Analytic Study of Periodic Chaos. II: Recursion Relations for the Power Spectra

Abstract

On the basis of the analytic formulas for the correlation functions of nonperiodic orbits, the recursion relations are obtained which fully characterize the spectral structure of periodic chaos generated by continuous one-dimensional maps with a single maximum. The power spectrum at the mth band-splitting point consists of two parts, a line spectrum and a continuous spectrum, and each part is determined from the corresponding part at the (m-1)th band-splitting point through its own recursion relations, and thus the spectra near the chaotic critical point are completely determined. Moreover, it is shown that there exist the intrinsic relations between the intensities of the line and continuous spectra. The scaling law for the particlular orbits decisive of the correlation functions plays an essential role in deriving the recursion relations. The tent map has the exact recursion relations with the two rescaling factors dependent on m. The corresponding relations hold for the logistic map to an excellent approximation with the two rescaling factors replaced by α2 and -α, where α is Feigenbaum’s rescaling factor. Universal aspects of the recursion relations are also discussed.