The present paper is devoted to investigation of the multidimensional case of the logistic mapping on the plane to itself. In this paper we learnt the properties of the sets of Julia and Mandelbrot for some two-dimensional logistic mappings. The sets of Julia and Mandelbrot help to define asymptotical behavior of the trajectories of certain mappings. The analytical solutions of the equations for finding fixed and periodic points and the computational simulations for describing the sets of Julia and Mandelbrot are the main results of this paper
1. Clark R. Dynamical Systems: Stability, Symbolic Dynamics and Chaos. London,(1995).
2. Fatou, P. Sur les Equationes Fonctionelles. Bull. Soc. Math. France Vol. 48, Issue1, 33-94 (1920).
3. Ganikhodzhayev R.N., Narziyev N.B., Seytov Sh.J. Multi-dimensional case of the problem of Von Neumann - Ulam. Uzbek Mathematical Journal Vol. 3, Issue1, 11-23 (2015).
4. Julia, G. Memoire sur l’Iteration des Fonctions Rationelles J. Math Pures Appl.Vol. 4, 47-245 (1918).
5. Devaney R. L. A First Course In Chaotic Dynamical Systems: Theory And Experiment. Boston, (1992).
6. Devaney R. L. An Introduction to Chaotic Dynamical Systems. New York, (1989).
7. Devaney R. L. Complex Exponential Dynamics. Elsevier, Vol. 3, Issue 2, 125-224 (2010).
8. Devaney R. L. Mastering Differential Equations: The Visual Method. Boston,(2011).
9. Goong Chen. Yu Haung. Chaotic Maps. Dynamics, Fractals and Rapid Fluctuations Washington, (2011).
10. Feigenbaum M.J. Qualitative Universality for a Class of Nonlinear Transformations. J. Stat. Phys. Vol. 19, Issue 1, 25-52 (1978).
11. John Milnor, Dynamics in One Complex Variable. Princeton, (2006).
12. Kathleen T. Alligood, Tim D. Sauer, James A. Yorke Chaos: An Introduction to Dynamical Systems. Berlin, (2000).
13. Li, T., and Yorke, J. Period Three imply Chaos. American Mathematical Monthly Vol. 82, Issue 10, 985-992 (1975).
14. Mandelbrot, B. Fractals Aspects of Nonlinear dynamics, Annals of the New York Academy of science Vol. 357, Issue 3, 249-252 (1980).
15. Mandelbrot, B. The Fractal Geometry of Nature. New York, (1982).
16. Mandelbrot, B. Fractals and Chaos: The Mandelbrot Set and Beyond. New York, (2004)
17. Mustafa R.S. Kulenovic, Orlando Merino, Discrete Dynamical Systems and Difference Equations with Mathematica. New York, (2002).
18. Robert W. Easton, Geometric methods for discrete dynamical systems. Oxford,(1998).
19. Sharkovsky A. N. Attractors orbits and their basins. Kiev, (2013).
20. Sharkovsky A. N. Coexistence of Cycles of a Continuous Map of a Line into itself. Ukrain J. Math. Vol. 16, Issue 3, 61-71 (1964).
21. Sharkovsky A. N., Maistrenko Yu. L., Romanenko E. Yu. Difference Equations and Their Applications. Netherlands, (1993).
22. Sharkovsky A. N., Kolyada S. F., Sivak A. G., Fedorenko V. V. Dynamics of One-Dimensional Maps. Kiev, (1989).
23. Van-der-Varden B. L. Algebra. Moscow, (1976).
Ganikhodzhaev, R.N.; Seytov, Sh.J; Obidjonov, I.N; and Sadullayev, L.O
"The sets of Julia and Mandelbrot for multi-dimensional case of logistic mapping,"
Central Asian Problems of Modern Science and Education: Vol. 2020
, Article 4.
Available at: https://uzjournals.edu.uz/capmse/vol2020/iss3/4