Research On Fractional Order Chaotic Systems With Coexisting Attractors And Their Synchronous Control

As an important part of nonlinear science,chaos theory has important application value in the fields of circuit engineering,image encryption and secure communication.With the deep research on chaos in academia,it is founded that many integer-order systems have the property of fractal dimension.Fractional-order systems have more complex dynamic behaviors than integer-order systems because their orders can be adjusted flexibly.Therefore,fractional calculus can be used to describe the nonlinear behavior of actual chaotic systems more objectively and is more suitable for engineering practice.Aiming at the problems of single dynamic behaviors of attractors and complex design of synchronous controller for existing fractional systems,taking fractional order system as a research object,this thesis studies the design of novel systems and their synchronous control.The main works are as follows:1.A novel three-dimensional integer-order chaotic system and its corresponding fractional-order system with coexisting attractors are designed and analyzed.An analog circuit of fractional-order system is given and the system's synchronization problem with unknown parameters is studied.Theoretical analysis and simulation results show that:(1)Integer-order system has the characteristic of coexistence of two isolated two-wing chaotic attractors and one four-wing chaotic attractor.There are also two isolated two-wing chaotic attractors in fractional order system.(2)For the fractional-order system,adaptive synchronization control can be effectively achieved through the designed synchronization controller.2.A novel three-dimensional integer-order chaotic system and its corresponding fractional-order system with multiple coexisting attractors are designed and analyzed.An analog circuit of fractional-order system is given and the system's stabilization of unstable equilibrium points is studied.Theoretical analysis and simulation results show that:(1)Integer-order system has the characteristic of coexistences of multiple attractors,such as the coexistence of two period-1 limit cycles,the coexistence of two period-2 limit cycles,the coexistence of one-wing and one-wing chaotic attractors,and the coexistence of two two-wing chaotic attractors and one four-wing chaotic attractor.(2)Fractional-order system also has multiple coexistences of attractors,such as the coexistence of periodic and periodic attractors,the coexistence of one-wing and one-wing chaotic attractors,and the coexistence of two two-wing chaotic attractors and one four-wing chaotic attractor.(3)The fractional order system can achieve asymptotic stability through the linear feedback scalar controller composed of single state variable.