Behaviour of trajectories near a two-cycle heteroclinic network

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.