Kiyotaka Tanikawa

Fractal basin boundary constructed by superposition of the Weierstrass function

Abstract The structure and properties of the basin boundary of a two-dimensional map proposed by Grebogi, Ott and Yorke are investigated. Functional analysis gives the exact shape of the boundary. Using perturbation analysis it is shown that the basin boundary is constructed by superposition of the Weierstrass function which is everywhere continuous but nowhere differentiable. The Lyapunov exponent and the fractal dimension of the boundary are also estimated in terms of the results given by the perturbation method. Perturbation analysis gives a good explanation for the fractal basin boundary.