Abstract
Basin boundaries are the boundaries between the basins of attraction of coexisting attractors. When one of the attractors breaks up and becomes a transient repelling structure the basin boundary also disappears. Nevertheless, it is possible to distinguish the two types of dynamics in phase space and to define and identify a remnant of the basin boundary, the edge of chaos. We here demonstrate the concept using a two-dimensional (2D) map, and discuss properties of the edge of chaos and its invariant subspaces, the edge states. The discussion is motivated and guided by observations on certain shear flows like pipe flow and plane Couette flow where the laminar profile and a transient turbulent dynamics coexist for certain parameters, and where the notions of edge of chaos and edge states proved to be useful concepts to characterize the transition to chaos. As in those cases we use the lifetime, i.e. the number of iterations needed to approach the laminar state, as an indicator function to track the edge of chaos and to identify the invariant edge states. The 2D map captures many of the features identified in laboratory experiments and direct numerical simulations of hydrodynamic flows. It illustrates the rich dynamical behavior in the edge of chaos and of the edge states, and it can be used to develop and test further characterizations.
Highlights
The transition to turbulence in systems like plane Couette flow or pipe flow differs from the better understood examples of Taylor-Couette or Rayleigh-Benard flow in that turbulent dynamics is observed while the laminar flow is still linearly stable (Grossmann 2000, Kerswell 2005, Eckhardt et al 2007, Eckhardt 2008)
Various bifurcations of saddle-node type have been found in these systems (Nagata 1990, Nagata 1997, Clever & Busse 1997, Waleffe 2003, Wang et al 2007, Eckhardt et al 2002, Faisst & Eckhardt 2003, Wedin & Kerswell 2004, Pringle & Kerswell 2007, Eckhardt et al 2008) but at least in pipe flow they differ from the standard phenomenology in that the node state is not stable but has unstable directions as well: it is like a saddle-node bifurcation in an unstable subspace
In the third part of the paper we turn to the case of a chaotic repellor in the turbulent dynamics which mimics turbulent transients decaying to a laminar flow profile: Section 5 deals with the case of a chaotic saddle coexisting with a fixed point attractor
Summary
The transition to turbulence in systems like plane Couette flow or pipe flow differs from the better understood examples of Taylor-Couette or Rayleigh-Benard flow in that turbulent dynamics is observed while the laminar flow is still linearly stable (Grossmann 2000, Kerswell 2005, Eckhardt et al 2007, Eckhardt 2008). In the case of pipe flow numerical evidence suggests that the saddle state is not a single fixed point or a travelling wave, but that it rather carries a chaotic dynamics (Schneider et al 2007). The present study is motivated by observations on the turbulence transition in situations where the laminar profile is linearly stable, and will use descriptions like ‘laminar’ and ‘turbulent’ to describe the two dominant state between which we would like to determine the basin boundary or edge of chaos. In the third part of the paper we turn to the case of a chaotic repellor in the turbulent dynamics which mimics turbulent transients decaying to a laminar flow profile: Section 5 deals with the case of a chaotic saddle coexisting with a fixed point attractor.
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