Amplification induced by oscillating mass and multiplicative white noise on parametrically amplified regions

We studied the amplification of the solution of a Mathieu-like equation with multiplicative white noise. This equation has a periodic varying mass term. The exponents were calculated numerically by solving the stochastic differential equations by symplectic method. It was shown that the exponent increases with a parameter α in the range of large α, where the value of α is determined by the intensity of noise and the strength of the coupling between physical variable (the solution) and noise. We found that the exponent as a function of α has one minima on parametrically amplified regions of α = 0. This indicates the suppression of the amplification by white noise.