Evaluation of Sustained Stochastic Oscillations by Means of a System of Differential Equations

Several approaches exist to model the evolution of dynamical systems with large populations. These approaches can be roughly divided into microscopic ones, which are usually stochastic and discrete, and macroscopic ones, which are obtained as the limit behaviour when the populations tend to infinity and are usually deterministic and continuous. We study the dynamics obtained by both approaches in systems with one deterministic equilibrium. We show that such dynamics can exhibit rather different behaviour around the deterministic equilibrium, in particular, the limit behaviour can tend to an equilibrium while the stochastic discrete dynamics oscillates indefinitely. To evaluate such stochastic oscillations quantitatively, we propose a system of differential equations on polar coordinates. The solution of this system provides several measures of interest related to the stochastic oscillations, such as average amplitude and frequency.