Intermittent Synchronization Of Fractional Order Chaotic Systems

Fractional calculus was a branch of mathematics as early as the 17 th century,but it has been slow to develop because it has not been supported by the research of physics and other disciplines.In recent years,since the fractional differential system can better explain various phenomena that occur in reality,such as when describing complex physical-mechanical problems,compared with the integer-order model,the physical meaning of the fractional model is clearer and the representation is more concise.Fractional order has a wide range of applications in fields such as chaos,electrochemistry,cell engineering,neuroscience,etc.Therefore,the research on fractional order has received the attention of many researchers.Fractional differential equations have also become a hot spot in current research.Due to the complexity of the theory of fractional differential equations,the results of synchronous research on fractional order chaotic systems are still not many,and there is still much room for research.Most of the currently known methods for the synchronization of fractional order chaotic systems are continuous control.Intermittent synchronization is a special type of discontinuous control,which is achieved in the synchronous control of integer order chaotic systems.There are two main ideas for how to realize the intermittent synchronization of fractional order chaotic systems.One idea is to transform the fractional interval chaotic error system into an integer order intermittent chaotic system for synchronization research;another approach is to directly used the existing theory of fractional order chaotic systems for synchronization research.There are mainly the following four methods:(1)Laplace transform method and time-frequency domain transformation method are used to solve the frequency domain,and the frequency domain expansion is obtained.Then the frequency domain is transformed into the time domain integer-order equation of state.The fractional-order chaotic system is approximately expressed as an integer-order chaos.system,and then intermittent synchronization study;(2)Reconstruct the controlled response system,and based on the Lyapunov stability theory,obtain the intermittent synchronization stability condition of the fractional-order chaotic system.At the same time,select the Chen system for numerical simulation;(3)Determine the stability of the fractional-order intermittent chaotic system based on the characteristics of the solution of the fractional differential equation,and select the Rossler system for verification;(4)Using the Mittag-Leffler stability and the fractional Lyapunov method to study the synchronization of the fractional-intermittent chaotic system,the Lorenz system is used for verification.