A method of investigation of the evolution of a nonlinear system driven by pulsed noise

A numerical method is proposed for determining the evolution of nonlinear systems subjected to noise. The method is based on a recurrence equation for the probability density which has been obtained analytically due to the choice of noise in the form of discrete series of random pulses. The method is applied to a dynamical system which describes the motion of a particle in a plane-wave field. The evolution of the probability density in phase and energy space is obtained. It is shown that because of noise effects, the region in phase space where particles can be found rapidly reaches the separatrix and then spreads over the phase space, mainly along the separatrix. In the energy spectrum a new peak appears at the separatrix's energy. This peak grows in time, while the main peak corresponding to the initial energy drops in time and shifts to lower energy. The moments of motion were analyzed. The character of their evolution indicates a high rate of chaotization. The growth of the fraction of energetic particles is very rapid (exponential at the beginning), whereas the mean energy grows linearly.