Shear-flow transition: the basin boundary

TL;DR

This paper investigates the structure of the basin boundary in a shear flow model, revealing that a periodic orbit mediates transition to turbulence and explaining the 'edge of chaos' through invariant sets.

Contribution

It identifies the role of a periodic orbit in the basin boundary and offers a new interpretation of the 'edge of chaos' in shear flow transition models.

Findings

The basin boundary contains both an equilibrium point and a periodic orbit.

Transition to turbulence is mediated by the periodic orbit P.

Near the equilibrium point, the basin of attraction region is extremely narrow.

Abstract

The structure of the basin of attraction of a stable equilibrium point is investigated for a dynamical system (W97) often used to model transition to turbulence in shear flows. The basin boundary contains not only an equilibrium point Xlb but also a periodic orbit P, and it is the latter that mediates the transition. Orbits starting near Xlb relaminarize. We offer evidence that this is due to the extreme narrowness of the region complementary to basin of attraction in that part of phase space near Xlb. This leads to a proposal for interpreting the 'edge of chaos' in terms of more familiar invariant sets.