Dynamics near a heteroclinic network

We study the dynamical behaviour of a smooth vector field on a three-manifold near a heteroclinic network. Under some generic assumptions on the network, we prove that every path on the network is followed by a neighbouring trajectory of the vector field - there is switching on the network. We also show that near the network there is an infinite number of hyperbolic suspended horseshoes. This leads to the existence of a horseshoe of suspended horseshoes with the shape of the network. Our results are motivated by an example constructed by Field (1996 Lectures on Bifurcations, Dynamics, and Symmetry (Pitman Research Notes in Mathematics Series vol 356) (Harlow: Longman)), where we have observed, numerically, the existence of such a network.