The Study Of Synchronization In Coupled Hamiltonian Systems

The study of synchronization has had a long history since Huygens observed the synchronization of a coupled pair of clock pendulum in 17 century. The chaos synchronization has been a topic of great attention by the pioneering work of Pecora and Carroll in 1990 and its application in various fields.The phenomenon of measure synchronization is a collective behavior which the coupled conservative systems display. It is a novel property of coupled conservative systems, be similar as the conception of synchronization in dissipative system. Measure synchronization should provide a new way in investigating the Hamiltonian systems as well as Quantum chaos.The fist chapter is a brief review about the theory of chaos, including the chaos controlling and synchronization.In the second chapter, we summarize the recent progress of chaos controlling and synchronization, especially, the progress of chaos synchronization and phase synchronization.In the third chapter, we firstly research the property of measure synchronization using global coupling standard map and show the relation of measure synchronization critical coupling strength and nonlinear parameter. We found the critical slowing down phenomenon in measure synchronization for chaotic systems. So, a new method is proposed for judging the critical point of measure synchronization. It is very convenience to determine measure synchronization of chaotic systems. We observed the effect introducing the Gauss white noise in coupled systems and found measure synchronization is steady for weak noise. In this chapter, we also investigate the measure synchronization by a four-dimension symplectic map which is a coupled version of two standard maps, and found measure synchronization still exists when the nonlinear parameters of two subsystems have a small mismatch. It is a better model for studying measure synchronization.In the forth chapter, the behavior of phase difference between two coupled standard map was researched. We have found some significant scaling in critical point neighborhood in numerical and analytical way. For coupled system in periodic or quasi-periodic motion, the measure synchronization is accompanied with the phase synchronization. When the coupled systems are chaotic, that is, the largest Lyapunov exponent is positive, the measure synchronization does not go with the phase synchronization, the phase locking changes random walk. Therefore, the largest Lyapunov exponent of coupled systems has relations with the phase synchronization in measure synchronization.In the fifth chapter, we discussed a coupled pair of high cycle orbit in regular region. A new phenomenon is, when the coupled systems which are composed of high cycle orbit reach measure synchronization, the coupled systems turn into chaos from regular, i.e., the largest Lyapunov exponent become positive. It is different from above result in three chapter. Furthermore, the phase behavior of coupled high cycle orbitsalso is different from that of a coupled pair of quasi-periodic orbits. We also discussed the many body coupling systems. For the coupled systems composed of three different quasi-periodic orbits, the largest Lyapunov exponent also become positive when the coupled systems achieved measure synchronization. These phenomena show measure synchronization has a complicated nature and many characters need to be explored yet.