Study On Stable Theorem And Synchronizing Approach About Fractional Chaotic System

Chaos universally exists in natural sciences and social sciences. Chaos is usually harmful. In controllingarea, it will cut down the controllability of the equipment controlled and exacerbate the damage toequipment. In testing area, the measuring results will be mixed by chaotic noise, and then it necessary toremove chaotic noise to ensure measure precision. Hence, it is necessary to study how to remove chaoticnoise to improve measure precision. This paper puts emphasis on studying the approach of removing noiseby synchronizing chaotic systems and the universally approach of synchronizing fractional chaotic system.As the stable theory for fractional chaotic systems is a developing area and imperfect, it hinders thedevelopment of the approach synchronizing and controlling fractional chaotic system. The maincontribution of this paper is about fractional system as follows:1.Fractional systems are distinguished into proper fraction systems and improper fraction systems andthe idea that different types of systems are studied respectively is proposed.2.Three stable theories about proper fractional nonlinear system are proposed and proved: 1)the theorem that a fractional system with order not more than 1 is stable if integer order formof the fractional system is stable; 2) h function stable theorem; 3) Lyapunov function stabletheorem.3.Integer order Chua's chaotic system and integer order Lorenz chaotic are analyzed in this paper.Fractional physical models of Chua's chaotic system and Lorenz chaotic system are respectivelyintroduced according to "middle processing" theorem. The relations between fractional chaotic system andfractional order are respectively analyzed.4.1) Configurating special matrix approach is proposed. Synchronizing not only fractional chaoticLorenz system with unknown parameters but also fractional chaotic CYQY system withfractional chaotic Lorenz system with unknown parameters is realized based on this approach.2) Backstepping approach is extended from integer order system to fractional order system. Synchronizingfractional order chaotic Newton-Leipnik system is realized by designing controllerbased on backstepping approach. 3) Linear feedback approach is used to synchronize fractionalchaotic system with unknown parameters and unknown parameter update rule is also designed. Theproblem how to fix on feedback coefficient is also resolved。5.Fractional order chaotic Chua's circuit is designed in this paper. First, viriable-scale reduction, fractional to integral are made on the Chua's non-dimension state equation. Then each module circuit basedon the state equation is designed and linked with the other circuits by corresponding relationships betweenstate variables. Then each module circuit based on the state equation is designed and linked with the othercircuits by corresponding relationships between state variables. The circuit consists of inverted adder,fractional differential, inverter and controller, and has symmetric structure. Hardware experiment results arein good agreement with the computer simulations and synchronizing fractional chaotic Chua's system isrealized with circuit.