Suppression of Growth by Multiplicative White Noise in a Parametric Resonant System

The growth of the amplitude in a Mathieu-like equation with multiplicative white noise is studied. To obtain an approximate analytical expression for the exponent at the extremum on parametric resonance regions, a time-interval width is introduced. To determine the exponents numerically, the stochastic differential equations are solved by a symplectic numerical method. The Mathieu-like equation contains a parameter alpha determined by the intensity of noise and the strength of the coupling between the variable and noise; without loss of generality, only non-negative alpha can be considered. The exponent is shown to decrease with alpha, reach a minimum and increase after that. The minimum exponent is obtained analytically and numerically. As a function of alpha, the minimum at alpha not equal 0, occurs on the parametric resonance regions of alpha=0. This minimum indicates suppression of growth by multiplicative white noise.