Ermakov systems with multiplicative noise

Using the Euler–Maruyama numerical method, we present calculations of the Ermakov–Lewis invariant and the dynamic, geometric, and total phases for several cases of stochastic parametric oscillators, including the simplest case of the stochastic harmonic oscillator. The results are compared with the corresponding numerical noiseless cases to evaluate the effect of the noise. Besides, the noiseless cases are analytic and their analytic solutions are briefly presented. The Ermakov–Lewis invariant is not affected by the multiplicative noise in the three particular examples presented in this work, whereas there is a shift effect in the case of the phases.