随机非线性动力学的Volterra级数方法：白噪声驱动下Van der Pol振子的线性响应
Volterra-series approach to stochastic nonlinear dynamics: linear response of the Van der Pol oscillator driven by white noise
摘要：
Van der Pol方程是弛豫振荡的典型模型。这种显著的非线性自持振荡运动现象是自然界和电气工程中重要的节律过程的基础。真实系统中的弛豫振荡通常与环境噪声耦合，这进一步丰富了它们的动力学，但使得对此类系统的理论分析和方程参数值的确定成为一项困难的任务。在另一篇文章中，我们提出了一个类似的问题的另一个经典的非线性模型，D-Duffing振子的分析方法。在这里，我们扩展我们的技术的情况下，由白噪声驱动的范德波尔方程。我们分析了解决方案的统计，并提出了一种方法来估计参数值从振荡器的时间序列。我们使用生物系统中主动振荡的实验数据来演示我们的方法如何适用于实际观测，以及如何将其推广到更复杂的模型。
英文摘要：
The Van der Pol equation is a paradigmatic model of relaxation oscillations. This remarkable nonlinear phenomenon of self-sustained oscillatory motion underlies important rhythmic processes in nature and electrical engineering. Relaxation oscillations in a real system are usually coupled to environmental noise, which further enriches their dynamics, but makes theoretical analysis of such systems and determination of the equation's parameter values a difficult task. In a companion paper we have proposed an analytic approach to a similar problem for another classical nonlinear model, the bistable Duffing oscillator. Here we extend our techniques to the case of the Van der Pol equation driven by white noise. We analyze the statistics of solutions and propose a method to estimate parameter values from the oscillator's time series. We use experimental data of active oscillations in a biological system to demonstrate how our method applies to real observations and how it can be generalized for more complex models.
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