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Keywords = Mandelbrot set of fractional order

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13 pages, 7880 KiB  
Article
Mandelbrot Set as a Particular Julia Set of Fractional Order, Equipotential Lines and External Rays of Mandelbrot and Julia Sets of Fractional Order
by Marius-F. Danca
Fractal Fract. 2024, 8(1), 69; https://doi.org/10.3390/fractalfract8010069 - 19 Jan 2024
Cited by 4 | Viewed by 2992
Abstract
This paper deepens some results on a Mandelbrot set and Julia sets of Caputo’s fractional order. It is shown analytically and computationally that the classical Mandelbrot set of integer order is a particular case of Julia sets of Caputo-like fractional order. Additionally, the [...] Read more.
This paper deepens some results on a Mandelbrot set and Julia sets of Caputo’s fractional order. It is shown analytically and computationally that the classical Mandelbrot set of integer order is a particular case of Julia sets of Caputo-like fractional order. Additionally, the differences between the fractional-order Mandelbrot set and Julia sets from their integer-order variants are revealed. Equipotential lines and external rays of a Mandelbrot set and Julia sets of fractional order are determined. Full article
(This article belongs to the Special Issue Advances in Fractional Integral and Derivative Operators with Applications)
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21 pages, 31507 KiB  
Article
On the Fractional-Order Complex Cosine Map: Fractal Analysis, Julia Set Control and Synchronization
by A. A. Elsadany, A. Aldurayhim, H. N. Agiza and Amr Elsonbaty
Mathematics 2023, 11(3), 727; https://doi.org/10.3390/math11030727 - 1 Feb 2023
Cited by 3 | Viewed by 1769
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Recent Advances in Chaos, Fractal and Complex Dynamics in Nonlinear Systems)
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10 pages, 1492 KiB  
Article
On the Stability Domain of a Class of Linear Systems of Fractional Order
by Marius-F. Danca
Fractal Fract. 2023, 7(1), 49; https://doi.org/10.3390/fractalfract7010049 - 31 Dec 2022
Cited by 5 | Viewed by 1899
Abstract
In this paper, the shape of the stability domain Sq for a class of difference systems defined by the Caputo forward difference operator Δq of order q∈(0,1) is numerically analyzed. It is shown numerically that [...] Read more.
In this paper, the shape of the stability domain Sq for a class of difference systems defined by the Caputo forward difference operator Δq of order q∈(0,1) is numerically analyzed. It is shown numerically that due to of power of the negative base in the expression of the stability domain, in addition to the known cardioid-like shapes, Sq could present supplementary regions where the stability is not verified. The Mandelbrot map of fractional order is considered as an illustrative example. In addition, it is conjectured that for q<0.5, the shape of Sq does not cover the main body of the underlying Mandelbrot set of fractional order as in the case of integer order. Full article
(This article belongs to the Special Issue Advances in Fractional-Order Embedded Systems)
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