Bifocal homoclinic orbits in four dimensions

The authors study the bifurcations which occur as they perturb four-dimensional systems of ordinary differential equations having homoclinic orbits that are bi-asymptotic to a fixed point with a double-focus structure. They give several methods of understanding the geometry of the invariant set that exists close to the homoclinic orbit and introduce a multi-valued one-dimensional map which can be used to predict the behaviour and bifurcation patterns which may occur. They argue that, although local strange behaviour is likely to occur, in a global sense (i.e. for large enough perturbations) the whole sequence of bifurcations produces a single periodic orbit, just as in the three-dimensional saddle-focus case.