In this work we consider the Allen--Cahn equation, a prototypical model problem in nonlinear dynamics that exhibits bifurcations corresponding to variations of a deterministic bifurcation parameter. Going beyond the state-of-the-art, we introduce a random coefficient function in the linear reaction part of the equation, thereby accounting for random, spatially-heterogeneous effects. Importantly, we assume a spatially constant, deterministic mean value of the random coefficient. We show that this mean value is in fact a bifurcation parameter in the Allen--Cahn equation with random coefficients. Moreover, we show that the bifurcation points and bifurcation curves become random objects. We consider two distinct modelling situations: (i) for a spatially homogeneous coefficient we derive analytical expressions for the distribution of the bifurcation points and show that the bifurcation curves are random shifts of a fixed reference curve; (ii) for a spatially heterogeneous coefficient we employ a generalized polynomial chaos expansion to approximate the statistical properties of the random bifurcation points and bifurcation curves. We present numerical examples in 1D physical space, where we combine the popular software package Continuation Core and Toolboxes (CoCo) for numerical continuation and the Sparse Grids Matlab Kit for the polynomial chaos expansion. Our exposition addresses both, dynamical systems and uncertainty quantification, highlighting how analytical and numerical tools from both areas can be combined efficiently for the challenging uncertainty quantification analysis of bifurcations in random differential equations.

翻译：本文研究Allen-Cahn方程——非线性动力学中一个典型模型问题，其分岔行为对应于确定性分岔参数的变化。为突破现有技术，我们在方程的线性反应项中引入随机系数函数，从而考虑随机空间异质效应。重要的是，我们假设该随机系数的空间恒定确定性均值成立，并证明该均值实际上构成了含随机系数Allen-Cahn方程的分岔参数。此外，我们揭示了分岔点与分岔曲线转变为随机对象。我们考虑两种不同的建模情形：（i）对于空间均匀系数，我们推导出分岔点分布的解析表达式，并证明分岔曲线是固定参考曲线的随机平移；（ii）对于空间异质系数，我们采用广义多项式混沌展开来逼近随机分岔点与分岔曲线的统计特性。我们在物理空间维度为一维时给出数值算例，将用于数值延拓的流行软件包Continuation Core and Toolboxes (CoCo)与用于多项式混沌展开的Sparse Grids Matlab Kit相结合。本文同时涉及动力系统与不确定性量化两个领域，重点展示了如何有效结合两个领域的解析与数值工具，以应对随机微分方程分岔分析中具有挑战性的不确定性量化问题。