Emergence of stability in a stochastically driven pendulum: beyond the Kapitsa effect

We consider a prototypical nonlinear system which can be stabilized by multiplicative noise: an underdamped non-linear pendulum with a stochastically vibrating pivot. A numerical solution of the pertinent Fokker-Planck equation shows that the upper equilibrium point of the pendulum can become stable even when the noise is white, and the “Kapitsa pendulum” effect is not at work. The stabilization occurs in a strong-noise regime where WKB approximation does not hold.