FRACTAL TRANSFORMATION OF THE ONE-DIMENSIONAL CHAOS PRODUCED BY LOGARITHMIC MAP

The logarithmic map given by the difference equation x(n+1) = ln(a\x(n)\) generates chaos for e-1 < a < e. The variation of x(n) in n-sequence 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 self-affinity can be found in the expansion of the n-sequence for various initial values x0.