Analytic study of chaos of the tent map: Band structures, power spectra, and critical behaviors

Abstract

Chaotic behaviors of the tent map (a piecewise-linear, continuous map with a unique maximum) are studied analytically throughout its chaotic region in terms of the invariant density and the power spectrum. As the height of the maximum is lowered, successive band-splitting transitions occur in the chaotic region and accumulate to the transition point into the nonchaotic region. The timecorrelation function of nonperiodic orbits and their power spectrum are calculated exactly at the band-splitting points and in the vicinity of these points. The method of eigenvalue problems of the Frobenius-Perron operator is used. 2 m−1 critical modes, wherem = 1,2, 3, ..., are found which exhibit the critical slowing-down near the 2 m−1-band to 2 m -band transition point. After the transition these modes become periodic modes which represent the cycling of nonperiodic orbits among 2 m bands together with the periodic modes generated by the preceding band splittings. Scaling laws near the transition point into the nonchaotic region are investigated and a new scaling law is found for the total intensity of the periodic part of the spectrum.