Universal description of stochastic oscillators - higher dimensional examples, extraction of the mapping from data, and networks of oscillators

许多重要的非平衡系统表现出由明显的相位和幅度波动所塑造的振荡行为。对于这种随机振子，已经提出了具有不同机制的不同的数学模型：(I)带有噪声的焦点不动点的动力学(具有热涨落的阻尼谐振子将是一个突出的例子)；(Ii)受白噪声扰动的极限环和弛豫振子；(Iii)由涨落驱动的可激发系统；(Iv)由噪声驱动的异宿系统。这些模型大多是强非线性的，因此很难在理论上处理。最近由我和我的合作者提出的一个理论，提供了一个用倒向Kolmogorov算子的一个本征函数来描述随机振子的通用框架。在将系统变量映射到一个新的复值变量后，振子的自发涨落统计量、振子的线性响应以及它与另一个与其耦合的随机振子的互相关都可以用简单而准确的解析表达式来捕捉。这种描述与随机振荡的机制无关。在这个项目中，我想在三个方面应用和推广这一理论：(I)它应该应用于比以前研究的二维更高的振荡器；(Ii)程序核心的非线性变换应该从(模拟或实验)数据中提取；(Iii)耦合随机振荡器的理论将更彻底地植根于，而且，扩展到随机振荡器网络的情况(两个以上的振荡器，前面研究的情况)。这些对框架的探索和扩展对于热力学平衡以外的许多系统的理论描述具有潜在的应用，特别是在生物学中。

Many important non-equilibrium systems display oscillatory behavior that is shaped by pronounced fluctuations in phase and amplitude. Different mathematical models with distinct mechanisms have been suggested for such stochastic oscillators: (i) dynamics with a focus fixed point endowed with noise (the damped harmonic oscillator with thermal fluctuations would be a prominent example); (ii) limit cycles and relaxation oscillators perturbed by white noise; (iii) excitable systems driven by fluctuations; (iv) heteroclinic systems driven by noise. Most of these models are strongly nonlinear and are thus difficult to treat theoretically. A recent theory, put forward by my collaborators and me, provides a universal framework for the description of stochastic oscillators in terms of one of the eigenfunctions of the backward Kolmogorov operator. After projecting the system's variables to a new complex-valued variable, the spontaneous fluctuation statistics, the linear response of the oscillator, and its cross-correlation with another stochastic oscillator to which it is coupled, are all captured by simple and exact analytical expressions. This description is independent of the mechanism for the stochastic oscillation. In this project I want to apply and generalize this theory in three respects: (i) it should be applied to oscillators that have higher dimensions than two going beyond what has been studied before; (ii) the nonlinear transformation at the core of the procedure should be extracted from (simulation or experimental) data; (iii) the theory for coupled stochastic oscillators will be more thoroughly grounded and, moreover, extended to the case of networks of stochastic oscillators (more than two oscillators, the case previously investigated). These explorations and extensions of the framework have potential applications for the theoretical description of many systems outside of thermodynamic equilibrium, in particular in biology.