Mandelbrot sets and their frames, Julia sets and the structure of fixed points of third-degree polynomials

V. S. Sekovanov^a, L. B. Rybina^b, I. V. Shaposhnikova<sup>c

a</sup> N. A. Nekrasov Kostroma State University
^b Kostroma State Agricultural Academy
^c Surgut State University

Abstract: This paper continues the research of R. Cronover, D. Milnor, H.-O. Peitgen, P. H. Richter, and the authors. Using mathematical methods and computer experiments, the Mandelbrot set framings of four families of third degree polynomials of a complex variable are revealed. The connection between the frames of Mandelbrot sets of the functions $f_{1}(z)=z^3+cz$, $f_{2}(z)=z^{3}+cz^2$, $f_{3}(z)=z^{3}+c$, $f_{4}(z)=z^3+cz^{2}+z $ with remarkable curves (lemniscate, epicycloid, and circle) is established. Algorithms for constructing the frames of Mandelbrot sets of the considered families of functions are developed. Algorithms for constructing Mandelbrot sets of the given families of functions are developed.

UDC: 517.988.5