Correlations and spectra of an intermittent chaos near its onset point (Q1073437)

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 \(m\epsilon^{-1}(\ln \epsilon^{-1}+1).\)