Giuggioli Luca, Neu Zohar
Department of Engineering Mathematics, University of Bristol, Woodland Road, Bristol BS8 1UB, UK.
Bristol Centre for Complexity Sciences, University of Bristol, Woodland Road, Bristol BS8 1UB, UK.
Philos Trans A Math Phys Eng Sci. 2019 Sep 9;377(2153):20180131. doi: 10.1098/rsta.2018.0131. Epub 2019 Jul 22.
Noise and time delays, or history-dependent processes, play an integral part in many natural and man-made systems. The resulting interplay between random fluctuations and time non-locality are essential features of the emerging complex dynamics in non-Markov systems. While stochastic differential equations in the form of Langevin equations with additive noise for such systems exist, the corresponding probabilistic formalism is yet to be developed. Here we introduce such a framework via an infinite hierarchy of coupled Fokker-Planck equations for the n-time probability distribution. When the non-Markov Langevin equation is linear, we show how the hierarchy can be truncated at n = 2 by converting the time non-local Langevin equation to a time-local one with additive coloured noise. We compare the resulting Fokker-Planck equations to an earlier version, solve them analytically and analyse the temporal features of the probability distributions that would allow to distinguish between Markov and non-Markov features. This article is part of the theme issue 'Nonlinear dynamics of delay systems'.
噪声和时间延迟，即历史依赖过程，在许多自然和人造系统中都起着不可或缺的作用。随机涨落与时间非局域性之间由此产生的相互作用是非马尔可夫系统中新兴复杂动力学的基本特征。虽然存在针对此类系统的带有加性噪声的朗之万方程形式的随机微分方程，但相应的概率形式体系尚未得到发展。在此，我们通过用于n时间概率分布的耦合福克 - 普朗克方程的无穷层级引入这样一个框架。当非马尔可夫朗之万方程为线性时，我们展示了如何通过将时间非局域朗之万方程转换为带有加性有色噪声的时间局域方程，在n = 2时截断该层级。我们将所得的福克 - 普朗克方程与早期版本进行比较，对其进行解析求解，并分析概率分布的时间特征，这些特征将有助于区分马尔可夫和非马尔可夫特征。本文是主题为“延迟系统的非线性动力学”的特刊的一部分。
