Twin Heteroclinic Connections of Reversible Systems

We examine smooth four-dimensional vector fields reversible under some smooth involution $L$ that has a smooth two-dimensional submanifold of fixed points. Our main interest here is in the orbit structure of such a system near two types of heteroclinic connections involving saddle-foci and heteroclinic orbits connecting them. In both cases we found families of symmetric periodic orbits, multi-round heteroclinic connections and countable families of homoclinic orbits of saddle-foci. All this suggests that the orbit structure near such connections is very complicated. A non-variational version of the stationary Swift – Hohenberg equation is considered, as an example, where such structure has been found numerically.