  The interplay between bifurcations and random switching processes of vector
fields is studied. More precisely, we provide a classification of piecewise
deterministic Markov processes arising from stochastic switching dynamics near
fold, Hopf, transcritical and pitchfork bifurcations. We prove the existence of
invariant measures for different switching rates. We also study, when the
invariant measures are unique, when multiple measures occur, when measures have
smooth densities, and under which conditions finite-time blow-up occurs. We
demonstrate the applicability of our results for three nonlinear models arising
in applications.
