On response of nonlinear oscillators with random frequency of excitation

Some types of nonlinear oscillators, for which the frequency of excitation is stochastic, are investigated. The paper consists of two parts. In the first part equations of motion are linearized. With the aid of stochastic averaging differential equations for the mean and variance of the process are obtained and integrated numerically. This approach is applicable for weakly nonlinear oscillators. se of strong nonlinearity is considered in the second part. Making use of computer simulation, a number of stochastic realizations of the process are computed. The stochastic process is characterized by the mean and standard deviation of these realizations. Calculations have been carried out for the Duffing, Ueda and van der Pol equations and for forced vibrations of a pendulum. These calculations show that if attractors exist then the deterministic vibrations (which may be chaotic) turn regular by adding noise and the motion terminates in a stable fixed point or on a limit cycle.