1/f Noise in Fractal Quaternionic Structures

TL;DR

This paper investigates the fractal structures arising from quaternionic logistic maps and Mandelbrot set projections, revealing that the distribution of fractal circle radii exhibits characteristic 1/f noise, indicating complex underlying dynamics.

Contribution

It introduces the analysis of 2D projections of quaternionic Mandelbrot sets and demonstrates that the radii of resulting fractal circles follow a 1/f noise pattern, a novel finding in quaternionic dynamics.

Findings

Fractal circles in quaternionic projections exhibit 1/f noise.

The point process of circle radii shows pure 1/f noise.

Quaternionic Mandelbrot set projections produce fractal structures with specific noise characteristics.

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.