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For piecewise-linear maps, the phenomenon that a branch of a one-dimensionalunstable manifold of a periodic solution is completely contained in its stablemanifold is codimension-two. Unlike codimension-one homoclinic corners, such`subsumed' homoclinic connections can be associated with stable periodicsolutions. The purpose of this paper is to determine the dynamics near ageneric subsumed homoclinic connection in two dimensions. Assuming theeigenvalues associated with the periodic solution satisfy $0 < |\lambda| < 1 <\sigma < \frac{1}{|\lambda|}$, in a two-parameter unfolding there exists aninfinite sequence of roughly triangular regions within which the map has astable single-round periodic solution. The result applies to both discontinuousand continuous maps, although these cases admit different characterisations forthe border-collision bifurcations that correspond to boundaries of the regions.The result is illustrated with a discontinuous map of Mira and thetwo-dimensional border-collision normal form.

Unfolding codimension-two subsumed homoclinic connections in two-dimensional piecewise-linear maps