The noisy Pais–Uhlenbeck oscillator

In this paper, we include simultaneously additive and multiplicative noise to the Pais–Uhlenbeck oscillator (PUO). We construct an integral of motion of the PUO with a time-dependent coefficient. Viewing the PUO as two coupled harmonic oscillators, we add noise to the corresponding frequencies. The systems are solved with the fourth-order stochastic Runge–Kutta method. Some graphics of the solutions and integrals of motion are presented, and the average deviations are calculated in order to quantify the noise influence.