A characteristic feature of the onset of turbulence in shear flows is the appearance of an "edge," a codimension-one invariant manifold that separates orbits that decay directly to the laminar state from orbits that decay more slowly. The object of this paper is to understand its structure. To this end we consider a succession of low-dimensional models. We hope in doing this to isolate geometric features that are robust under increase of dimension and are therefore candidates for explaining

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... ogous features in arbitrarily high dimension and therefore in realistic shear flows. We find in all cases considered that there are extensive ranges of the principal parameter (the Reynolds number R) in which all or part of the boundary of the basin of attraction of the laminar state has the character of an edge. This is a part of the boundary that passes through the basin without separating it from some other set. It is also the stable manifold of an edge state (an equilibrium point or a periodic orbit in the present paper). An important feature of the edge is that it lies in a region of phase space which, while unbounded in some directions, is bounded in others. The edge boundedness is associated to the presence of invariant upper branches objects. The edge can be not only extensive but also complicated, as we show in some detail.