  We explore a two-dimensional dynamical system modeling transition in shear
flows to try to understand the nature of an 'edge' state. The latter is an
invariant set in phase space separating the basin of attraction B of the
laminar state into two parts distinguished from one another by the nature of
relaminarizing orbits. The model is parametrized by R, a stand-in for Reynolds
number. The origin is a stable equilibrium point for all values of R and
represents the laminar flow. The system possesses four critical parameter
values at which qualitative changes take place, R_{sn}, R_h, R_{bh} and
R_\infty. The origin is globally stable if R < R_{sn} but for R > R_{sn} has
two further equilibrium points, X_{lb} and X_{ub} . Of these X_{lb} is unstable
for all values of R > R_{sn} whereas X_{ub} is stable for R < R_{bh} and
therefore possesses its own basin of attraction D. At R = R_h a homoclinic
bifurcation takes place with the simultaneous formation of a homoclinic loop
and an edge state.
  For R_h < R < R_{bh} the edge state, which is the stable manifold of X_{lb},
forms part of \partial B, the boundary of the basin of attraction of the
origin. The other part of \partial B is a periodic orbit P bounding D. P and D
shrink with increasing R. At R = R_{bh} there is a 'backwards Hopf' bifurcation
at which X_{ub} loses its stability and P and D disappear. For R_{bh} < R <
R_\infty the edge is 'pure' in the sense that it is the only phase space
structure that lies outside the basin of attraction of the origin. As R
increases the point X_{lb} recedes to progressively greater distances, with a
singularity at R = R_\infty where it becomes infinite. For R > R_\infty X_{lb}
has reappeared, the edge state has disappeared, and the geometrical structure
favors permanent transition from the laminar state, increasingly so for
increasing values of R.
