From travelling waves to mild chaos: a supercritical bifurcation cascade in pipe flow

We study numerically a succession of transitions in pipe Poiseuille flow that leads from simple travelling waves to waves with chaotic time-dependence. The waves at the origin of the bifurcation cascade possess a shift-reflect symmetry and are both axially and azimuthally periodic with wave numbers {\kappa} = 1.63 and n = 2, respectively. As the Reynolds number is increased, successive transitions result in a wide range of time dependent solutions that includes spiralling, modulated-travelling, modulated-spiralling, doubly-modulated-spiralling and mildly chaotic waves. We show that the latter spring from heteroclinic tangles of the stable and unstable invariant manifolds of two shift-reflect-symmetric modulated-travelling waves. The chaotic set thus produced is confined to a limited range of Reynolds numbers, bounded by the occurrence of manifold tangencies. The states studied here belong to a subspace of discrete symmetry which makes many of the bifurcation and path-following investigations presented technically feasible. However, we expect that most of the phenomenology carries over to the full state-space, thus suggesting a mechanism for the formation and break-up of invariant states that can sustain turbulent dynamics.