SIAM Journal on Applied Mathematics, Volume 80, Issue 1, Page 422-447, January 2020.
Stochastic oscillations are ubiquitous in many systems. For deterministic systems, the oscillator's phase has been widely used as an effective one-dimensional description of a higher dimensional dynamics, particularly for driven or coupled systems. Hence, efforts have been made to generalize the phase concept to the stochastic framework. One notion of phase due to Schwabedal and Pikovsky is based on the mean--return-time (MRT) of the oscillator but has so far been described only in terms of a numerical algorithm. Here we develop the boundary condition by which the partial differential equation for the MRT has to be complemented in order to obtain the isochrons (lines of equal phase) of a two-dimensional stochastic oscillator, and rigorously establish the existence and uniqueness of the MRT isochron function (up to an additive constant). We illustrate the method with a number of examples: the stochastic heteroclinic oscillator (which would not oscillate in the absence of noise), the isotropic Stuart--Landau oscillator, the Newby--Schwemmer oscillator, and the Stuart--Landau oscillator with polarized noise. For selected isochrons we confirm by extensive stochastic simulations that the return-time from an isochron to the same isochron (after one complete rotation) is always the mean--first-passage time (irrespective of the initial position on the isochron). Put differently, we verify that Schwabedal and Pikovsky's criterion for an isochron is satisfied. In addition, we discuss how to extend the construction to arbitrary finite dimensions. Our results will enable development of analytical tools to study and compare different notions of phase for stochastic oscillators.

中文翻译：

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平面随机振荡器均值-返回时间相位的偏微分方程

SIAM应用数学杂志，第80卷，第1期，第422-447页，2020年1月。
随机振荡在许多系统中无处不在。对于确定性系统，振荡器的相位已被广泛用作高维动力学的有效一维描述，尤其是对于驱动或耦合系统。因此，已经进行了努力以将阶段概念推广到随机框架。Schwabedal和Pikovsky提出的一种相位概念是基于振荡器的均值-返回时间（MRT），但到目前为止，仅根据数值算法进行了描述。在这里，我们开发了边界条件，必须通过该边界条件来补充MRT的偏微分方程，以获得二维随机振荡器的等时线（等相位线），并严格确定MRT等时线的存在性和唯一性函数（最大为加法常数）。我们通过许多示例来说明该方法：随机异质振荡器（在没有噪声的情况下不会振荡），各向同性的Stuart-Landau振荡器，Newby-Schwemmer振荡器和带偏振的Stuart-Landau振荡器噪声。对于选定的等时线，我们通过广泛的随机模拟确定，从等时线到同一等时线的返回时间（一个完整的旋转之后）始终是均值-第一通过时间（与等时线的初始位置无关）。换句话说，我们验证了等时同步的Schwabedal和Pikovsky准则。此外，我们讨论了如何将构造扩展到任意有限尺寸。我们的结果将有助于开发分析工具，以研究和比较随机振荡器的不同相位概念。