Analytic study of power spectra of the tent maps near band-splitting transitions

Abstract

Successive band-splitting transitions occur in the one-dimensional map x_i+1=g(x_i),i=0, 1, 2,... withg(x)=αx, (0 ⩽x ⩽ 1/2) −αx +α, (1/2 <x ⩽ 1) as the parameterα is changed from 2 to 1. The transition point fromN (=2ⁿ) bands to 2Nbands is given byα=(√2)^1/N (n=0, 1,2,...). The time-correlation functionξ _i=〈δx_iδx₀〉/〈(δx₀)²,δx_i≡ x_i−〈x_i〉 is studied in terms of the eigenvalues and eigenfunctions of the Frobenius-Perron operator of the map. It is shown that, near the transition pointα=√2,ξ _i−[(10−4√2)/17] δ_i,0-[(10√2-8)/51]δ_i,1 + [(7 + 4√2)/17](−1)ⁱe^−yi, whereγ≡√2(α−√2) is the damping constant and vanishes atα=√2, representing the critical slowing-down. This critical phenomenon is in strong contrast to the topologically invariant quantities, such as the Lyapunov exponent, which do not exhibit any anomaly atα=√2. The asymptotic expression forξ _i has been obtained by deriving an analytic form ofξ _i for a sequence ofα which accumulates to √2 from the above. Near the transition pointα=(√2)^1/N, the damping constant ofξ _i fori ⩾N is given byγ _N=√2(α^N-√2)/N. Numerical calculation is also carried out for arbitrary a and is shown to be consistent with the analytic results.