NUMERICAL DETECTION AND CONTINUATION OF CODIMENSION-2 HOMOCLINIC BIFURCATIONS

A numerical procedure is presented for the automatic accurate location of certain codimension-two homoclinic singularities along curves of codimension-one homoclinic bifurcations to hyperbolic equilibria in autonomous systems of ordinary differential equations. The procedure also allows for the continuation of multiple-codimension homoclinic orbits in the relevant number of free parameters. A systematic treatment is given of codimension-two bifurcations that involve a unique homoclinic orbit. In each case the known theoretical results are reviewed and a regular test-function is derived for a truncated problem. In particular, the test-functions for global degeneracies involving the orientation of a homoclinic loop are presented. It is shown how such a procedure can be incorporated into an existing boundary-value method for homoclinic continuation and implemented using the continuation code AUTO. Several examples are studied, including Chua's electronic circuit and the FitzHugh-Nagumo equations. In each case, the method is shown to reproduce codimension-two bifurcation points that have previously been found using ad hoc methods, and in some cases, to obtain new results.