1/f Noise in Fractal Quaternionic Structures

Abstract

We consider the logistic map over quaternions $\mathbb{H}\sim\mathbb{R}^4$ and different 2D projections of Mandelbrot set in 4D quaternionic space. The approximations (for finite number of iterations) of these 2D projections are fractal circles. We show that the point process defined by radiuses $R_j$ of those fractal circles exhibits pure 1/f noise.