For a=2.9 the fixed point has evolved to a value x>0.5. While the iterations are running, click somewhere in the plot to start a new iteration with initial condition the value of x at the mouse position.
All initial conditions lead to an orbit that converges to the fixed point which is stable. The orbits now spiral into the fixed point, corresponding to the negative slope of f here (successive deviations from the fixed point change sign as well as shrink).
The stability of the fixed point is easily seen going to the second iterate of the map f 2(x)=f(f(x)) by putting nf=1.
The slope of the curve at the intersection with the diagonal is less than unity, and the orbit staircases in to the fixed point.