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首页> 外文期刊>IEEE Transactions on Circuits and Systems. I, Regular Papers >Singularity induced bifurcation and the van der Pol oscillator
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Singularity induced bifurcation and the van der Pol oscillator

机译：奇异性引起的分叉和范德波尔振荡器

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摘要

In parameter dependent differential-algebraic models (DAEs) of the form x˙=f and 0=g, it has been shown recently that the generic codimension one local bifurcations are the well-known saddle node and Hopf bifurcations and a new bifurcation called the singularity induced bifurcation. The latter occurs generically when an equilibrium of the DAE system crosses the singular surface of noncausal points. In this paper, it is shown that when singularly perturbed models of the form x˙=f and ∈y˙=g are considered, the singularity induced bifurcation in the slow DAE system corresponds to oscillatory behavior in the singularly perturbed models. As an example, it is proved that the oscillations in the classical van der Pol oscillator arise when a stable equilibrium undergoes the singularity induced bifurcation in the slow DAE system, which in turn corresponds to the occurrence of supercritical Hopf bifurcations in the singularly perturbed models
机译：在形式为x = f和0 = g的参数相关的微分代数模型（DAE）中，最近发现，通用共维数一个局部分支是众所周知的鞍形结和Hopf分支，以及一个称为的新分支。奇异性引起分叉。后者通常在DAE系统的平衡穿过非因果点的奇异表面时发生。在本文中，表明当考虑形式为x＆= f和∈y＆= g的奇异摄动模型时，慢DAE系统中的奇异性引起的分叉对应于奇异摄动模型中的振荡行为。例如，证明了当稳定平衡在慢速DAE系统中经历奇异性引起的分叉时，经典van der Pol振荡器中出现了振荡，这又对应于奇摄动模型中超临界Hopf分叉的出现。

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