Scaling and fine structure of superstable periodic orbits in the logistic map

Abstract

The logistic map, defined by the recurrence xn+1 = αxn(1 − xn), has been a major tool in the emergence of chaos theory. When the parameter α is varied from 0 to 4, various bifurcations occur, with cycles of period 2, 4, 8 and so on, appearing successively, till cycles of period 2k are all unstable and chaos emerges. Even after chaos has emerged, stable periodic cycles of period p can appear in some parameter α windows, which include one superstable cycle, characterized by a zero first derivative. In this paper, all superstable values of α are obtained numerically for period p up to 21, and the two smallest superstable values are established for p values up to 1000, plus p = 1024 and p = 2048. These superstable parameters exhibit discrete fine structures, often different for even and odd p, combined with global and local scaling properties. The number of superstable orbits increases like 2p−1/p for large values of p, with a distribution of values approximately following a gamma distribution with mean αΜ = 4–0.0753 + 0.0026p. Our solution for the largest superstable value of α at a period p, written α = 4−εp, follows Lorenz scaling prediction εp = π2/4p, to an accuracy better than 10−3 %. The next superstable values α = 4−βp,j follow the (1 + 2j)2 multiplier to εp predicted by Simó and Tatjer up to values of j increasing with p. The width of the superstable orbits, the xn domain around initial point 1/2 with periodicity p, determined numerically, obeys a scaling law β/2p, for β sufficiently small. For p larger than 15, the periodicity of the calculated time-series sometimes displays numerical instabilities. Thus, the numerical logistic map differs from the mathematical ideal, which invites for caution, but it also provides a generic imperfect model closer to reality and experimental data when investigating the role of higher order periodic orbits as underlying structures of chaotic behaviours.