Phase Synchronization Of Chaotic System

Since synchronization phenomenon has been revealed,the effect of the chaotic synchronization between coupled subsystems has been one of the hot topics in the nonlinear science.Along with the continuous deepening of research,the basic concept of chaos from the initial perfect chaos synchronization expand to generalized chaos synchronization,chaotic phase synchronous,imperfect chaotic phase synchronization, and so on,from their field of physics into biology,electrical and electronic engineering,communications technology,and other disciplines.So far,despite the chaotic phase synchronization has been achieved certain results,but the exploration of weak effect between the chaotic subsystems is in the initial stage,and many of the issues such as the stability of phase synchronization,the synchronization mechanism, and the influence of different coupling,and so on,have yet to be studied further, in-depth study of chaotic phase synchronization of coupling system,not only it is theoretical significance to reveal the chaotic interaction mechanism,and the process of chaos,but also it is guiding value to the practical application.This paper focus on the theme of the chaotic phase synchronization of coupling system,phase synchronization of parametric excited Rossler system has been investigated in chapter 4.It has been demonstrated that the mean frequency of chaotic attractor and the frequency of the parametric excitation may be locked in different ratios for certain parameter conditions,implying phase synchronization can be observed.The evolution from non-synchronized state to phase synchronization has been discussed in details,which reveals different phase dynamics may exist during the process.With the variation of parameters,the imaging point on the Poincaréplane may finally settle down onto the attractor,which yields phase synchronized state.Phase synchronization of two coupled Rossler systems with 1:1 and 1:2 internal resonances is investigated in chapter 5.For the different internal resonances,with the increase of the coupling parameterε,both differences between the mean frequencies of the two sub-oscillators approach zero,implying perfect phase synchronization can be achieved for strong interaction between the two oscillators.In the process to perfect phase synchronization,for the primary resonance,the amplitudes of the fluctuations of the difference seem much smaller comparing to the case with frequency ratio 1:2,even with weak coupling strength.Further investigation reveals that the states from non-synchronization to imperfect phase synchronization until perfect phase synchronization are related to the critical variations of the Lyapunov exponents,which can also be demonstrated by the diffuse clouds.Phase synchronization between nonlinearly coupled systems with 1:1 and 1:2 resonances is investigated in chapter 6.It demonstrates that for the different internal resonances,with relatively small parameterε,both differences between the mean frequencies of the two sub-oscillators approach zero,implying phase synchronization can be achieved for weak interaction between the two oscillators.With the increase of the coupling strength,fluctuations of the frequency difference can be observed,and for the primary resonance,the amplitudes of the fluctuations of the difference seem much smaller comparing to the case with frequency ratio 1:2,even with weak coupling strength.Unlike the enhance effect on the synchronization for linear coupling,the increase of nonlinear coupling strength results in the transition from phase synchronization to non-synchronized state.Further investigation reveals that the states from phase synchronization to non-synchronization are related to the critical changes of the Lyapunov exponents,which can also be explained by the diffuse clouds.At last,Phase synchronization between linearly and nonlineady coupled systems with fundamental resonance is investigated in this paper.It demonstrates that the detuning parameterσbetween the two natural frequenciesω₁ andω₂ affects phase dynamics,and with the increase of the linear coupling strength,the effect of phase synchronization between two sub-systems was enhanced,while decayed as nonlinear coupling strength increases.Further investigation reveals that the transition of phase states between the two oscillators are related to the critical changes of the Lyapunov exponents,which can also be explained by the diffuse clouds.