The influence of external fluctuations on self-sustained temporal oscillations

The influence of external fluctuations on the bifurcational behavior of two-dimensional dynamical systems exhibiting limit cycles is investigated. Studying both exactly and approximately solvable examples it is shown that the variances of the external fluctuations occur as additional bifurcation parameters. The threshold values for soft as well as for hard self-excitation of oscillations are affected by the external fluctuations. To classify bifurcations of dynamical systems in the presence of fluctuations some aspects of catastrophe theory are applied to the corresponding stationary probability distributions.