  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.
