On low-order non-linear stochastic-dynamic systems

A low-order system of equations derived from the advective equation with forcing and dissipation is investigated. A stochastic-dynamic treatment of these equations based on the closure assumption of the neglect of third and higher moments is carried out. For this system, the steady states, including those created by the closure assumption, are determined, and their linear stability properties are found. It turns out that 5 steady states exist in addition to the 3 related to the deterministic equations. Among the 8 steady states, 3 are stable. 2 of them are the steady states coming from the deterministic problem.

The asymptotic solutions in a stochastic-dynamic sense are determined without any closure assumption. It is shown that the 3 stable steady states for the system with closure form a subset of the continuum of asymptotic states for the system without closure. Several examples showing the differences between the asymptotic solutions of the 2 systems are calculated.

Numerical experiments based on backward integrations in time show that the system with closure also has a limit cycle characterized by a closed curve in the mean values and in each point a degenerate uncertainty ellipse consisting of a line segment.