Stabilization by multiplicative noise

Abstract

We consider a dynamical model containing a short-time scale ${\ensuremath{\tau}}_{s}$ and a long-time scale ${\ensuremath{\tau}}_{l}$ 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 ${\ensuremath{\tau}}_{c}\ensuremath{\ll}{\ensuremath{\tau}}_{l}$ 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 $\ensuremath{\lambda}=\frac{{\ensuremath{\tau}}_{s}}{{\ensuremath{\tau}}_{c}}$. If the fluctuations of the control parameter are Gaussian, and if $\frac{{\ensuremath{\tau}}_{c}}{{\ensuremath{\tau}}_{l}}\ensuremath{\rightarrow}0$, $\frac{{\ensuremath{\tau}}_{s}}{{\ensuremath{\tau}}_{l}}\ensuremath{\rightarrow}0$, on the scale ${\ensuremath{\tau}}_{l}$, the fluctuations act as a Stratonovich noise source for ${\ensuremath{\tau}}_{c}\ensuremath{\gg}{\ensuremath{\tau}}_{s}$ and as an It\^o noise source for ${\ensuremath{\tau}}_{c}\ensuremath{\ll}{\ensuremath{\tau}}_{s}$. The intermediate regime ${\ensuremath{\tau}}_{s}=\ensuremath{\lambda}{\ensuremath{\tau}}_{c}$ with arbitrary $\ensuremath{\lambda}$ is analyzed and found to be observable by the noise-induced shift of the threshold associated with it.