Anomalous diffusion in a class of round-off walks.

摘要

The round-off errors introduced by discretization in numerical study of a dynamical system are usually treated as random noise and assumed to resemble an elementary random walk, with a corresponding Gaussian probability distribution function (p.d.f) for displacements and power-law behavior of moments. In this paper we apply a walk model to study the reversible and non-dissipative dynamics of the round-off errors in a special class of uniformly discretized linear maps of the plane. Using localization and embedding, the round-off maps on the lattice are conjugated to piecewise affine maps (return maps) on polygonal domains in the local space. With recursive partitions of scaling domains, the return maps correspond to families of periodic substitution walks. The density of walks within each family forms a scaling sequence. The trajectories of the substitution walks are fractals with different space and time scaling factors. The p.d.f. for the displacements on the substitution walks exhibits scaling in time and space. The possible displacements in the family of walks during a finite time belong to a finite set of periodic walks below some critical level. Contributions to the p.d.f. from the displacements in the periodic walks above the critical level can be calculated by a geometric series. The subtle competition between the increasing p.d.f. and the decreasing density of the walks produces asymptotic power law behavior of moments with periodic correction in the logarithm of time. In the case of substitution walks families with different scaling factors, the correction is quasi-periodic in the logarithm of time.