<p>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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$f_{1}(z)=z^3+cz$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>f</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mi>z</mi> <mn>3</mn> </msup> <mo>+</mo> <mi>c</mi> <mi>z</mi> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$f_{2}(z)=z^{3}+cz^2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>f</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mi>z</mi> <mn>3</mn> </msup> <mo>+</mo> <mi>c</mi> <msup> <mi>z</mi> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$f_{3}(z)=z^{3}+c$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>f</mi> <mn>3</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mi>z</mi> <mn>3</mn> </msup> <mo>+</mo> <mi>c</mi> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$f_{4}(z)=z^3+cz^{2}+z $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>f</mi> <mn>4</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mi>z</mi> <mn>3</mn> </msup> <mo>+</mo> <mi>c</mi> <msup> <mi>z</mi> <mn>2</mn> </msup> <mo>+</mo> <mi>z</mi> </mrow> </math></EquationSource> </InlineEquation> 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.</p>

MANDELBROT SETS AND THEIR FRAMES, JULIA SETS AND THE STRUCTURE OF FIXED POINTS OF THIRD-DEGREE POLYNOMIALS

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发表日期：

2026年3月25日

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V. S. Sekovanov,
L. B. Rybina,
I. V. Shaposhnikova

摘要

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 $f_{1}(z)=z^3+cz$ $f_{1} (z) = z^{3} + c z$ , $f_{2}(z)=z^{3}+cz^2$ $f_{2} (z) = z^{3} + c z^{2}$ , $f_{3}(z)=z^{3}+c$ $f_{3} (z) = z^{3} + c$ , $f_{4}(z)=z^3+cz^{2}+z $ $f_{4} (z) = z^{3} + c z^{2} + z$ 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.