The Research On Control Of Several Typical Fractional-Order Chaotic Systems

Since the rise of the chaos,classical chaotic systems have always been a hot research spot because of their representation and particularity,such as Chua circuit,Lorenz system,Chen system,Lü system,BLDCM system,etc.For such classical chaotic systems or its deformation,a variety of stability control schemes and synchronization methods have been given,but most of the researches are based on integer order systems.In recent years,with the deepening of the fractional calculus theory research and application,the fractional differential equation can be used to more precisely model many real physical processes,e.g.,electrode-electrolyte polarization,dielectric polarization,heat conduction,reaction-diffusion,super-diffusion.Study shows that the fractional order system has more complex dynamic behavior,including the familiar chaotic behavior.People began to extend the classical integer order to fractional order chaotic systems,and its control problem is studied.Researches about chaotic control and synchronization of fractional order dynamic systems have potential applications in many fields of physics and engineering science,such as complex network control,information transmission,authentication encryption,image hiding,secure communication,and so on.Motivated by the above-mentioned classical integer order chaos system,this paper constructs the corresponding fractional order system,and innovatively put forward the single input stability control method and compound generalized function projective synchronization scheme.First of all,a modified fractional-order Chua’s chaotic circuit is reported,and the chaotic attractor is given.Based on the Mittag-Leffler function in two parameters and Gronwall’s Lemma,a modified fractional-order Chua’s chaotic circuit can be stable via a fractional-order scalar controller,which can be determined by the Caputo fractional derivative of a single input.Then a fractional-order BLDCM chaotic system is introduced.Based on the fractional-order extension of the Lyapunov direct method,we propose a control scheme determined by a single-state variable,and the unstable equilibrium points of the fractional-order brushless DC motors are stabilized in the sense of Lyapunov.Finally,we present a new synchronization scheme for fractionalorder chaotic system and call this type of synchronization compound generalized function projective synchronization,or briefly denote it by CGFPS.There are one scaling-drive system,more than one base-drive system,and one response system,and the scaling function matrix comes from multidrive systems.To verify its effectiveness,we achieve the CGFPS between three drive systems and one response system.Numerical simulations suggest that the presented CGFPS scheme works well.The strict mathematical deduction of control and synchronization scheme is given,all the results of numerical simulation are consistent with theoretical analysis.The all simulation results show that the proposed scheme is effective.