Fractal Transformation of the OneDimensional Chaos Produced by Logarithmic Map
Abstract
The logarithmic map given by the difference equation x_{x+1} = ln(ax_{n}) generates chaos for e(1) < a < e. The variation of x_{n} in nsequence of a chaos region represents characteristic shapes depending on parameter a. The entropy and Lyapunov exponent for the system are obtained as a function of a. On repeating transformation for the case a = 1.0 by which a point stretched unlimitedly by this dynamical equation is squeezed in the region (0, 1), a fractal behaviour characterized by selfaffinity can be found in the expansion of the nsequence for various initial values x_{0}.
 Publication:

Progress of Theoretical Physics
 Pub Date:
 April 1991
 DOI:
 10.1143/ptp/85.4.759
 Bibcode:
 1991PThPh..85..759K