The intersection surfaces in a 4-dimensional homoclinic/heteroclinic tangle

We study the homoclinic/heteroclinic tangle in a three degrees of freedom system corresponding to a 4-dimensional Poincaré map. In particular, we look at the 2-dimensional homoclinic/heteroclinic intersection surfaces between the stable and unstable manifolds of the most important codimension-2 invariant sets and study how the internal structures of these invariant sets are transported into the intersection surfaces. This provides a pictorial overview of the 2-dimensional continuum of homoclinic/heteroclinic connections via this particular intersection surface. As an example of demonstration, we use a model for a barred galaxy containing a three degrees of freedom version of a ternary symmetric horseshoe. © 2022, The Author(s), under exclusive licence to Springer Nature B.V.