An example of ℤ2-transversality in a system of four coupled oscillators

The existence of structurally stable homoclinic or heteroclinic cycles for equivariant vector fields depends on the existence of stable non-transverse intersections of invariant manifolds. This chapter investigates a model system of four coupled oscillators where this breakdown of transversality occurs in a more complicated way than in our previous examples. It devotes to the description of some simple examples of Gan₂-transversality that one use in the construction of model system. It is possible to develop a theory of equivariant transversality that satisfies this openness condition as well as the other characteristic properties of transversality. The chapter shows that the phenomenon of T-transversality and the associated twisting around the invariant manifolds should bear some relationship to the inclination-flip bifurcations previously studied by a number of authors.