  Non-linear interactions among the inertial modes of a rotating fluid can be
described by a network of coupled oscillators. We use such a description for an
incompressible fluid to study the development of the r-mode instability of
rotating neutron stars. A previous hydrodynamical simulation of the r-mode
reported the catastrophic decay of large amplitude r-modes. We explain the
dynamics and timescale of this decay analytically by means of a single three
mode coupling. We argue that at realistic driving and damping rates such large
amplitudes will never actually be reached. By numerically integrating a network
of nearly 5000 coupled modes, we find that the linear growth of the r-mode
ceases before it reaches an amplitude of around 10^(-4). The lowest parametric
instability thresholds for the r-mode are calculated and it is found that the
r-mode becomes unstable to modes with 13<n<15 if modes up to n=30 are included.
Using the network of coupled oscillators, integration times of 10^6 rotational
periods are attainable for realistic values of driving and damping rates.
Complicated dynamics of the modal amplitudes are observed. The initial
development is governed by the three mode coupling with the lowest parametric
instability. Subsequently a large number of modes are excited, which greatly
decreases the linear growth rate of the r-mode.
