Research On Fractional-order R(?)ssler System With The Largest Lyapunov Exponent

Fractional chaotic dynamical systems have richer and more complex dynamics behaviors than that of integer-order systems. In recent years, dynamical systems which based on fractional differential and integral fractional have been researched extensively, involving fractional circuit, fractional digital signal processing, fractional dynamics control systems and fractional-Chaos and super-chaos phenomenon, Chaos Control and fractional chaos synchronization, and security communications. And a lot of research results are obtained. In this thesis, calculate the largest Lyapunov exponents of fractional systems, and period doubling bifurcation of fractional R?ssler system is analyzed based on these. The main contents of this thesis are as follows:1,Research on chaos of fractional chaotic dynamics systemsFirstly, the solution method of fractional system is introduced. Comparing with the merits and drawbacks of solutions in time domain and frequency domain, the solution of fractional system is constructed with the time domain and iteration. With the Matlab software platform, a simulation of a model based on fractional R?ssler system is done.2,Research on estimate the largest Lyapunov exponents of fractional systemsLyapunov exponent is one of the most important characteristic quantities to perform chaos. One positive Lyapunov exponent indicates the existence of chaos. But computing method which is suitable in the integer-order systems cannot be used directly in fractional systems since its own Characteristics. Consequently the calculate methods of estimate the largest Lyapunov exponent mainly focus on analyzing the time sequence of fractional systems. Firstly, phase space reconstruction and small data sets in time sequence are introduced. Phase diagram of fractional R?ssler attractor with proper parameters is given by using computer simulation. Then C-C method is choosed to determine the parameters of phase space reconstruction, the largest lyapunov exponent is calculated by using the small data sets. The simulate result is positive, chaos is proved. Furthermore, robustness of small data set is discussed.3,Chaotic characteristics of fractional R?ssler systemResearch on chaotic characteristics always is a basal aspect of chaos, and phase plot, bifurcation diagram and Lyapunov exponent diagram are the useful tools. The fractional order as the parameter can affect the dynamic behavior in fractional system. In this thesis, the system parameter's influence to system moving Path is discussed firstly, and then the fractional order as parameter is studied too. Under the Matlab platform, bifurcation diagram is given with these parameters. With these, the clear evolutionary process which achieves chaos through period doubling bifurcation is shown. Then small data sets method is used to compute the largest Lyapunov exponents according to the bifurcation diagram. Calculated result consonant with phase plot and bifurcation diagram.