Stabilization by multiplicative noise

We consider a dynamical model containing a short-time scale ${\tau }_s$ and a long-time scale ${\tau }_l$ and exhibiting a continuous instability depending on a control parameter. We study how the threshold is shifted if the control parameter is noisy with a correlation time ${\tau }_c\ll {\tau }_l$ with the use of both qualitative and systematic methods of adiabatic elimination. We find a noise-induced increase of the threshold of instability depending on the ratio $\lambda =\frac{{\tau }_s}{{\tau }_c}$. If the fluctuations of the control parameter are Gaussian, and if $\frac{{\tau }_c}{{\tau }_l}\to 0$, $\frac{{\tau }_s}{{\tau }_l}\to 0$, on the scale ${\tau }_l$, the fluctuations act as a Stratonovich noise source for ${\tau }_c\gg {\tau }_s$ and as an Itô noise source for ${\tau }_c\ll {\tau }_s$. The intermediate regime ${\tau }_s=\lambda {\tau }_c$ with arbitrary $\lambda$ is analyzed and found to be observable by the noise-induced shift of the threshold associated with it.