Numerical Detection and Continuation of Codimension-Two Homoclinic Bifurcations

摘要

A numerical procedure is presented for the automatic accurate location of certaincodimension-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. All known codimension-two bifurcations that involve a unique homoclinic orbit are considered. In each case the known theoretical results are reviewed and a regular test function is derived. 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 codim 2 bifurcation points that have previously been found using ad hoc methods, and, in some cases to obtain new results.