Quasi-Mandelbrot sets for perturbed complex analytic maps: visual patterns

TL;DR

This paper explores how small perturbations to the complex quadratic map affect the shape of quasi-Mandelbrot sets, revealing both abrupt and smooth transitions in their visual patterns.

Contribution

It introduces the concept of quasi-Mandelbrot sets for perturbed complex maps and describes their continuous and sharp morphological changes.

Findings

Quasi-Mandelbrot sets can change sharply at critical perturbation values.

Smooth transitions from classical to linear-structured forms are observed.

Two examples of continuous evolution demonstrate the range of visual pattern changes.

Abstract

We consider perturbations of the complex quadratic map $ z \to z^2 +c$ and corresponding changes in their quasi-Mandelbrot sets. Depending on particular perturbation, visual forms of quasi-Mandelbrot set changes either sharply (when the perturbation reaches some critical value) or continuously. In the latter case we have a smooth transition from the classical form of the set to some forms, constructed from mostly linear structures, as it is typical for two-dimensional real number dynamics. Two examples of continuous evolution of the quasi-Mandelbrot set are described.