Slow-fast dynamics in the localized turbulent pipe flow with transitional Reynolds number

This paper explores the slow-fast dynamics in localized turbulence for an experimentally validated model (Barkley et al. (Nature, 2015)), which describes the onset of turbulent pipe flow through a reaction-diffusion system. By geometric singular perturbation theory and the generalized rotated vector field method, we demonstrate the existence and wave speed of traveling pulses for transitional Reynolds numbers, which correspond to the homoclinic orbits connecting the unique equilibrium in the slow-fast traveling wave system. Subsequently, we present the spectral stability of traveling pulses based on asymptotic theory and the Evans function. Additionally, we investigate the existence of periodic wave trains corresponding to periodic orbits in the slow-fast traveling wave system. Furthermore, the coexisting geometric structures of periodic and homoclinic orbits can be demonstrated for the transitional Reynolds number.