时滞诱导振荡的非线性相位-振幅减缩
Nonlinear phase-amplitude reduction of delay-induced oscillations
摘要：
时滞引起的自发振荡在许多实际系统中被观察到。延迟微分方程（DDE）描述的极限环振子的相位约化理论已被发展用于分析其同步性质，但它只适用于施加在振子上的扰动足够弱的情况。在这项研究中，我们制定了一个非线性相位振幅减少理论的基础上描述的有限环振荡器的DDE的Floquet定理，这是适用于当振荡器受到中等强度的扰动。我们提出了一种数值方法来评估领先的Floquet本征值，本征函数，和伴随本征函数的必要的减少，并推导出一组低维非线性相位振幅方程近似描述的振荡器动力学。通过分析一个具有立方非线性项的解析易处理的振子模型，我们证明了在无限维状态空间中振子状态的渐近相位可以近似估计，并且在导出的相位-振幅方程的基础上可以预测由中等强度周期扰动引起的振幅的非平凡双稳性.我们进一步分析了一个基因调控振子模型，说明了简化方程可以阐明其在非弱扰动下的复杂动力学机制，这可能与昼夜节律睡眠障碍等实际生理现象有关。
英文摘要：
Spontaneous oscillations induced by time delays are observed in many real-world systems. Phase reduction theory for limit-cycle oscillators described by delay-differential equations (DDEs) has been developed to analyze their synchronization properties, but it is applicable only when the perturbation applied to the oscillator is sufficiently weak. In this study, we formulate a nonlinear phase-amplitude reduction theory for limit-cycle oscillators described by DDEs on the basis of the Floquet theorem, which is applicable when the oscillator is subjected to perturbations of moderate intensity. We propose a numerical method to evaluate the leading Floquet eigenvalues, eigenfunctions, and adjoint eigenfunctions necessary for the reduction and derive a set of low-dimensional nonlinear phase-amplitude equations approximately describing the oscillator dynamics. By analyzing an analytically tractable oscillator model with a cubic nonlinearity, we show that the asymptotic phase of the oscillator state in an infinite-dimensional state space can be approximately evaluated and non-trivial bistability of the oscillation amplitude caused by moderately strong periodic perturbations can be predicted on the basis of the derived phase-amplitude equations. We further analyze a model of gene-regulatory oscillator and illustrate that the reduced equations can elucidate the mechanism of its complex dynamics under non-weak perturbations, which may be relevant to real physiological phenomena such as circadian rhythm sleep disorders.
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