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

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. © IOP Publishing Ltd and Deutsche Physikalische Gesellschaft.