  Differential equations with random parameters have gained significant
prominence in recent years due to their importance in mathematical modelling
and data assimilation. In many cases, random ordinary differential equations
(RODEs) are studied by using Monte-Carlo methods or by direct numerical
simulation techniques using polynomial chaos (PC), i.e., by a series expansion
of the random parameters in combination with forward integration. Here we take
a dynamical systems viewpoint and focus on the invariant sets of differential
equations such as steady states, stable/unstable manifolds, periodic orbits,
and heteroclinic orbits. We employ PC to compute representations of all these
different types of invariant sets for RODEs. This allows us to obtain fast
sampling, geometric visualization of distributional properties of invariants
sets, and uncertainty quantification of dynamical output such as periods or
locations of orbits. We apply our techniques to a predator-prey model, where we
compute steady states and stable/unstable manifolds. We also include several
benchmarks to illustrate the numerical efficiency of adaptively chosen PC
depending upon the random input. Then we employ the methods for the Lorenz
system, obtaining computational PC representations of periodic orbits,
stable/unstable manifolds and heteroclinic orbits.
