Theoretical Investigations of Chaotic Dynamics

Abstract

Abstract : An important component of the work for AFOSR was the discovery and investigation of 'riddled basins'. A riddled basin for a chaotic attractor's basin is arbitrarily close to points in another attractor's basin (the first basin is riddled with holes). When an attractor has a riddled basin there is an extreme end-state sensitivity to initial conditions in their sense that for any initial condition in the riddled basin an arbitrarily small error in computation can result in the erroneous prediction-of which attractor the initial condition is eventually attracted to. This contrasts with the more usual situation of a chaotic attractor with a non-riddled basin where any error in computation propagates exponentially but one can reliably say which attractor the initial condition is attracted to. Since the researchers discovery of the phenomenon of riddled basins, physical examples have been found in scattering, statistical mechanical, and ecological models. As can be seen from the bibliography, they have also done extensive work in other areas of dynamics, including the properties of indecomposable continua occurring in models of turbulent fluid flow.