Correlations and spectra of an intermittent chaos near its onset point
type: Item
instance of: scholarly article
title: Correlations and spectra of an intermittent chaos near its onset point
zbMATH Open document ID: 0588.58053
DOI: 10.1007/BF01010988
author: Byon-Chyol So
author: Nobuyuki Yoshitake
author: H. Okamoto
author: Hazime Mori
published in: Journal of Statistical Physics
publication date: 1984-01-01T00:00:00Z
review text: A one-parameter family of piecewise-linear discontinuous maps, which bifurcates from a periodic state of period m \((m=2,3,...)\), to an intermittent chaos, is studied as a new model for the onset of turbulence via intermittency. The onset of chaos of this model is due to the excitation of an infinite number of unstable periodic orbits and hence differs from Pomeau-Manneville's mechanism, which is a collapse of a pair of stable and unstable periodic orbits. The invariant density, the time- correlation function, and the power spectrum are analytically calculated for an infinite sequence of values of the bifurcation parameter \(\beta\) which accumulate to the onset point \(\beta_ c\) from the chaos side \(\epsilon \equiv \beta -\beta_ c>0\). The power spectrum near \(\epsilon =0\) is found to consist of a large number of Lorentzian lines with two dominant peaks. The highest peak lies around frequency \(\omega =2\pi /m\) with the power-law envelope \(1/| \omega -(2\pi /m)|^ 4.\) The second-highest peak lies around \(\omega =0\) with the envelope \(1/| \omega |^ 2\). The width of each line decreases as \(\epsilon\), and the separation \(\Delta\) \(\omega\) between lines decreases as \(\epsilon/\ln \epsilon^{-1}\). It is also shown that the Lyapunov exponent takes the form \(\lambda\) \(\simeq \epsilon /m\) and the mean lifetime of the periodic state in the intermittent chaos is given by \(m\epsilon^{-1}(\ln \epsilon^{-1}+1).\)
Mathematics Subject Classification ID: 37G99
zbMATH DE Number: 3944920
zbMATH Keywords: burst
zbMATH Keywords: ordered motion
zbMATH Keywords: turbulence
zbMATH Keywords: ergodicity
zbMATH Keywords: Perron-Frobenius operator
zbMATH Keywords: eigenfunction expansion
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3944920