Singular Features of Large Fluctuations in Oscillating Chemical Systems†

We investigate the way in which large fluctuations in an oscillating, spatially homogeneous chemical system take place. Starting from a master equation, we study both the stationary probability density of such a system far from its limit cycle and the optimal (most probable) fluctuational paths in its space of species concentrations. The flow field of optimal fluctuational paths may contain singularities, such as switching lines. A “switching line” separates regions in the space of species concentrations that are reached, with high probability, along topologically different sorts of fluctuational paths. If an unstable focus lies inside the limit cycle, the pattern of optimal fluctuational paths is singular and self-similar near the unstable focus. In fact, a switching line spirals down to the focus. The logarithm of the stationary probability density has a self-similar singular structure near the focus as well. For a homogeneous Selkov model, we provide a numerical analysis of the pattern of optimal fluctuational paths and compare it with analytic results.