The basin of attraction of a stable equilibrium point is investigated for a dynamical system (W97) that has been used to model transition to turbulence in shear flows. The basin boundary contains a linearly unstable equilibrium point X_lb which, in the self-sustaining scenario, plays a role in mediating the transition in that transition orbits cluster around its unstable manifold. However we find—for W97 with canonical parameter values—that this role is played not by X_lb but rather by a periodic orbit also lying on the basin boundary. Moreover, it appears via numerical computations that all orbits beginning near X_lb relaminarize. We offer numerical evidence that the parameter values of W97 are post-critical in the following sense: for some, subcritical parameter values, the basin boundary coincides with the stable manifold of X_lb and only a subset of nearby orbits relaminarize, whereas for supercritical values the basin boundary is the union of two stable manifolds, one belonging to the periodic orbit and dominating the basin boundary, and the other belonging to X_lb and detectable only as edge separating relaminarizing orbits of different characters. The periodic orbit appears at the critical parameter value via a homoclinic connection. This further leads to a proposal for the structure of the 'edge of chaos' somewhat different from that which has previously been proposed.