Homoclinic Orbits to Streamwise-Localized Solution in Pipe Flow

We do numerical study on transition to turbulence in pipe flow by investigating a streamwise-localized solution. We explore the trajectory along the unstable manifold of the lower branch of this time-periodic solution and found the presence of homoclinic orbits related to it. These homoclinic orbits are extracted by performing a bisection method. The presence of homoclinic orbits implies the existence of a Smale horseshoe which generates chaos and can be evidence to support a theoretical description of the onset of spatially localized transient turbulence -- i.e., turbulent puff -- observed in pipe flow experiments.