Synchronization Of A Class Of Fractional Order Chaotic Systems With A Bizarre Number Of Equilibrium Points

Recently, many research results show that a large number of fractional orderdynamic systems exist chaos. Because of the potential application value of fractionalorder chaotic systems in many fields, in recent decades, research on fractional orderchaotic systems began to rise. The main research directions about fractional orderchaotic systems include kinetic analysis of chaotic systems, chaos control, chaossynchronization and so on. However，many fractional order chaotic systems in previousstudies have a determining number of equilibrium points, there are few papers aboutfractional order chaotic systems with a bizarre number of equilibrium points(noneequilibrium point and an infinite number of equilibrium points). In addition, the study ofcontrol and synchronization of fractional order chaotic systems was relatively late in ourcountry, control theory and synchronization method related to fractional order chaoticsystems need to be further developed and improved.Motivated by the above discussions, according to some classic fractional orderchaotic systems, this thesis constructs a class of fractional order chaotic systems with abizarre number of equilibrium points. Including a class of fractional order chaoticsystems with none equilibrium point and a class of fractional order chaotic systems withan infinite number of equilibrium points. In this thesis, the Lyapunov exponents andchaotic attractor of fractional order chaotic systems with a bizarre number ofequilibrium points are analyzed. A circuit simulation for a fractional order chaoticsystem with none equilibrium point is implemented. Synchronization of fractional orderchaotic systems is further considered. Based on the stability theory of fractional ordernonlinear systems and the idea of tracking control, suitable controllers are designed,adaptive synchronization of a fractional order chaotic system with none equilibriumpoint and an unknown parameter, complete synchronization of a fractional order chaoticsystem with an infinite number of equilibrium points and generalized synchronizationbetween two fractional order chaotic systems with a bizarre number of equilibriumpoints are realized. Moreover, a new synchronization method named scaling-base drivefunction projective synchronization of fractional order chaotic systems is proposed. Toillustrate the effectiveness of the proposed synchronization method, a rigoroustheoretical analysis and mathematical proof are provided, numerical simulations are performed by MATLAB. Simulation results are consistent with the results of theoreticalanalysis, demonstrating the proposed synchronization method is feasible and effective.