Synchronization And Anti-synchronization Of Typical Chaotic Systems And Coupled Neurons

Chaos control and synchronization of chaotic systems is the premise for applying them in communication systems. On the one hand, due to the strong confidentiality and anti-interference ability of chaos communication, studying of chaos control and synchronization has some practical significance, when chaos is used in the industrial systems' communication; On the other hand, there are also chaotic phenomena in biological system, such as the brain system and visual system. In order to learn the communication mechanism of neurons, analyzing of the synchronization between them has some theoretical significance. In this paper, based on nonlinear control methods, synchronization and anti-synchronization of a class of typical chaotic systems is studied via theoretical analysis, and also the synchronization between coupled neurons. Finally, the results are illustrated by numerical simulation. The major work of this paper is summarized as follows:(1) Synchronization of the chaotic and hyper-chaotic systems with identical or different structures is studied, which is based on direct construction method. Firstly, an appropriate controller for the response system is designed by direct construction method. And then transform the error system with the controller into triangular structure, so it can be obtained that the error system is asymptotically stable at the origin, which shows the drive system and response system achieve synchronization. Through numerical simulation, the effectiveness of this method is proved further.(2) The dynamics of fractional order hyper-chaotic Chen system is studied. The results show:when the order equal to or greater than 0.95, the fractional order hyper-chaotic Chen system is hyper-chaotic;and with the order increasing, the chaos becomes evident increasingly. The author designs a control strategy to realize self-synchronization of two fractional order hyper-chaotic Chen systems by active control method, case in the order is 0.95 and 0.98, respectively. The results of numerical simulation show the effectiveness of the strategy to a class of fractional order hyper-chaotic Chen systems.(3) The anti-synchronization of two hyper-chaotic Lorenz systems is studied, as well as hyper-chaotic Lorenz system and hyper-chaotic Chen system with different structures. Nonlinear control method is used to the two hyper-chaotic Lorenz systems with certain parameters, and adaptive control method is used to the hyper-chaotic Lorenz system and hyper-chaotic Chen system with uncertain parameters, respectively. The Lyapunov function and parameters'update laws are selected, which can identify the unknown parameters successfully. The numerical simulation shows the effectiveness of the strategies.(4) The chaotic synchronization of two electrical coupled FitzHugh-Nagumo (FHN) neurons with unknown parameters via adaptive control is investigated. Based on the Lyapunov stability theory, an adaptive controller and parameters'update laws are designed, which can achieve the synchronization of the two gap junction coupled FHN neurons when the individual neuron is chaotic, but without considering the coupling strength. The controller is robust to the uncertainties such as approximate error, ionic channel noise and external disturbances, and the parameters can be identified successfully. The numerical simulation results confirm the effectiveness of the designed controller.