Critical Behavior of Complex Logarithmic Map

Abstract

It is exhibited, for a complex logarithmic map (Remark: Graphics omitted.), that there exists a k-period region near the point with Ω=1⁄k on the boundary curve of the cycloid C=exp (i2πΩ)−i2πΩ where a Hopf bifurcation takes place, and that a circle map is embedded into the curve. The Devil’s staircase emerges just within the boundary and the Farey tree is constructed. The area of the k-period region in a periodic sequence 2→3→4→… is predicted to be scaled by k⁻⁶ for large k, and which is also numerically ascertained.