  We examine nonlinear dynamical systems of ordinary differential equations or
differential algebraic equations. In an uncertainty quantification, physical
parameters are replaced by random variables. The inner variables as well as a
quantity of interest are expanded into series with orthogonal basis functions
like the polynomial chaos expansions, for example. On the one hand, the
stochastic Galerkin method yields a large coupled dynamical system. On the
other hand, a stochastic collocation method, which uses a quadrature rule or a
sampling scheme, can be written in the form of a large weakly coupled dynamical
system. We apply projection-based methods of nonlinear model order reduction to
the large systems. A reduced-order model implies a low-dimensional
representation of the quantity of interest. We focus on model order reduction
by proper orthogonal decomposition. The error of a best approximation located
in a low-dimensional subspace is analysed. We illustrate results of numerical
computations for test examples.
