  We present a path - integral approach to treat a 2D model of a quantum
bifurcation. The model potential has two equivalent minima separated by one or
two saddle points, depending on the value of a continuous parameter. Tunneling
is therefore realized either along one trajectory or along two equivalent
paths. Zero point fluctuations smear out the sharp transition between these two
regimes and lead to a certain crossover behavior. When the two saddle points
are inequivalent one can also have a first order transition related to the fact
that one of the two trajectories becomes unstable. We illustrate these results
by numerical investigations. Even though a specific model is investigated here,
the approach is quite general and has potential applicability for various
systems in physics and chemistry exhibiting multi-stability and tunneling
phenomena.
