Abstract: The equilibrium of a pole supported on a parabola wall is related to its length, slope angle and the frictional factor between them. Under the action of gravity, the pole can move clockwise under the focus of parabola, and move counterclockwise or slide down at two ends over the focus. The state of pole, i.e. disequilibrium, stable and unstable equilibrium with friction, is determined on the support force at the boundary of friction cone. The equilibrium set is a pitchfork bifurcation with width related to the frictional factor.