  We are concerned with random ordinary differential equations (RODEs). Our
main question of interest is how uncertainties in system parameters propagate
through the possibly highly nonlinear dynamical system and affect the system's
bifurcation behavior. We come up with a methodology to determine the
probability of the occurrence of different types of bifurcations based on the
probability distribution of the input parameters. In a first step, we reduce
the system's behavior to the dynamics on its center manifold. We thereby still
capture the major qualitative behavior of the RODEs. In a second step, we
analyze the reduced RODEs and quantify the probability of the occurrence of
different types of bifurcations based on the (nonlinear) functional appearance
of uncertain parameters. To realize this major step, we present three
approaches: an analytical one, where the probability can be calculated
explicitly based on Mellin transformation and inversion, a semi-analytical one
consisting of a combination of the analytical approach with a moment-based
numerical estimation procedure, and a particular sampling-based approach using
unscented transformation. We complement our new methodology with various
numerical examples.
