Periodic orbits near onset of chaos in plane Couette flow

TL;DR

This paper investigates the bifurcation structure of plane Couette flow, revealing a cascade of periodic orbits leading to turbulence, and introduces a symbolic dynamics approach to describe these orbits.

Contribution

It demonstrates the existence of a periodic doubling cascade ending in a crisis bifurcation and applies symbolic dynamics to classify orbits in turbulent flow.

Findings

Periodic orbits follow a universal sequence similar to unimodal maps.

Orbits cover much of turbulent dynamics when projected onto energy variables.

A symbolic dynamics framework is introduced for classifying flow orbits.

Abstract

We track the secondary bifurcations of coherent states in plane Couette flow and show that they undergo an incomplete periodic doubling cascade that ends with a crisis bifurcation. We introduce a symbolic dynamics for the orbits and show that the ones that exist fall into the universal sequence described by Metropolis, Stein and Stein for unimodal maps. The periodic orbits cover much of the turbulent dynamics in that their temporal evolution overlaps with turbulent motions when projected onto a plane spanned by energy production and dissipation.