We present a numerical method for branch switching between homoclinic orbits to equilibria of ODEs computed via numerical continuation. Starting from a 1-homoclinic orbit our method allows us to find and follow an N-homoclinic orbit, for any N>1 (if it exists nearby). This scheme is based on Lin's method and it is robust and reliable.
The method is implemented in AUTO/HomCont. A system of ordinary differential equations introduced by Sandstede featuring inclination and orbit flip bifurcations and homoclinic-doubling cascades, is used as a test bed for the algorithm. It is also successfully applied to reliably find multi-hump travelling wave solutions in the FitzHugh-Nagumo nerve-axon equations and in a 4th-order Hamiltonian system arising as a model for water waves.
Additional information: Preprint of a paper later published by World Scientific (2003), International Journal of Bifurcation and Chaos, 13(10), pp.2977-2999, ISSN 0218-1274