## Christof Jung (UNAM Cuernavaca, Mexico)

### A ternary symmetric heteroclinic tangle in a three degrees of freedom system

When the effective potential of a 3-dof system has two symmetrically placed index-1 saddles, then we find the corresponding two symmetry related codim-2 normally hyperbolic invariant manifolds (NHIMs) over these saddles. In an appropriate Poincar\'e section the stable and unstable manifolds of these NHIMs form the 4-dimensional generalisation of a ternary symmetric horseshoe. The elementary intersection structure in this heteroclinic tangle are 2-dimensional surfaces with the topology of a sphere $S^2$. Each point on such a sphere corresponds to a heteroclinic trajectory connecting individual substructures on the two NHIMs. Numerical examples are shown for the motion of a test particle in the effective potential of a barred galaxy in the corotating frame of reference. The stable and unstable manifolds of the NHIMs are directly related to observable patterns in the galaxy.