  Using various techniques from dynamical systems theory, we rigorously study
an experimentally validated model by [Barkley et al., Nature, 526:550-553,
2015], which describes the rise of turbulent pipe flow via a PDE system of
reduced complexity. The fast evolution of turbulence is governed by
reaction-diffusion dynamics coupled to the centerline velocity, which evolves
with advection of Burgers' type and a slow relaminarization term. Applying to
this model a spatial dynamics ansatz and geometric singular perturbation
theory, we prove the existence of a heteroclinic loop between a turbulent and a
laminar steady state and establish a cascade of bifurcations of various
traveling waves mediating the transition to turbulence. The most complicated
behaviour can be found in an intermediate Reynolds number regime, where the
traveling waves exhibit arbitrarily long periodic-like dynamics indicating the
onset of chaos. Our analysis provides a systematic mathematical approach to
identifying the transition to spatio-temporal turbulent structures that may
also be applicable to other models arising in fluid dynamics.
