Chaotic double cycling

We study the dynamics of a generic vector field in the neighbourhood of a heteroclinic cycle of non-trivial periodic solutions whose invariant manifolds meet transversely. The main result is the existence of chaotic double cycling: there are trajectories that follow the cycle making any prescribed number of turns near the periodic solutions, for any given bi-infinite sequence of turns. Using symbolic dynamics, arbitrarily close to the cycle, we find a robust and transitive set of initial conditions whose trajectories follow the cycle for all time and that is conjugate to a Markov shift over a finite alphabet. This conjugacy allows us to prove the existence of infinitely many heteroclinic and homoclinic subsidiary connections, which give rise to a heteroclinic network with infinitely many cycles and chaotic dynamics near them, exhibiting themselves switching and cycling. We construct an example of a vector field with Z ₃ symmetry in a five-dimensional sphere with a heteroclinic cycle having this property.