Power Spectra and Mixing Properties of Strange Attractors

ABSTRACT: Edward Lorenz used the phrase "deterministic nonperiodic flow" to describe the first example of what is now known as a "strange" or "chaotic" attractor. Nonperiodicity, as reflected by a braodband component in a power spectrum of a time series, is the characteristic by which chaos is currently experimentally identified. In principle, this identification is straightforward: Systems that are periodic or quasiperiodic have power spectra composed of delta functions; any dynamical system whose spectrum is not composed of delta functions is chaotic.

We have found that, to the resolution of our numerical experiments, some strange attractors have power spectra that are superpositions of delta functions and broad backgrounds, As we shall show, strange attractors with this property, which we call phase coherence, are chaotic, yet, nonetheless, at least approach being periodic or quasi-periodic in a statistical sense. Under various names, this property has also been noted by Lorenz ("noisy periodicity"), Ito et al. ("nonmixing chaos"), and the authors. The existence of phase coherence can make it difficult to discriminate expermentally between chaotic and periodic behavior by means of a power spectrum. In this paper, we investigate the geometric basis of phase coherence and demonstrate that this phenomenon is closely related to the mixing properties of attractors.