Bifurcation Scenarios in the Atmosphere
Bifurcation is relevant to the atmosphere because it effects the predictability of the evolution of the atmosphere.  I can't speak to its merits with climate models, but with weather it is quite important.  As Lorenz discovered, the atmosphere is chaotic and numerical prediction is very sensitive to the initial conditions.  If the initial conditions are close to the bifurcation point, the solution can be wildly different and this leads to poor predictability.  

In particular, weather models assimilate observations and each observation is a value and an uncertainty.  When the uncertainty contains the bifurcation point that essentially means for that particular scenario the instrument error is such that the instrument can provide an observation on either side of the bifurcation. This can be explored with ensemble modeling where many models are run with slightly different initial conditions (or slightly different models, or both) and leads to a large spread in the ensemble and low confidence in the forecast.  Our atmosphere can be more sensitive than the instruments we use to measure it can detect.  

I'll try to dig up some references to presentations I've seen that show hurricane dissapation / rapid strengthening determined by differences in gulf of Mexico observations smaller than the instrument error reporting them.  For more NWP-centric studies, you can search for articles on predictability.  I know there should be a couple by Dr. Fuqing Zhang (PSU) and I'll edit when I can point to some specific articles.