Basin boundary, edge of chaos, and edge state in a two-dimensional model

TL;DR

This paper investigates the boundary between laminar and turbulent states in shear flows using a two-dimensional map, revealing complex dynamics at the edge of chaos and its relation to basin boundaries.

Contribution

It introduces a two-dimensional map model capturing key features of shear flow transitions, enabling analysis of edge states and basin boundaries in transient turbulence.

Findings

Different dynamical regimes coexist at the edge of chaos.

The model reproduces features observed in experiments and simulations.

It provides a framework for further characterization of flow transitions.

Abstract

In shear flows like pipe flow and plane Couette flow there is an extended range of parameters where linearly stable laminar flow coexists with a transient turbulent dynamics. When increasing the amplitude of a perturbation on top of the laminar flow, one notes a a qualitative change in its lifetime, from smoothly varying and short one on the laminar side to sensitively dependent on initial conditions and long on the turbulent side. The point of transition defines a point on the edge of chaos. Since it is defined via the lifetimes, the edge of chaos can also be used in situations when the turbulence is not persistent. It then generalises the concept of basin boundaries, which separate two coexisting attractors, to cases where the dynamics on one side shows transient chaos and almost all trajectories eventually end up on the other side. In this paper we analyse a two-dimensional map which captures many of the features identified in laboratory experiments and direct numerical simulations of hydrodynamic flows. The analysis of the map shows that different dynamical situations in the edge of chaos can be combined with different dynamical situations in the turbulent region. Consequently, the model can be used to develop and test further characterisations that are also applicable to realistic flows.