Stability of the inverted pendulum - a topological explanation

An explanation and a proof of stability of the inverted pendulum whose suspension point undergoes vertical oscillations is given. The main idea of the argument is topological: as it turns out, existence of stable regimes can be proven with little effort using only very crude qualitative information about the system. More precisely, let n be the number of times the pendulum becomes vertical during one forcing period. If n changes by more than 4 with the change of a parameter μ, then for an open interval of intermediate values of μ the pendulum will be stable.