Invariant patterns in a non-invertible chaotic map by the directional coloring

Abstract

We propose a visualization method called the directional coloring for chaotic attractors in planer discrete systems. A color in the hue circle is assigned to the argument determined by the current point and its n-th mapped point. Some unstable n-periodic points embedded in the chaotic attractor become visible as radiation points and they can be accurately detected by combination of this coloring and the Newton's method. For a chaotic attractor in a non-invertible map, we find out invariant patterns around the fixed point and detect its nearest unstable n-periodic point. The computed results of their locations show a fractal property of the system.