平面随机振子平均返回时相位的偏微分方程
A Partial Differential Equation for the Mean--Return-Time Phase of Planar Stochastic Oscillators
摘要：
随机振荡在许多系统中普遍存在。对于确定性系统，振子的相位已被广泛用作高维动力学的有效一维描述，特别是对于驱动或耦合系统。因此，已作出努力，推广阶段的概念，随机框架。由于Schwabedal和Pikovsky的相位的一个概念是基于振荡器的平均返回时间（MRT），但迄今为止仅在数值算法方面进行了描述。在这里，我们开发的边界条件，偏微分方程的MRT必须补充，以获得等时线（线的等相位）的二维随机振荡器，并严格建立的MRT等时函数的存在性和唯一性（直到一个附加常数）。我们说明的方法与一些例子：随机异宿振荡器（不会振荡的情况下，噪声）;各向同性的斯图尔特-朗道振荡器，纽比-Schwemmer振荡器，和斯图尔特-朗道振荡器与极化噪声。对于选定的等时线，我们确认广泛的随机模拟，从一个等时线到同一等时线的返回时间（一个完整的旋转后）总是平均首次通过时间（无论初始位置的等时线）。换句话说，我们验证了Schwabedal和Pikovsky的标准的等时线是满意的。此外，我们还讨论了如何将该构造推广到任意有限维。我们的研究结果将使分析工具的发展，研究和比较不同的概念，随机振荡器的相位。
英文摘要：
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.
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