Exact dynamics of stochastic linear delayed systems: Application to spatiotemporal coordination of comoving agents
Exact dynamics of stochastic linear delayed systems: Application to spatiotemporal coordination of comoving agents
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DOI：
10.1103/physreve.90.042135
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发表时间：
2014-10-22
影响因子：
2.4
通讯作者：
Giuggioli, Luca
中科院分区：
文献类型：
--
作者：
McKetterick, Thomas John;Giuggioli, Luca
延迟动力学是由能量、质量或信息形式的信号的有限传输速度引起的。在随机系统中，由于包含单调、振荡和不稳定演化的丰富行为库，所产生的滞后动态挑战了我们的理解。尽管有大量的文献，量化这种丰富的行为是有限的，缺乏明确的高维随机延迟系统的分析研究。在这里，我们填补了这个空白的系统由线性朗之万方程的任何数量的延迟和空间维度加性高斯噪声。利用拉普拉斯变换，我们能够得到一个精确的时间相关的朗之万方程的解析解。通过使用特征泛函，我们能够构建作为延迟和非延迟随机变量的函数的随机过程的多变量概率分布的全时间依赖性。作为一个应用程序，我们认为在动物集体运动的相互作用，超越了传统的假设瞬时对齐。我们提出了协调演习的协同运动的代理人适用于最近的实证研究结果，在鸽子和蝙蝠，个人复制他们的邻居的标题有一定的延迟模型。我们强调可能的策略，个别对可能采取减少他们的速度差异和/或在他们的空间分离的方差。我们还表明，在长时间的空间分离的方差的最小值可以实现一定的测量反应延迟的比率。
Delayed dynamics result from finite transmission speeds of a signal in the form of energy, mass, or information. In stochastic systems the resulting lagged dynamics challenge our understanding due to the rich behavioral repertoire encompassing monotonic, oscillatory, and unstable evolution. Despite the vast literature, quantifying this rich behavior is limited by a lack of explicit analytic studies of high-dimensional stochastic delay systems. Here we fill this gap for systems governed by a linear Langevin equation of any number of delays and spatial dimensions with additive Gaussian noise. By exploiting Laplace transforms we are able to derive an exact time-dependent analytic solution of the Langevin equation. By using characteristic functionals we are able to construct the full time dependence of the multivariate probability distribution of the stochastic process as a function of the delayed and nondelayed random variables. As an application we consider interactions in animal collective movement that go beyond the traditional assumption of instantaneous alignment. We propose models for coordinated maneuvers of comoving agents applicable to recent empirical findings in pigeons and bats whereby individuals copy the heading of their neighbors with some delay. We highlight possible strategies that individual pairs may adopt to reduce the variance in their velocity difference and/or in their spatial separation. We also show that a minimum in the variance of the spatial separation at long times can be achieved with certain ratios of measurement to reaction delay.