Characteristics of power spectra for regular and chaotic systems

Power spectra for chaotic and quasiperiodic systems are analyzed in detail. Differences in convergence with increasing T are carefully examined and a number of important results are emphasized. First, we demonstrate that a finite time power spectrum is incapable, in principle, of providing an unambiguous distinction between a grassy power spectrum associated with a chaotic system and a singular power spectrum associated with quasiperiodic motion. By contrast the statistics of power spectra at fixed energy are shown to provide an adequate method for making such a distinction. Second we show two routes to linking the power spectrum to the physically meaningful spectral density. One route results from a correct analysis of the Wiener–Khinchin theorem wherein the power spectrum and spectral density are shown to be related through a weak limit over frequencies. The second route emerges from a study of the asymptotic behavior of averages over chaotic power spectrum leading to a method for extracting, via extrapolation to the long time limit, the smooth spectral density from the grassy power spectra associated with chaotic behavior. Computations on the cat map, and Henon–Heiles and NaClK Hamiltonians are provided throughout.

Characteristics of power spectra for regular and chaotic systems Journal Articles