  We study behaviour of trajectories near a type Z heteroclinic network which
is a union of two cycles. Analytical and numerical studies indicate that
attractiveness of this network can be associated with various kinds of dynamics
in its vicinity, one or both of these cycle being fragmentarily asymptotically
stable, or both being completely unstable. In the latter case trajectories can
switch irregularly between the cycles, or they can make a certain number of
turns around one of them before switching to the other one. Regular behaviour
of trajectories near a heteroclinic network can be described using the notion
of an omnicycle, which we introduce in this paper. In particular, we use it to
prove that the network can be fragmentarily asymptotically stable even if both
cycles are completely unstable.
