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An extended turbulent state can coexist with the stable laminar state in pipeflows. We focus here on short pipes with additional discrete symmetriesimposed. In this case, the boundary between the coexisting basins ofattraction, often called the edge of chaos, is the stable manifold of an edgestate, which is a lower-branch traveling wave solution. We show that alow-dimensional submanifold of the edge of chaos can be constructed fromvelocity data using the recently developed theory of spectral submanifolds(SSMs). These manifolds are the unique smoothest nonlinear continuations ofnonresonant spectral subspaces of the linearized system at stationary states.Using very low dimensional SSM-based reduced-order models, we predicttransitions to turbulence or laminarization for velocity fields near the edgeof chaos.

Capturing the Edge of Chaos as a Spectral Submanifold in Pipe Flows