2002/11/08 by N. Korabel, Nickolay Korabel, Rainer Klages +3
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #Anomalous diffusion #Chaotic Dynamics (nlin.CD) #Complex Systems and Time Series Analysis #Diffusion #FOS: Physical sciences #Fractal #Geometry #Mathematical Dynamics and Fractals #Mathematical analysis #Mathematics #Nonlinear system #Parameter space #Physics #Sine #Statistical Mechanics (cond-mat.stat-mech) #Statistical physics #Theoretical and Computational Physics #cond-mat.stat-mech #nlin.CD
paper · pdf · doi:10.48550/arxiv.nlin/0211012
31 pages in Latex, Elsevier style; 12 figures, some of the figures are in reduced quality
arxiv created 2002/11/08 · openalex publication_date 2002/11/08 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
The nonlinear climbing sine map is a nonhyperbolic dynamical system exhibiting both normal and anomalous diffusion under variation of a control parameter. We show that on a suitable coarse scale this map generates an oscillating parameter-dependent diffusion coefficient, similarly to hyperbolic maps, whose asymptotic functional form can be understood in terms of simple random walk approximations. On finer scales we find fractal hierarchies of normal and anomalous diffusive regions as functions of the control parameter. By using a Green-Kubo formula for diffusion the origin of these different regions is systematically traced back to strong dynamical correlations. Starting from the equations of motion of the map these correlations are formulated in terms of fractal generalized Takagi functions obeying generalized de Rham-type functional recursion relations. We finally analyze the measure of the normal and anomalous diffusive regions in the parameter space showing that in both cases it is positive, and that for normal diffusion it increases by increasing the parameter value.