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” a t t ra~tor . ’ -~ Nonperiodicity, as reflected by a broadband 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 peri~dicity”),~ Ito et al. (“nonmixing ~ h a o s ” ) , ~ and the authors6 The existence of phase coherence can make it difficult to discriminate experimentally 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. The theory of dynamical systems provides two useful measures of chaos: the Kolmogorov entropy’ and the Lyapunov characteristic exponent^.^,^ The application of this theory to strange attractors is not well understood, but it is at least a commonly expressed conjecture, supported by numerical evidence,” that these quantities can be defined for strange attractors, and that the appropriately normalized Kolmogorov entropy is equal to the sum of the positive Lyapunov characteristic exponents.”.’* The presence of chaotic behavior in a dynamical system is signaled by a positive Kolmogorov entropy. We have discovered, however, that neither of these quantities distinguish between phase coherent and phase incoherent chaos. Since attractors with a long term average periodicity are intuitively “more orderly” than incoherent attractors, these quantities provide an inadequate measure of the chaotic properties of a strange attractor. Phase coherent strange attractors may be the correct models for many physical systems. In Couette flow, for example, through certain parameter ranges, wavy Taylor