Critical exponents for power-spectra scaling at mergings of chaotic bands.

Abstract

The properties of the power spectra of a broad class of smooth d-dimensional (d\ensuremath{\ge}2) dynamical systems exhibiting a reverse sequence of pairwise mergings of chaotic bands is discussed. In particular, the width of the power pairwise spectral feature associated with the pairwise merging of chaotic bands scales as \ensuremath{\Vert}p-${p}_{m}$${\ensuremath{\Vert}}^{\ensuremath{\gamma}}$, where p is a system parameter, the mth merging occurs at p=${p}_{m}$, and \ensuremath{\gamma} is the critical exponent. Numerical experiments are performed which verify theoretical predictions for \ensuremath{\gamma}. For example, near the accumulation of period doublings the result for the critical exponent reduces to \ensuremath{\gamma}${=(1/2+2}^{\mathrm{\ensuremath{-}}m}$\ensuremath{\eta}/ln(1/J), where \ensuremath{\eta} is a universal number, J is the Jacobian determinant, and ${2}^{m}$ is the number of bands (before merging). (In the trivial case of smooth one-dimensional maps J\ensuremath{\rightarrow}0 and the exponent is (1/2.) .AE