Kinetics of dichotomous noise-induced transitions in a multistable multivariate system

We study the stochastic evolution of a multivariate and deterministically multistable system subjected to an additive Markovian dichotomous noise. To this end, a steady-state probability density support is defined in such a way that no stochastic trajectory can escape. An appropriate boundary condition is then imposed in order to numerically evaluate this distribution. When the noise amplitude is large enough, the system may evolve from one deterministic attractor to another. A partition of the support in disjoint species is proposed. It is then possible to study the kinetics of the noise-induced transitions between species. Projection operator techniques are used to obtain a phenomenological kinetic law valid when the interspecies transition time scale is much longer than all the other time scales characterizing the system. We also develop a fast algorithm permitting the numerical evaluation of the phenomenological transition rate. As an example, we consider a bivariate system exhibiting two deterministic stable fixed points and a saddle point. The results confirm the existence of a phenomenological law insofar as the noise amplitude is large enough and its correlation time, small enough.