Van der Pol时滞弱耦合系统多尺度分析

Multiple Scale Analysis of weakly Coupled Van der Pol System with Time Delay

对于非线性耦合项中带有时滞的van der Pol系统,采用多尺度法对该系统进行定性以及定量的分析。研究结果表明,对于van der Pol时滞耦合系统,时滞的存在影响了系统的稳定性,使系统的周期解发生了静态分岔和Hopf分岔。研究还发现,对于耦合强度较弱的情形,利用多尺度法对系统进行定量分析是合理可靠的。我们取不同的耦合强度作出了系统的时间历程图、相图和分岔图,分析了解析解与数值解之间产生误差的原因。本文所研究的系统来源于耦合的激光振荡器。

The method of multiple scale is used to obtain qualitative and quantitative analysis of nonlinear coupled van der Pol system with time delay. The research results indicate that the time delay has an effect on the stability of the motion,and the static bifurcation or the Hopf bifurcation is happened upon the periodic solutions of the system. It is also found reasonable and dependable by utilizing the method of multiple scale to make a quantitative analysis for weakly coupled system. Time history diagrams.phase diagrams and bifurcation diagrams of the system with different coupling level were made to analysis by reason caused the the error of approximate solutions and numerical solutions. The time delay coupled van der Pol system corresponds to the laser oscillators in engineering.