Synchronization of four coupled van der Pol oscillators
Synchronization of four coupled van der Pol oscillators
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DOI：
10.1007/s11071-008-9402-y
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发表时间：
2009-06-01
影响因子：
5.6
通讯作者：
Sen, Mihir
中科院分区：
文献类型：
--
作者：
Barron, Miguel A.;Sen, Mihir
由于轴的弹性耦合，叶轮机械叶片中的自激振动有可能发生同步。本文给出了四个耦合van der Pol振子的同步简化模型。对于拟线性振动，通过将原方程在未扰动的极限环附近线性化并将其转化为Hill方程的分析，得到了一个稳定性条件。对于非线性情形，数值模拟表明在参数空间中存在两个明确定义的相关系区域，其中嵌入了多个周期吸引子。这些区域的大小强烈地依赖于振子和耦合常数的值。对于低于某一临界值的耦合常数，存在一个相移吸引子多样性的区域，而对于高于该临界值的相移吸引子，则以同相吸引子为主。观察到亚临界区反相吸引子的存在与耦合振子周期的突变有关。在几个初始条件下，讨论了耦合系统收敛到特定周期吸引子的问题。将研究扩展到非全同振子，发现即使在很大范围内振子常数的差异也是同步的。
It is possible that self-excited vibrations in turbomachine blades synchronize due to elastic coupling through the shaft. The synchronization of four coupled van der Pol oscillators is presented here as a simplified model. For quasilinear oscillations, a stability condition is derived from an analysis based on linearizing the original equation around an unperturbed limit cycle and transforming it into Hill's equation. For the nonlinear case, numerical simulations show the existence of two well-defined regions of phase relationships in parameter space in which a multiplicity of periodic attractors is embedded. The size of these regions strongly depends on the values of the oscillator and coupling constants. For the coupling constant below a critical value, there exists a region in which a diversity of phase-shift attractors is present, whereas for values above the critical value an in-phase attractor is predominant. It is observed that the presence of an anti-phase attractor in the subcritical region is associated with sudden changes in the period of the coupled oscillators. The convergence of the coupled system to a particular periodic attractor is explored using several initial conditions. The study is extended to non-identical oscillators, and it is found that there is synchronization even over a wide range of difference among the oscillator constants.