Effect of a parametric driving force on noise-induced transitions: Analytical results

The suppression by a parametric harmonic action of noise-induced oscillations in an underdamped pendulum with nonlinear friction, recently reported by Landa et al. [Phys. Rev. E 56, 1465 (1997)], is studied in an approximately soluble model system. In the high-frequency limit, a process of consecutive averaging over two widely different relevant time scales reveals the analogy of the problem with a noise-induced transition whose critical point is changed by the driving term. The obtainment of analytical results for the probability distribution function and the spectrum allows us to understand and control the effect.