Research On Several Problems Of Generalized Synchronization And Phase Synchronization

Nonlinear science has gradually become a hot topic of scientific research during the past 20 years. With its vigorous development, applied mathematics, mechanics and physics has achieved substantial progress. The application of nonlinear science also related to many fields of natural sciences, engineering and social science. Chaos, which is the main one of the branches of nonlinear science, has been widely studied and applied. Chaos synchronization is an important research direction, and it has showed a good prospect of application in many fields such as communications security, medical science, ecology and even economy. In this paper, two common kinds of chaos synchronization: generalized synchronization (GS) and chaotic phase synchronization (CPS) are studied.The origin and development of chaos, some related notions,theory and methods of chaos synchronization are given in the introduction. The relevant contents and progress of GS and CPS are focused. We also introduced the development of chaos synchronization in complex networks. The existence and applications of GS and CPS are more extensive compare to other synchronizations. GS and CPS of drive-response chaotic systems as the coupling strength changing are first studied. Most studies of CPS were based on numerical simulation and appropriate quantitative indexes. So quantitative indexes to measure CPS in amplitude coupled systems and complex dynamical networks are investigated respectively. Numerical simulations are given. At last, the conclusions and prospects are given. The detailed works are as follows:(1) By using the auxiliary system approach and calculating the average winding number, we study GS and CPS of drive-response chaotic systems as the coupling strength changing. Find that in some condition, GS can be weaker than CPS, and CPS is a kind of GS.(2) In order to investigate CPS by using quantitative indexes, the complex frequency order parameter of dynamics is defined. We use it to study CPS in amplitude coupled systems. We find that there being a correspondence between CPS and the complex frequency order parameter. It is an appropriate quantitative characteristic for measuring CPS.(3) Appropriate quantitative indexes are investigated to measure CPS in the chaotic oscillator networks. The mean phase locking value and mean frequency difference with adjacent nodes of the network are defined. Lorenz chaotic oscillators with several rotational centers are chosen as networks nodes. The chaotic oscillator networks are formed. We find that the mean phase locking value and mean frequency difference have some connection with CPS.For all the above studies, we use Matlab to program and obtain simulation results. The simulation results are in agreement with the conclusions.