Quasi-Mandelbrot sets for perturbed complex analytic maps: visual   patterns
  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. 
A. V. Toporensky