Dynamic Description of the Critical 2  Attractor and 2m-Band Chaos

Motion on the critical 2∞ attractor at the accumulation point of period doubling is studied by the sum of the local expansion rates of nearby orbits over time n; Sn(X0)≡∑n-1t=0λ1(Xt). The sum Sn(X0) fluctuates with n, depending upon the initial value X0. The Liapunov number Λ∞≡lim n→∞Sn(X0)/nis zero, indicating that the critical attractor has no mixing. Then it is found that, as nincreases, the variance ≪{Sn(X)}2 > exhibits a remarkable self-similar temporal structure which can be described by a universal memory function. Further, the algebraic exponent βn(X0)≡Sn(X0)/log nis introduced for large n, and its fluctuation spectrum ψβ(β) is shown to exhibit a linear part with a universal slope sc≃-0.756. These two phenomena characterize the critical attractor clearly. The critical attractor is also studied as the limit of the cascade of the band-splitting bifurcations of band chaos. It is found for the 2m-band attractor that, as M≡2m→∞, the spectrum ψβ(β) for n≪Mreduces to that of the critical 2∞ attractor, whereas the fluctuation spectrum ψ(Λ) of (Λn)(X0)≡Sn(X0)/nfor n≫Mtends to satisfy a critical scaling law ψ(Λ)=M-1 φ(M(Λ- Λ∞)) with a universal scaling function φ(z) which has two linear parts with slope sα=-1 and sβ≃1.37. Thus we obtain a unifying description of the critical 2∞ attractor and the 2m-band attractor in terms of the fluctuations of the sum Sn(X0).