Singular Hopf bifurcation in a differential equation with large state-dependent delay
Singular Hopf bifurcation in a differential equation with large state-dependent delay
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具有大状态相关延迟的微分方程中的奇异 Hopf 分岔
DOI：
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
T. Erneux
中科院分区：
文献类型：
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作者：
G. Kozyreff;T. Erneux
研究了一类受控制理论启发的经典状态依赖时滞微分方程的持续振荡问题。由于考虑的大时滞，Hopf分岔是奇异的，振荡迅速获得一个不稳定的阈值后的轮廓。使用渐近技术，我们明确地捕捉到从近正弦振荡的逐渐变化。延迟对解的依赖可以是线性的或非线性的，至少具有二次依赖。在前一种情况下，与瑞利振子的渐近连接。在后者中，货车der Pol的方程推导出的小振幅振荡。SDD微分方程是目前研究的热点，目的是建立或修正常时滞微分方程的一般定理，但其解的解析构造很少。本文说明了奇异摄动技术的使用和不寻常的方式，其中可解性条件可能会出现SDD问题的大延迟。
We study the onset of sustained oscillations in a classical state-dependent delay (SDD) differential equation inspired by control theory. Owing to the large delays considered, the Hopf bifurcation is singular and the oscillations rapidly acquire a sawtooth profile past the instability threshold. Using asymptotic techniques, we explicitly capture the gradual change from nearly sinusoidal to sawtooth oscillations. The dependence of the delay on the solution can be either linear or nonlinear, with at least quadratic dependence. In the former case, an asymptotic connection is made with the Rayleigh oscillator. In the latter, van der Pol’s equation is derived for the small-amplitude oscillations. SDD differential equations are currently the subject of intense research in order to establish or amend general theorems valid for constant-delay differential equation, but explicit analytical construction of solutions are rare. This paper illustrates the use of singular perturbation techniques and the unusual way in which solvability conditions can arise for SDD problems with large delays.