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Search Results (3)

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Keywords = Julia sets of fractional order

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20 pages, 10684 KB  
Article
Control and Synchronization of Julia Sets of the Discrete Three-Dimensional Fractional HCV Model
by Miao Ouyang, Yang Chen, Yuan Jiang, Junhua Li and Shutang Liu
Fractal Fract. 2026, 10(3), 207; https://doi.org/10.3390/fractalfract10030207 - 22 Mar 2026
Viewed by 401
Abstract
This paper investigates the fractal dynamical behavior of a discrete Caputo fractional-order hepatitis C virus model. First, we analyze the stability of the system by using spectral radius and design the fractional-order controller based on coordinate transformation. Then, a nonlinear coupling controller is [...] Read more.
This paper investigates the fractal dynamical behavior of a discrete Caputo fractional-order hepatitis C virus model. First, we analyze the stability of the system by using spectral radius and design the fractional-order controller based on coordinate transformation. Then, a nonlinear coupling controller is constructed to achieve synchronization between two fractional-order models with different parameters and different fractional orders, and the synchronization is supported by rigorous mathematical proof. Numerical simulations are used to verify the effectiveness of control and synchronization. Full article
(This article belongs to the Special Issue Advances in Dynamics and Control of Fractional-Order Systems)
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13 pages, 7880 KB  
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 8 | Viewed by 4790
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 KB  
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 5 | Viewed by 2471
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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