### Homoclinic-to-Saddle-Node Orbits

For a homoclinic orbit to a saddle-node equilibrium, the extended defining system undergoes some small changes. Now has to be placed in the center-unstable subspace. Analogously, must be in the center-stable subspace. This again is implemented by requiring that the vector is orthogonal to the orthogonal complement of the corresponding space. So the equations (93) themselves do not really change; the changes happen in the computation of the matrices . The defining system now has one equation less than in the HHS case (, with the dimension of the stable, and of the unstable eigenspace); the number of equations is restored however, by adding the constraint that the equilibrium must be a saddle-node. For this we use the bordering technique, as described in section 4.2.1 of [22].