Numerical continuation of homoclinic orbits to non-hyperbolic equilibria in planar systems

In this paper we develop numerical algorithms for the continuation of degen erate homoclinic orbits to non-hyperbolic equilibria in planar systems. The first situation corresponds to a saddle-node equilibrium (a zero eigenvalu e) and the second one is the so-called cuspidal loop (double-zero eigenvalu e). The methods proposed may deal with codimension-two and -three homoclini c connections. Application of the algorithms to several examples supports i ts validity and demonstrates its usefulness.