ON A CLOSENESS OF THE JULIA SETS OF NOISE-PERTURBED COMPLEX QUADRATIC MAPS

Abstract

In the present work, we extend the results of the study of the structural stability of the Julia sets of noise-perturbed complex quadratic maps in the presence of dynamic and output noise both for the additive and the multiplicative cases. The critical values of the strength of the noise for which the Julia set of a family of noise-perturbed complex quadratic maps completely loses its original Julia structure were also calculated. Using graphical tools we demonstrate how one can localize the regions of the Julia sets that are affected by the presence of noise in each case. Finally, two numerical invariants for the Julia set of noise-perturbed complex quadratic maps are proposed for the study of the noise effect.