PROBABILISTIC ANALYSIS OF A NONLINEAR PENDULUM

We investigate the reliability of a nonlinear pendulum forced by a res onant harmonic excitation and interacting in a random environment. Two types of random perturbations are considered: additive weak noise, an d random phase fluctuations of the harmonic resonant forcing. Our goal is to predict the probability of a transition of the response from os cillatory regime to rotatory regime. In the first stage, the noise-fre e system is analyzed by an averaging method in view of predicting peri od-1 resonant orbits. By averaging the fast oscillations of the respon se, these orbits are mapped into equilibrium points in the space of th e energy and resonant phase variables. In the second stage, the random fluctuating terms exciting the averaged system are evaluated, leading to a Fokker-Planck-Kolmogorov equation governing the probability dens ity function of the energy and phase variables. This equation is solve d asymptotically in the form of a WKB approximation p similar to exp(- Q/epsilon) as the parameter epsilon characterizing the smallness of th e random perturbations tends to zero. The quasipotential Q is solution of a Hamilton-Jacobi equation, and can be obtained numerically by a m ethod of characteristics. Of critical importance is the evaluation of the minimum difference of quasipotential between the equilibrium point and the boundary across which the transitions occur. We show that thi s minimum difference determines to logarithmic accuracy the mean first -passage time to the critical boundary and hence the probability of fa ilure of the oscillatory regime. The effects of the two types of rando m perturbations are analyzed separately.