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Symmetry
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Bifurcation and Chaos in Fractional-Order Systems
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Bifurcation and Chaos in Fractional-Order Systems

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: closed (30 November 2020) | Viewed by 24719

Printed Edition Available!
A printed edition of this Special Issue is available here.

Special Issue Editors

Prof. Dr. Marius-F. Danca

E-Mail Website
Guest Editor
Romanian Instituite of Science and Technology, 400487 Cluj-Napoca, Romania
Interests: nonlinear dynamics; continuous/non-smooth chaotic dynamical systems of integer/fractional order; fractals
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Guanrong (Ron) Chen

E-Mail Website
Guest Editor
Department of Electrical Engineering, City University of Hong Kong, Hong Kong, China
Interests: nonlinear dynamics; complex networks and control systems
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

The concept of fractional differentiation first emerged in 1965 in a historical correspondence between the Marquise de L'Hospital and mathematician Leibnitz.  In the sequel, famous mathematicians such as Euler, Laplace, Abel, Liouville, and Riemann further developed essential technical details. It was realized recently that many scientific phenomena with complex dynamics cannot be well modeled by differential equations using integer-order derivatives. As a result, there has been an increasing interest to merge the mathematical fundamentals of fractional calculus into scientific and engineering applications such as an interdisciplinary approach, which has started to demonstrate some advantages over conventional integer-order systems. Although the previous decade witnessed significant development in this research area, many theoretical and technical problems remain to be further explored, including particularly fractional-order chaotic systems. On the other hand, finding hidden attractors in continuous-time and discrete-time fractional-order chaotic systems represents a new trend of research, one which is equal parts exciting and challenging. Of particular interest are those systems with symmetry, in which bifurcations can lead to a family of conjugate attractors all related by symmetry. Therefore, this research direction of bifurcation and chaos in fractional-order dynamical systems opens up a corpus of opportunities with great promises in such scientific fields as complex dynamics, systems and networks, and signal processing, to name a few. The present Special Issue calls for new contributions to these important and interesting topics of research, and we sincerely hope that you will be happy to participate.

Prof. Dr. Marius-F. Danca
Prof. Dr. Guanrong Chen
Guest Editors

Manuscript Submission Information

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Keywords

  • fractional-order derivative
  • fractional chaotic system
  • hidden attractor
  • symmetry
  • complex dynamics

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Published Papers (8 papers)

