  We consider a classical spring-mass model of human running which is built
upon an inverted elastic pendulum. Based on our previous results concerning
asymptotic solutions for large spring constant (or small angle of attack), we
construct analytical approximations of solutions in the considered model. The
model itself consists of two sets of differential equations - one set describes
the motion of the centre of mass of a runner in contact with the ground
(support phase), and the second set describes the phase of no contact with the
ground (flight phase). By appropriately concatenating asymptotic solutions for
the two phases we are able to reduce the dynamics to a one-dimensional apex to
apex return map. We find sufficient conditions for this map to have a unique
stable fixed point. By numerical continuation of fixed points with respect to
energy, we find a transcritical bifurcation in our model system.
