A novel investigation of quaternion Julia and Mandelbrot sets using the viscosity iterative approach

Abstract

This paper presents a novel technique for visualizing quaternion Julia and Mandelbrot sets of a quaternion-valued polynomial mapping $T\left(q\right)=q^n+mq+c$, where $q$ is a quaternion variable, $n\in N\setminus \left\{1\right\}$, and $m,c$ are quaternion parameters, by employing the viscosity approximation method. The investigation begins with a study of a new escape criterion, specifically designed for generating quaternion Julia and Mandelbrot sets using the viscosity approximation technique. Based on this result, two dimensions and three dimensions cross-sections of quaternion Julia and Mandelbrot sets are created. The paper also examines how variations in the parameters of the iterative methods impact the resulting sets’ characteristics, such as shape, size, symmetry, and color.