Chaos Characteristics Analysis Of Fractional Nonlinear Systems With Synchronous Control Algorithm

Study of chaotic dynamics performances and its applications of fractional-order nonlinear chaotic systems is a hot topic in the field of nonlinear science. In this thesis, taking the fractional-order simplified Lorenz system as the model, adopting the methods of theory analysis and numerical simulation, the dynamic characteristic, synchronization control and its application of fractional-order simplified Lorenz system were invevstigated. Research results are important to study chaotic theory and its application further.By using the time-domain solution of the fractional-order calculus, and the Matlab platform, the bifurcations and chaotic behaviors of the fractional-order simplified Lorenz system were analyzed and discussed numerically by means of phase portraits, bifurcation diagrams, and the largest Lyapunov exponent. It showed that both the system parameter and the fractional orders can be taken as bifurcation parameters. It has complex chaotic dynamics with interesting characteristics are presented for a wide range of fractional orders, both less than and greater than three. The range of existing chaos is different with different parameters and orders. The lowest order we found for the fractional-order simplified Lorenz system to yield chaos is 2.66.Synchronization approaches, including projective synchronization, generalized projective synchronization, function projective synchroniz-ation, and modified function projective, were applied to the fractional-order chaotic system with different initial values. Based on Routh-Hurwitz stability criterion and Lyapunov stability principle, synchronization theorems were obtained, and the range of control parameters was determined. The relations between the synchronization performances and the control parameters were discussed. The corresponding numerical simulations based on Matlab were done to verify the effectiveness of the proposed schemes for fractional-order chaotic system. Chaotic masking schemes for secure communication were designed and realized successfully based on generalized projective synchronization and modified function projective synchronization respectively. The results of the numerical simulation and performance analysis verified the effective and correct of the proposed secure communication schemes, and it lays a foundation for the applications of fractional-order chaotic system.