Fractional Chaos System Synchronization With Non-equilibrium Points

The fractional order system is a dynamic system which is described by differential equations of arbitrary real numbers and even complex numbers.It can describe the complex phenomena appeared in the actual system more accurately and effectively.It has been widely applied in mathematical science,chemical engineering science and nonlinear dynamic systems and control and many other fields.Chaos is a special kind of nonlinear system.Chaotic system has caught the interest of scholars since it was put forward by the early twentieth century include such traits as its inherent randomness and extremely sensitive to the initial condition.It is worth noting that these special systems are different from normal chaotic ones because the classical criterion cannot be used to prove chaos of such systems.The research in this article focuses on the following aspects:First,this paper study basic theory of fractional calculus and the theory of non-linear system.Study the chaotic phenomena combination of natural and social sciences and introduces the basic method synchronizing of integer order chaotic system.Then a new non-equilibrium fractional chaos system is analyzed,and the solution sequence is obtained by using fractional differential transformation.Study the Kaplan Yorke dimension and dissipation of fractional order chaotic systems without equilibrium,then obtained the maximum Lyapunov characteristic index by QR decomposition based on the discrete mapping of system.According the maximum of Lyapunov index.It can be judged whether the system keep chaos.Then,an all-state adaptive control method is presented which can track the expected trajectory of the system.Finally,the feasibility of the control method is verified by numerical simulation.In addition,the synchronization problem of the chaotic system with uncertain parameters and no equilibrium is studied.Based on the idea of adaptive control,adaptive control method is introduced into the chaotic synchronization.Use a new class of no equilibrium of fractional order chaotic system as an example,design the corresponding synchronous controller and parameter adaptive law achieve the synchronization of no equilibrium between the same structures' fractional order chaotic systems with different parameters.At the same time,active control method is applied to construct the active controller to study the system.Unprofitable nonlinear term in the error of the system through the design of the controller offset,puts forward a kind of simplified fractional order chaotic system synchronization controller design method,Through compare the two methods by the numerical simulation,we can verify the effectiveness of the two methods of synchronization.By comparing the two methods,we can draw a conclusion that the improved adaptive control method has faster speed and higher control efficiency,and active control method has more simple calculation process.In the end,the fractional order with a new fractional order chaotic systems using adaptive method to synchronize hyper chaos system research,design the corresponding synchronous controller and parameter adaptive law,achieve no balance fractional order chaotic systems synchronization between different structures.The effectiveness of this method is verified by simulation.