The behavior of an ordinary differential equation for the low wave number velocity
mode is analyzed. This equation was derived in [5]
by an iterative process on the two-dimensional Navier-Stokes equations (NSE). It
resembles the NSE in form, except
that the kinematic viscosity is replaced by an iterated viscosity
which is a partial sum, dependent on the low-mode velocity. The convergence of
this sum as the number of iterations is taken to be arbitrarily large is explored.
This leads to a limiting dynamical system which displays
several unusual mathematical features.