The genesis of streamwise-localized solutions from globally periodic travelling waves in pipe flow

At intermediate Reynolds numbers, pipe flow exhibits spatio-temporal turbulence, where localized patches of turbulence may spread, split or decay. To construct a dynamical systems framework for this behaviour requires streamwise-localized solutions embedded within the turbulent dynamics. To date a single localized periodic orbit has been found, in contrast to the large number of known downstream-periodic solutions. Here we find the origin of this localized solution in a symmetry-breaking Hopf bifurcation from a known downstream periodic travelling wave. This bifurcation structure is found in a second symmetry subspace leading to new localized solutions. Our results indicate that localized versions of every downstream-periodic travelling wave should be expected.