Ruelle-Pollicott Resonances of Stochastic Systems in Reduced State Space. Part I: Theory
Ruelle-Pollicott Resonances of Stochastic Systems in Reduced State Space. Part I: Theory
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DOI：
10.1007/s10955-020-02535-x
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发表时间：
2020-04-15
影响因子：
1.6
通讯作者：
Neelin, J. David
中科院分区：
文献类型：
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作者：
Chekroun, Mickael D.;Tantet, Alexis;Neelin, J. David
提出了随机微分系统的Ruelle-Pollicott(RP)共振理论。这些共振被定义为给定随机系统的生成元(Kolmogorov算子)的本征值。然后利用马尔可夫半群理论，推导出关联函数和功率谱密度(PSD)关于RP共振的分解公式。对于一大类随机微分方程(SDE)，这些公式描述了Rp共振如何表征相关性的衰减以及由PSD中的峰值所表现的信号的振荡分量。结果表明，只要在简化的状态空间V内部分观察到动力学，就可以严格定义简化的RP共振。这些简化的共振是由作用于状态空间V的函数的简化的马尔可夫算符的谱元获得的，并且可以从级数估计。它们告诉我们一些粗粒度版本的SDE生成器的光谱元素。当从V中的部分观测中收集跃迁的时间滞后足够小或足够大时，证明了约化的RP共振近似于V中条件期望的生成器的(弱)RP共振，即V中通过平均未观测变量的贡献而获得的最优约化系统。该方法在一个随机慢-快系统上进行了说明，结果表明，即使在时间尺度分离较弱的情况下，减小的RP共振也可以很好地重建关联函数和PSD。配套的文章，第二部分[114]和第三部分[113]，论述了这一贡献中提出的理论的更多实际方面。一个重要的副产品是由RP共振提供的随机动力学的诊断有用性。这在第二部分中以随机Hopf分叉为例进行了说明。结果表明，这种分叉具有明显的表现形式，即在左半平面中沿离散抛物线的RP共振的几何组织。这种由(减少的)RP共振形成的几何特征可以从时间序列中提取，从而允许提供嵌入随机背景中的非线性振荡的明确的“特征”。第三部分根据本文提出的减少RP共振的理论，解决了高维随机系统中这种振荡的检测和特征问题，即受噪声影响的厄尔尼诺-南方振荡的Cane-Zebiak模型，该模型模拟了快速大气波动。
A theory of Ruelle-Pollicott (RP) resonances for stochastic differential systems is presented. These resonances are defined as the eigenvalues of the generator (Kolmogorov operator) of a given stochastic system. By relying on the theory of Markov semigroups, decomposition formulas of correlation functions and power spectral densities (PSDs) in terms of RP resonances are then derived. These formulas describe, for a broad class of stochastic differential equations (SDEs), how the RP resonances characterize the decay of correlations as well as the signal's oscillatory components manifested by peaks in the PSD. It is then shown that a notion reduced RP resonances can be rigorously defined, as soon as the dynamics is partially observed within a reduced state space V. These reduced resonances are obtained from the spectral elements of reduced Markov operators acting on functions of the state space V, and can be estimated from series. They inform us about the spectral elements of some coarse-grained version of the SDE generator. When the time-lag at which the transitions are collected from partial observations in V, is either sufficiently small or large, it is shown that the reduced RP resonances approximate the (weak) RP resonances of the generator of the conditional expectation in V, i.e. the optimal reduced system in V obtained by averaging out the contribution of the unobserved variables. The approach is illustrated on a stochastic slow-fast system for which it is shown that the reduced RP resonances allow for a good reconstruction of the correlation functions and PSDs, even when the time-scale separation is weak. The companions articles, Part II [114] and Part III [113], deal with further practical aspects of the theory presented in this contribution. One important byproduct consists of the diagnosis usefulness of stochastic dynamics that RP resonances provide. This is illustrated in the case of a stochastic Hopf bifurcation in Part II. There, it is shown that such a bifurcation has a clear manifestation in terms of a geometric organization of the RP resonances along discrete parabolas in the left half plane. Such geometric features formed by (reduced) RP resonances are extractable from time series and allow thus for providing an unambiguous "signature" of nonlinear oscillations embedded within a stochastic background. By relying then on the theory of reduced RP resonances presented in this contribution, Part III addresses the question of detection and characterization of such oscillations in a high-dimensional stochastic system, namely the Cane-Zebiak model of El Nino-Southern Oscillation subject to noise modeling fast atmospheric fluctuations.