[论文解读] Two-time, response-excitation moment equations for a cubic half-oscillator under Gaussian and cubic-Gaussian colored excitation. Part 1: The monostable case

G. A. Athanassoulis, Ivi C. Tsantili+1 more|arXiv (Cornell University)|Apr 8, 2013

Spectroscopy and Quantum Chemical Studies参考文献 60被引用 4

一句话总结

本文提出了一种新颖的两时间、响应-激励矩方程方法，用于非线性系统在色噪声、非高斯激励下的响应分析。通过结合精确的时间闭合条件与高斯矩闭合方法，推导出一个因果的、非局部的单时间矩系统，能够准确预测响应统计特性——经蒙特卡洛模拟验证，表明激励相关时间与形状对单稳态三次半振子在三次-高斯色噪声下的响应协方差具有显著影响。

ABSTRACT

In this paper a new method is presented for the formulation and solution of two-time, response-excitation moment equations for a nonlinear dynamical system excited by colored, Gaussian or non-Gaussian processes. Starting from equations for the two-time moments (e.g. for Cxy(t,s), Cxx(t,s)), the method uses an exact time-closure condition, in addition to a Gaussian moment closure, in order to obtain a closed, non-local in time (causal) subsystem for the one-time (t=s) moments. After solving this causal system, the two-time moments can be calculated for all (t,s) pairs as well. The present method differs essentially from the classical Itô/FPK approach since it does not involve any specific assumptions regarding the correlation structure of the excitation. In the case where the input random process can be obtained as the solution of an Itô equation (as, e.g., happens with an Ornstein-Uhlenbeck process), the proposed non-local system is localized, leading to moment equations identical with the usual ones. The closed, non-local in time, moment system is numerically solved by means of an appropriate, two-scale, iterative scheme, and numerical results are presented for two families of colored stochastic excitations. The results are confirmed by means of extensive Monte Carlo simulations. It is found that both the correlation time and the details of the shape of the input random function affect appreciable the response covariance. In the present paper we focus on a monostable cubic half-oscillator, excited by a smoothly-correlated, linear-plus-cubic-Gaussian (non-Gaussian) random input. The bistable case, as well as more general nonlinear systems can be treated by the same method, provided that a more elaborate moment closure will be used.

研究动机与目标

• 为在色噪声、非高斯激励下非线性动力系统的矩系统建立一种无需假设特定相关结构的闭式、因果矩系统。
• 通过避免对激励相关性的假设，将传统基于矩的方法扩展至伊tô/福克-普朗克框架之外。
• 通过结合时间闭合与高斯闭合技术，实现对非线性系统响应协方差的精确预测。
• 通过与大量蒙特卡洛模拟对比，对单稳态三次半振子进行方法的数值验证。
• 展示系统响应对激励相关时间与非高斯形状特征的敏感性。

提出的方法

• 推导了受色噪声、非高斯激励驱动的三次半振子的两时间矩方程（例如，Cxy(t,s)、Cxx(t,s)）。
• 应用精确的时间闭合条件，将无穷阶矩方程层次简化为有限的、非局部的因果时间系统。
• 实施高斯矩闭合，进一步闭合系统，从而实现单时间矩的数值求解。
• 采用两尺度、迭代的数值格式求解所得的非局部矩系统。
• 从求解得到的单时间矩重构所有(t,s)对的两时间矩。
• 针对两类色激励（高斯型与三次-高斯型）与蒙特卡洛模拟结果进行对比验证。

实验结果

研究问题

• RQ1 如何在不假设特定相关结构的前提下，为在色噪声、非高斯激励下非线性系统的矩方程推导出一个闭合的因果系统？
• RQ2 非高斯色激励的相关时间与形状在多大程度上影响非线性振子的响应协方差？
• RQ3 所提出的方法能否在单稳态三次半振子上重现与蒙特卡洛模拟一致的结果？
• RQ4 与经典的伊tô/福克-普朗克方法相比，该方法在通用性与准确性方面表现如何？
• RQ5 时间闭合条件在实现非线性随机动力学因果、非局部矩系统中起到何种作用？

主要发现

• 所提出的方法成功构建了一个闭合的、非局部的、因果的矩方程系统，能够准确捕捉非线性振子的统计响应。
• 响应协方差显著受激励相关时间与输入激励形状的影响，尤其在非高斯三次-高斯色噪声下表现明显。
• 对于奥恩斯坦-乌伦贝克型激励，该非局部系统可实现局部化，并退化为经典矩方程，验证了与已有理论的一致性。
• 矩系统数值解与大量蒙特卡洛模拟结果高度一致，验证了方法的准确性。
• 该方法展现出鲁棒性与通用性，通过采用更高级的矩闭合技术，可进一步推广至双稳态及更复杂的非线性系统。
• 结果强调了在随机动力学中考虑非高斯与色噪声激励特征的重要性，因其显著改变系统响应的统计特性。