Numerical Continuation of Homoclinic Orbits to Non-Hyperbolic Equilibria in Planar Systems

In this paper we develop numerical algorithms for thecontinuation of degenerate homoclinic orbits to non-hyperbolicequilibria in planar systems. The first situation corresponds to asaddle-node equilibrium (a zero eigenvalue) and the second one is theso-called cuspidal loop (double-zero eigenvalue). The methods proposedmay deal with codimension-two and -three homoclinic connections.Application of the algorithms to several examples supports its validityand demonstrates its usefulness.