  Robbin and Salamon showed that attractor-repellor networks and Lyapunov maps
are equivalent concepts and illustrate this with the example of linear flows on
projective spaces. In these examples the fixed points are linearly ordered with
respect to the Smale order which makes the attractor-repellor network overly
simple. In this paper we provide a class of examples in which the
attractor-repellor network and its lattice structure can be explicitly
determined even though the Smale order is not total. They are associated with
special flows on complex flag manifolds. In the process we show that the Smale
order on the set of fixed points can be identified with the well-known Bruhat
order. This could also be derived from results of Kazhdan and Lusztig, but we
give a new proof using the Lambda-Lemma of Palis. For the convenience of the
reader we also introduce the flag manifolds via elementary dynamical systems
using only a minimum of Lie theory.
