  Dynamical properties of limit cycles in a two-dimensional max-plus dynamical
system are discussed. We apply a Poincare map method to the limit cycles in
order to reveal their stabilities. This method reduces the two dimensional
system to a one-dimensional piecewise linear discrete dynamical system composed
of the Poincare map and its cross section. Basins for one of the limit cycles
are derived by considering the inverse system of the original model. It is
found that the obtained basins show a hierarchic structure. Relationship
between the Poincare map method and the method of piecewise linear mapping
studied in integrable system theory for the limit cycles is discussed.

Chaotic Dynamics;Dynamical Systems