ISSN:
1572-9613
Keywords:
Chaos
;
mapping
;
ergodic
;
mixing
;
time-correlation function
;
chaos-chaos transition
;
Frobenius-Perron operator
Source:
Springer Online Journal Archives 1860-2000
Topics:
Physics
Notes:
Abstract Successive band-splitting transitions occur in the one-dimensional map xi+1=g(xi),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=〈δxiδx0〉/〈(δx0)2,δxi≡ xi−〈xi〉 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] δi,0-[(10√2-8)/51]δi,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(αN-√2)/N. Numerical calculation is also carried out for arbitrary a and is shown to be consistent with the analytic results.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01009682
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