Chaos In Fractional-Order Systems And Their Synchronization

The study on fractional-order calculus theory has a history more than 300 years, however, the scholars studying on fractional-order calculus theory mainly exist in the field of math during the past long period. In 1983, Mandelbort initially suggested that there are plenty of fractal dimension phenomena in the field of nature and the field of science and technology, and there are self-similarities between integer and fraction; then, as the dynamics foundation of geometry and fractal dimension, fractional-order calculus gained new development and become a hot issue of the international. The present article sdudies chaos and chaos synchronization of fractional-order system to the fractional-order dynamics systems . The main achievements of the present article are as fellows:1. Two conversion methods are introduced to study the chaos phenomena of two classical fractional-order systems. First, time-domain and frequency-domain algorithm is introduced to study fractional-order unified system and fractional-order Liu system chaotic dynamic performance systematically, it is revealed that these two fractional-order still have chaotic attractors when the system order is less than 3, and the lowest order is 0.3. Secondly, predictor - corrector method was adopted to study the chaotic phenomenon of fractional-order Liu system, first, time series of predictor -corrector of the system was calculated, and then computer simulation was introduced, phase diagram from different orders and different parameter were presented, and find out the pass by which fractional-order Liu system from periodicity to chaos, the lowest fraction order is 2.61, the existing of chaos was confirmed by computing the max Lyapunov indexes.2. As to the single output fractional-order chaotic system which has a specific state variable as a system output conditions, a controller S was designed according to the fractional-order chaotic system non-linear observer theory and stability theory, with this control input, just by control the state variable x_i and it fractional-order derivative x_i^(α),x_i^(2α) we can achieve the state variable synchronization of controller and observing fractional-order system state variable. Both theoretical analysis and simulation confirmed the validity of this synchronization design.3. The synchronization control of fractional-order system is the focus of recent researches, but chaotic synchronization methods are limited, and few of fractional-order Synchronization of Different Systems was reported. The presented article studied the Synchronization of Different Systems, based on fractional-order linear system and stability theory, using Active control technology, the synchronization of fractional-order unified system and fractional-order Liu system was achieved.