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The dynamics of two coupled van der Pol oscillators with delay coupling.

机译：具有延迟耦合的两个耦合范德波尔振荡器的动力学。

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摘要

In this work, we investigate the dynamics of two weakly coupled van der Pol oscillators in which the coupling terms have time delay t . Our work is motivated by applications to laser dynamics and the coupling of microwave oscillators. The governing equations are x&d3;1+x 1-e1-x 21 x&d2;1=e ax&d2; 2t-t , x&d3;2+x 2-e1-x 22 x&d2;2=e ax&d2; 1t-t , where the coupling is chosen to be through the damping terms because this form of coupling occurs in radiatively coupled microwave oscillator arrays. We use the method of averaging to obtain the approximate simplified system of three slow-flow equations R&d2;1= 12R 11-R2 14+a R2cos f+t , R&d2;2= 12R 21-R 224 +aR1 cosf-t , f&d2; =a2 -R2R 1sinf +t-R1 R2 sinf-t . Equilibria of these slow-flow equations correspond to periodic motions in the original equations. In the examination of the stability and bifurcation of the equilibria of these equations, we found that the in-phase and out-of-phase modes coexisted and were both stable in the parameter range for which the delay is about ¼ of the unperturbed limit cycle period. We also found that the in-phase mode ceased to exist if the delay was about ½ of the unperturbed period and the coupling was strong enough. Similarly the out-of-phase mode ceased to exist if the delay was approximately the same as that of the unperturbed period. We also found that if the coupling was sufficiently small, various other motions were predicted to exist besides the in-phase and out-of-phase modes. These additional motions were predicted to change their form through a series of elaborate bifurcations. Nevertheless all these motions were predicted to be periodic, and we did not observe chaos for any parameter values.;In order to check the validity of the approximations, we numerically integrated the original differential delay equations for the case e1 and t=O1 and compared their predictions regarding the stability of the in-phase and out-of-phase modes with those of the slow-flow analysis. The two sets of results showed excellent agreement.
机译：在这项工作中，我们研究了两个弱耦合的范德波尔振荡器的动力学，其中耦合项具有时间延迟t。我们的工作受到激光动力学和微波振荡器耦合应用的推动。控制方程为x＆d3; 1 + x 1-e1-x 21 x＆d2; 1 = e ax＆d2; 2t-t，x＆d3; 2 + x 2-e1-x 22 x＆d2; 2 = e ax＆d2; 1t-t，其中选择通过阻尼项进行耦合，因为这种耦合形式发生在辐射耦合微波振荡器阵列中。我们使用求平均值的方法来获得三个慢流方程的近似简化系统R＆d2; 1 = 12R 11-R2 14 + a R2cos f + t，R＆d2; 2 = 12R 21-R 224 + aR1 cosf-t，f＆d2 ; = a2-R2R 1sinf + t-R1 R2 sinf-t。这些慢流量方程的平衡对应于原始方程中的周期性运动。在检查这些方程式的稳定性和分叉性时，我们发现同相和异相模式共存，并且在延迟约为无扰动极限周期的1/4的参数范围内均稳定。期。我们还发现，如果延迟约为无扰动周期的1/2，并且耦合足够强，则同相模式将不复存在。类似地，如果延迟与无扰动周期的延迟大致相同，则异相模式将不复存在。我们还发现，如果耦合足够小，则可以预测除同相和异相模式外还会存在其他各种运动。预计这些额外的运动会通过一系列复杂的分叉来改变其形式。尽管如此，所有这些运动都被预测为周期性的，并且对于任何参数值我们都没有观察到混乱。为了检查逼近的有效性，我们对e 1和t = O1并将其关于同相和异相模式稳定性的预测与慢流分析的稳定性进行了比较。两组结果显示出极好的一致性。

著录项

  • 作者

    Wirkus, Stephen Allen.;

  • 作者单位

    Cornell University.;

  • 授予单位 Cornell University.;
  • 学科 Mathematics.;Engineering General.
  • 学位 Ph.D.
  • 年度 1999
  • 页码 146 p.
  • 总页数 146
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类 数学;工程基础科学;
  • 关键词

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