Actually this is ok. I had a wrong sign there, but u still have a mistake for x=0 so check it. We have ether one unstable at x=0 or one stable at x=0 and two unstable at +-sqrt(h-mg-1/2). Bifurcation looks much better now :) Thoughts?
I have the same problem for my maths assignment :) check ur calculations carefully. I have a single unstable fixed point at x=0 for h<mg/k+0.05, second derivative is >0 and 3 stable fixed points for h>mg/k+0.05, second derivative<0. Why do u have the opposite?