Generalized Logistic Maps in the Complex Plane: Structure, Symmetry, and Escape-Time Dynamics
In this paper, we introduce a generalised formulation of the logistic map extended to the complex plane and correspondingly redefine the classical Mandelbrot and Julia sets within this broader framework. Central to our approach is the development of an escape criterion based on the Picard orbit, whi...
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MDPI AG
2025-05-01
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| author | Krzysztof Gdawiec Muhammad Tanveer |
| author_facet | Krzysztof Gdawiec Muhammad Tanveer |
| author_sort | Krzysztof Gdawiec |
| collection | DOAJ |
| description | In this paper, we introduce a generalised formulation of the logistic map extended to the complex plane and correspondingly redefine the classical Mandelbrot and Julia sets within this broader framework. Central to our approach is the development of an escape criterion based on the Picard orbit, which underpins the escape-time algorithms employed for graphical approximations of these sets. We analyse the structural and dynamical properties of the resulting Mandelbrot and Julia sets, emphasising their inherent symmetries through detailed visualisations. Furthermore, we examine how variations in a key parameter of the generalised map affect two critical numerical metrics: the average escape time and the non-escaping area index. Our computational study reveals that, particularly for Julia sets, these dependencies are characterised by intricate, highly non-linear behaviour—highlighting the profound complexity and sensitivity of the system under this generalised mapping. |
| format | Article |
| id | doaj-art-84fee1c8b1a845ec828af12361b12f8e |
| institution | Kabale University |
| issn | 2075-1680 |
| language | English |
| publishDate | 2025-05-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj-art-84fee1c8b1a845ec828af12361b12f8e2025-08-20T03:26:25ZengMDPI AGAxioms2075-16802025-05-0114640410.3390/axioms14060404Generalized Logistic Maps in the Complex Plane: Structure, Symmetry, and Escape-Time DynamicsKrzysztof Gdawiec0Muhammad Tanveer1Institute of Computer Science, University of Silesia in Katowice, Bedzinska 39, 41-200 Sosnowiec, PolandDepartment of Mathematics and Statistics, Sub-Campus Depalpur, University of Agriculture Faisalabad, Faisalabad 38000, PakistanIn this paper, we introduce a generalised formulation of the logistic map extended to the complex plane and correspondingly redefine the classical Mandelbrot and Julia sets within this broader framework. Central to our approach is the development of an escape criterion based on the Picard orbit, which underpins the escape-time algorithms employed for graphical approximations of these sets. We analyse the structural and dynamical properties of the resulting Mandelbrot and Julia sets, emphasising their inherent symmetries through detailed visualisations. Furthermore, we examine how variations in a key parameter of the generalised map affect two critical numerical metrics: the average escape time and the non-escaping area index. Our computational study reveals that, particularly for Julia sets, these dependencies are characterised by intricate, highly non-linear behaviour—highlighting the profound complexity and sensitivity of the system under this generalised mapping.https://www.mdpi.com/2075-1680/14/6/404logistic mapMandelbrot setJulia setescape criterionsymmetry |
| spellingShingle | Krzysztof Gdawiec Muhammad Tanveer Generalized Logistic Maps in the Complex Plane: Structure, Symmetry, and Escape-Time Dynamics Axioms logistic map Mandelbrot set Julia set escape criterion symmetry |
| title | Generalized Logistic Maps in the Complex Plane: Structure, Symmetry, and Escape-Time Dynamics |
| title_full | Generalized Logistic Maps in the Complex Plane: Structure, Symmetry, and Escape-Time Dynamics |
| title_fullStr | Generalized Logistic Maps in the Complex Plane: Structure, Symmetry, and Escape-Time Dynamics |
| title_full_unstemmed | Generalized Logistic Maps in the Complex Plane: Structure, Symmetry, and Escape-Time Dynamics |
| title_short | Generalized Logistic Maps in the Complex Plane: Structure, Symmetry, and Escape-Time Dynamics |
| title_sort | generalized logistic maps in the complex plane structure symmetry and escape time dynamics |
| topic | logistic map Mandelbrot set Julia set escape criterion symmetry |
| url | https://www.mdpi.com/2075-1680/14/6/404 |
| work_keys_str_mv | AT krzysztofgdawiec generalizedlogisticmapsinthecomplexplanestructuresymmetryandescapetimedynamics AT muhammadtanveer generalizedlogisticmapsinthecomplexplanestructuresymmetryandescapetimedynamics |