Generalized Mandelbrot Sets of a Family of Polynomials Pnz=zn+z+c;n≥2
In this paper, we study the general Mandelbrot set of the family of polynomials Pnz=zn+z+c;n≥2, denoted by GM(Pn). We construct the general Mandelbrot set for these polynomials by the escaping method. We determine the boundaries, areas, fractals, and symmetry of the previous polynomials. On the othe...
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Wiley
2022-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/2022/4510088 |
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| author | Salma M. Farris |
| author_facet | Salma M. Farris |
| author_sort | Salma M. Farris |
| collection | DOAJ |
| description | In this paper, we study the general Mandelbrot set of the family of polynomials Pnz=zn+z+c;n≥2, denoted by GM(Pn). We construct the general Mandelbrot set for these polynomials by the escaping method. We determine the boundaries, areas, fractals, and symmetry of the previous polynomials. On the other hand, we study some topological properties of GMPn. We prove that GMPn is bounded and closed; hence, it is compact. Also, we characterize the general Mandelbrot set as a union of basins of attraction. Finally, we make a comparison between the properties of famous Mandelbrot set Mz2+c and our general Mandelbrot sets. |
| format | Article |
| id | doaj-art-4f45e8af698f4fb5b622a1120146ff38 |
| institution | OA Journals |
| issn | 1687-0425 |
| language | English |
| publishDate | 2022-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-4f45e8af698f4fb5b622a1120146ff382025-08-20T02:05:38ZengWileyInternational Journal of Mathematics and Mathematical Sciences1687-04252022-01-01202210.1155/2022/4510088Generalized Mandelbrot Sets of a Family of Polynomials Pnz=zn+z+c;n≥2Salma M. Farris0Department of MathematicsIn this paper, we study the general Mandelbrot set of the family of polynomials Pnz=zn+z+c;n≥2, denoted by GM(Pn). We construct the general Mandelbrot set for these polynomials by the escaping method. We determine the boundaries, areas, fractals, and symmetry of the previous polynomials. On the other hand, we study some topological properties of GMPn. We prove that GMPn is bounded and closed; hence, it is compact. Also, we characterize the general Mandelbrot set as a union of basins of attraction. Finally, we make a comparison between the properties of famous Mandelbrot set Mz2+c and our general Mandelbrot sets.http://dx.doi.org/10.1155/2022/4510088 |
| spellingShingle | Salma M. Farris Generalized Mandelbrot Sets of a Family of Polynomials Pnz=zn+z+c;n≥2 International Journal of Mathematics and Mathematical Sciences |
| title | Generalized Mandelbrot Sets of a Family of Polynomials Pnz=zn+z+c;n≥2 |
| title_full | Generalized Mandelbrot Sets of a Family of Polynomials Pnz=zn+z+c;n≥2 |
| title_fullStr | Generalized Mandelbrot Sets of a Family of Polynomials Pnz=zn+z+c;n≥2 |
| title_full_unstemmed | Generalized Mandelbrot Sets of a Family of Polynomials Pnz=zn+z+c;n≥2 |
| title_short | Generalized Mandelbrot Sets of a Family of Polynomials Pnz=zn+z+c;n≥2 |
| title_sort | generalized mandelbrot sets of a family of polynomials pnz zn z c n≥2 |
| url | http://dx.doi.org/10.1155/2022/4510088 |
| work_keys_str_mv | AT salmamfarris generalizedmandelbrotsetsofafamilyofpolynomialspnzznzcn2 |