On the Fractional-Order Complex Cosine Map: Fractal Analysis, Julia Set Control and Synchronization

A. A. Elsadany (), A. Aldurayhim, H. N. Agiza and Amr Elsonbaty
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A. A. Elsadany: Department of Mathematics, Faculty of Science and Humanities in Al-Kharj, Prince Sattam bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia
A. Aldurayhim: Department of Mathematics, Faculty of Science and Humanities in Al-Kharj, Prince Sattam bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia
H. N. Agiza: Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt
Amr Elsonbaty: Department of Mathematics, Faculty of Science and Humanities in Al-Kharj, Prince Sattam bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia

Mathematics, 2023, vol. 11, issue 3, 1-21

Abstract: In this paper, we introduce a generalized complex discrete fractional-order cosine map. Dynamical analysis of the proposed complex fractional order map is examined. The existence and stability characteristics of the map’s fixed points are explored. The existence of fractal Mandelbrot sets and Julia sets, as well as their fractal properties, are examined in detail. Several detailed simulations illustrate the effects of the fractional-order parameter, as well as the values of the map constant and exponent. In addition, complex domain controllers are constructed to control Julia sets produced by the proposed map or to achieve synchronization of two Julia sets in master/slave configurations. We identify the more realistic synchronization scenario in which the master map’s parameter values are unknown. Finally, numerical simulations are employed to confirm theoretical results obtained throughout the work.

Keywords: complex cosine map; discrete fractional; fractal sets; Julia set control; Julia sets synchronization (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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