Noise-perturbed quaternionic Mandelbrot sets

Abstract

Quaternionic Mandelbrot sets (abbreviated as M sets) have been a focus on the research in high-dimensional fractals. This paper explores the topological structure and the fission evolution of the quaternionic M sets under noise perturbations as well as the boundaries of their regions of stability. Our experimental results indicate that the additive dynamic noise displaces the M sets from the origin position, and the multiplicative dynamic noise shrinks the regions of stability of the M sets in a certain proportion. In addition, the influence of the output noise is mainly on the inner structure of the regions of stability. The presence of such noises leads to a decrease of the box dimension of the M sets except for the additive dynamic noise. It can be concluded that the earlier-mentioned noises have a great impact on the M sets.

Keywords:

• Mandelbrot set
• noise perturbation
• bifurcation
• quaternion
• fractal

2000 AMS Subject Classifications :

• 28A80
• 37F45

Acknowledgements

This work is supported by a grant from the National Natural Science Foundation of China (No. 60573172), the Superior University Doctor Subject Special Scientific Research Foundation of China (No. 20070141014) and the National Natural Science Foundation of Liaoning province (No. 20082165). We thank the referees for their helpful comments.