受扰动梯度系统中的最可能逃逸路径
Most probable escape paths in perturbed gradient systems
AI总结：
本文基于大偏差理论和 Melnikov 理论，研究了非梯度扰动随机梯度系统的最可能逃逸路径，并确定了最优逃逸路径为异宿轨道的条件。
摘要：
随机系统被用于对噪声起关键作用的各种现象进行建模。在这些模型中，一个潜在目标是确定噪声能否诱导状态间的跃迁，若能，则计算离开吸引子的最可能逃逸路径。在小噪声极限下，Freidlin-Wentzell 大偏差理论提供了一个计算这些路径的变分框架。本工作致力于使用大偏差理论来计算具有非梯度扰动的随机梯度系统的此类路径。对于梯度系统，最可能逃逸路径由时间反转的异宿轨道组成，而对于一般系统，这可能是一项具有挑战性的计算。通过将 Melnikov 理论应用于重写为 Hamiltonian 形式的 Euler-Lagrange 方程，我们确定了最优逃逸路径何时为受扰动系统的异宿轨道的条件。我们提供了一个数值示例，以说明计算出的最可能逃逸路径与理论结果的比较情况。
英文摘要：
Stochastic systems are used to model a variety of phenomena in which noise plays an essential role. In these models, one potential goal is to determine if noise can induce transitions between states, and if so, to calculate the most probable escape path from an attractor. In the small noise limit, the Freidlin-Wentzell theory of large deviations provides a variational framework to calculate these paths. This work focuses on using large deviation theory to calculate such paths for stochastic gradient systems with non-gradient perturbations. While for gradient systems the most probable escape paths consist of time-reversed heteroclinic orbits, for general systems it can be a challenging calculation. By applying Melnikov theory to the resulting Euler-Lagrange equations recast in Hamiltonian form, we determine a condition for when the optimal escape path is the heteroclinic orbit for the perturbed system. We provide a numerical example to illustrate how the computed most probable escape path compares with the theoretical result.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
