method, the dynamical behaviors of fractional-order complex Lorenz system with single time-delay and its self time-delay synchronization, and the stability control of fractional-order nonlinear system with multiple time-delays. The main contributions and innovations in this thesis are as follows:1. A modified predictor-corrector method for the fractional-order equationsIn this part, a modified predictor-corrector method for the fractional-order differential equations is proposed. The accuracy of the modified method is improved through the modification in the prediction part of predictor-corrector method. It is proved that the numerical accuracy of this new method is superior to that of the classical predictor-corrector method by three numerical examples. The comparison with the corresponding results demonstrates that the modified predictor-corrector method is more accurate than the predictor-corrector method in solving fractional differential equations numerically.2. The dynamical behavior of fractional-order complex Lorenz system with single time-delay and its self time-delay synchronizationIn this part, by the means of phase portraits and the largest Lyapunov exponent, the dynamical behaviors of the fractional-order complex Lorenz system with single time –delay are investigated under various fractional orders and different values of time-delay. In particular, the effect of time-delay on the chaotic behavior is discussed. It is found that appropriate fractional orders and time-delays will enhance or suppress the emergence of chaos in the fractional-order complex Lorenz system with time-delay. Based on the above results, a self time-delay synchronization scheme between fractional –order complex Lorenz system and fractional-order complex Lorenz system with time-delay is presented. And the effectiveness of the proposed scheme is demonstrated by the corresponding numerical simulations.3. The stability control of fractional order nonlinear system with multiple time-delaysIn this part, efforts are paid on the stability control of fractional-order nonlinear system with multiple time-delays. Based on the fractional Lyapunov's direct method, we introduced an effective approach to realize the stability control of fractional-order nonlinear system with multiple time-delays using linear feedback control. To illustrate the proposed approach, three fractional-order chaotic systems with multiple time-delays are presented as examples. It is found that with appropriate controller parameters, the stability control of the above systems can be realized. And the effectiveness of the proposed approach is demonstrated by the corresponding numerical simulations.