论文标题
在对随机和确定性强迫的响应的某些方面
On Some Aspects of the Response to Stochastic and Deterministic Forcings
论文作者
论文摘要
操作员半群的扰动理论用于得出响应公式，以进行各种作用强迫和参考背景动力学的组合。在背景随机动力学的情况下，我们使用Koopman运算符发电机特征函数和相应的特征值分解响应公式，从而为识别物理相关系统中的宽松时间和模式提供了功能基础。对于领先顺序，线性响应对作用于随机或混乱的动力学系统的额外确定性强制而对期望值进行了校正。在考虑弱噪声的影响时，该响应是在背景随机动力学的（额外）噪声强度中线性的，而当参考动力学混乱时，二阶响应给出了领先阶校正。在后一种情况下，我们澄清说，当给出噪声的合适解释 - Stratonovich与ITO的合适解释时，可以将以前发表的分歧结果带到共同点。最后，通过扰动方法研究了两点相关对扰动的响应。我们的结果除其他外，还可以估算混乱动力学系统的相关性如何变化，这是增加随机强迫的结果。
The perturbation theory of operator semigroups is used to derive response formulas for a variety of combinations of acting forcings and reference background dynamics. In the case of background stochastic dynamics, we decompose the response formulas using the Koopman operator generator eigenfunctions and the corresponding eigenvalues, thus providing a functional basis towards identifying relaxation timescales and modes in physically relevant systems. To leading order, linear response gives the correction to expectation values due to extra deterministic forcings acting on either stochastic or chaotic dynamical systems. When considering the impact of weak noise, the response is linear in the intensity of the (extra) noise for background stochastic dynamics, while the second order response given the leading order correction when the reference dynamics is chaotic. In this latter case we clarify that previously published diverging results can be brought to common ground when a suitable interpretation - Stratonovich vs. Ito - of the noise is given. Finally, the response of two-point correlations to perturbations is studied through the resolvent formalism via a perturbative approach. Our results allow, among other things, to estimate how the correlations of a chaotic dynamical system changes as a results of adding stochastic forcing.
