Numerical detection and continuation of saddle-node homoclinic bifurcations of codimension one and two

An extension of an existing truncated boundary-value method for the numerical continuation of connecting orbits is proposed to deal with homoclinic orbits to a saddle-node equilibrium. In contrast to previous numerical work by Schecter and Friedman & Doedel, the method is based on (linear) projection boundary conditions. These boundary conditions, with extra defining conditions for a saddle-node, naturally allow the continuation of codimension-one curves of saddle-node homoclinic orbits. A new test function is motivated for detecting codimension-two points at which loci of saddle-nodes and homoclinic orbits become detached. Two methods for continuing such codim 2 points in three parameters are discussed. The numerical methods are applied to two example systems, modelling a DC Josephson Junction and CO-oxidation. For the former model existing numerical results are recovered and extended; for the latter, new dynamical features are uncovered. All computations are performed using AUTO.