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 and Doedel, the method is based on (linear) projection boundary conditions. These boundary conditions, with extra defining conditions for a saddle-node, 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.