Second Harmonics Effects in Random Duffing Oscillators

We consider a stochastic model for Duffing oscillators, where the nonlinearity and the randomness are scaled in such a way that they interact strongly. A typical feature is the appearance of second harmonics effects. An asymptotic statistical analysis for these oscillators is performed in the diffusion limit, when a suitable absorbing boundary condition is imposed, according to the underlying physical problem. The related Fokker–Planck equation has been numerically solved to obtain the first two moments of the oscillator’s displacement from its rest-position. Dependence on the nonlinearity strength and on the location of the absorbing boundary has also been investigated. Such results have been compared with those computed solving the corresponding stochastic Ito differential equations by a Monte Carlo method, where quasi-random sequences of numbers have been efficiently used.