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Research

23 pages, 9456 KiB  
Article
Chaotic Discrete Fractional-Order Food Chain Model and Hybrid Image Encryption Scheme Application
by Sameh Askar, Abdulrahman Al-khedhairi, Amr Elsonbaty and Abdelalim Elsadany
Symmetry 2021, 13(2), 161; https://doi.org/10.3390/sym13020161 - 21 Jan 2021
Cited by 12 | Viewed by 2580
Abstract
Using the discrete fractional calculus, a novel discrete fractional-order food chain model for the case of strong pressure on preys map is proposed. Dynamical behaviors of the model involving stability analysis of its equilibrium points, bifurcation diagrams and phase portraits are investigated. It [...] Read more.
Using the discrete fractional calculus, a novel discrete fractional-order food chain model for the case of strong pressure on preys map is proposed. Dynamical behaviors of the model involving stability analysis of its equilibrium points, bifurcation diagrams and phase portraits are investigated. It is demonstrated that the model can exhibit a variety of dynamical behaviors including stable steady states, periodic and quasiperiodic dynamics. Then, a hybrid encryption scheme based on chaotic behavior of the model along with elliptic curve key exchange scheme is proposed for colored plain images. The hybrid scheme combines the characteristics of noise-like chaotic dynamics of the map, including high sensitivity to values of parameters, with the advantages of reliable elliptic curves-based encryption systems. Security analysis assures the efficiency of the proposed algorithm and validates its robustness and efficiency against possible types of attacks. Full article
(This article belongs to the Special Issue Bifurcation and Chaos in Fractional-Order Systems)
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18 pages, 654 KiB  
Article
Generalized Bessel Polynomial for Multi-Order Fractional Differential Equations
by Mohammad Izadi and Carlo Cattani
Symmetry 2020, 12(8), 1260; https://doi.org/10.3390/sym12081260 - 30 Jul 2020
Cited by 33 | Viewed by 3516
Abstract
The main goal of this paper is to define a simple but effective method for approximating solutions of multi-order fractional differential equations relying on Caputo fractional derivative and under supplementary conditions. Our basis functions are based on some original generalization of the Bessel [...] Read more.
The main goal of this paper is to define a simple but effective method for approximating solutions of multi-order fractional differential equations relying on Caputo fractional derivative and under supplementary conditions. Our basis functions are based on some original generalization of the Bessel polynomials, which satisfy many properties shared by the classical orthogonal polynomials as given by Hermit, Laguerre, and Jacobi. The main advantages of our polynomials are two-fold: All the coefficients are positive and any collocation matrix of Bessel polynomials at positive points is strictly totally positive. By expanding the unknowns in a (truncated) series of basis functions at the collocation points, the solution of governing differential equation can be easily converted into the solution of a system of algebraic equations, thus reducing the computational complexities considerably. Several practical test problems also with some symmetries are given to show the validity and utility of the proposed technique. Comparisons with available exact solutions as well as with several alternative algorithms are also carried out. The main feature of our approach is the good performance both in terms of accuracy and simplicity for obtaining an approximation to the solution of differential equations of fractional order. Full article
(This article belongs to the Special Issue Bifurcation and Chaos in Fractional-Order Systems)
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13 pages, 2533 KiB  
Article
Bifurcations, Hidden Chaos and Control in Fractional Maps
by Adel Ouannas, Othman Abdullah Almatroud, Amina Aicha Khennaoui, Mohammad Mossa Alsawalha, Dumitru Baleanu, Van Van Huynh and Viet-Thanh Pham
Symmetry 2020, 12(6), 879; https://doi.org/10.3390/sym12060879 - 27 May 2020
Cited by 10 | Viewed by 2559
Abstract
Recently, hidden attractors with stable equilibria have received considerable attention in chaos theory and nonlinear dynamical systems. Based on discrete fractional calculus, this paper proposes a simple two-dimensional and three-dimensional fractional maps. Both fractional maps are chaotic and have a unique equilibrium point. [...] Read more.
Recently, hidden attractors with stable equilibria have received considerable attention in chaos theory and nonlinear dynamical systems. Based on discrete fractional calculus, this paper proposes a simple two-dimensional and three-dimensional fractional maps. Both fractional maps are chaotic and have a unique equilibrium point. Results show that the dynamics of the proposed fractional maps are sensitive to both initial conditions and fractional order. There are coexisting attractors which have been displayed in terms of bifurcation diagrams, phase portraits and a 0-1 test. Furthermore, control schemes are introduced to stabilize the chaotic trajectories of the two novel systems. Full article
(This article belongs to the Special Issue Bifurcation and Chaos in Fractional-Order Systems)
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13 pages, 3008 KiB  
Article
On Two-Dimensional Fractional Chaotic Maps with Symmetries
by Fatima Hadjabi, Adel Ouannas, Nabil Shawagfeh, Amina-Aicha Khennaoui and Giuseppe Grassi
Symmetry 2020, 12(5), 756; https://doi.org/10.3390/sym12050756 - 6 May 2020
Cited by 27 | Viewed by 2883
Abstract
In this paper, we propose two new two-dimensional chaotic maps with closed curve fixed points. The chaotic behavior of the two maps is analyzed by the 0–1 test, and explored numerically using Lyapunov exponents and bifurcation diagrams. It has been found that chaos [...] Read more.
In this paper, we propose two new two-dimensional chaotic maps with closed curve fixed points. The chaotic behavior of the two maps is analyzed by the 0–1 test, and explored numerically using Lyapunov exponents and bifurcation diagrams. It has been found that chaos exists in both fractional maps. In addition, result shows that the proposed fractional maps shows the property of coexisting attractors. Full article
(This article belongs to the Special Issue Bifurcation and Chaos in Fractional-Order Systems)
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13 pages, 4884 KiB  
Article
Fractional Levy Stable and Maximum Lyapunov Exponent for Wind Speed Prediction
by Shouwu Duan, Wanqing Song, Carlo Cattani, Yakufu Yasen and He Liu
Symmetry 2020, 12(4), 605; https://doi.org/10.3390/sym12040605 - 11 Apr 2020
Cited by 5 | Viewed by 2786
Abstract
In this paper, a wind speed prediction method was proposed based on the maximum Lyapunov exponent (Le) and the fractional Levy stable motion (fLsm) iterative prediction model. First, the calculation of the maximum prediction steps was introduced based on the maximum Le. The [...] Read more.
In this paper, a wind speed prediction method was proposed based on the maximum Lyapunov exponent (Le) and the fractional Levy stable motion (fLsm) iterative prediction model. First, the calculation of the maximum prediction steps was introduced based on the maximum Le. The maximum prediction steps could provide the prediction steps for subsequent prediction models. Secondly, the fLsm iterative prediction model was established by stochastic differential. Meanwhile, the parameters of the fLsm iterative prediction model were obtained by rescaled range analysis and novel characteristic function methods, thereby obtaining a wind speed prediction model. Finally, in order to reduce the error in the parameter estimation of the prediction model, we adopted the method of weighted wind speed data. The wind speed prediction model in this paper was compared with GA-BP neural network and the results of wind speed prediction proved the effectiveness of the method that is proposed in this paper. In particular, fLsm has long-range dependence (LRD) characteristics and identified LRD by estimating self-similarity index H and characteristic index α. Compared with fractional Brownian motion, fLsm can describe the LRD process more flexibly. However, the two parameters are not independent because the LRD condition relates them by αH > 1. Full article
(This article belongs to the Special Issue Bifurcation and Chaos in Fractional-Order Systems)
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11 pages, 3517 KiB  
Article
Dynamical Properties of Fractional-Order Memristor
by Shao Fu Wang and Aiqin Ye
Symmetry 2020, 12(3), 437; https://doi.org/10.3390/sym12030437 - 9 Mar 2020
Cited by 7 | Viewed by 2747
Abstract
The properties of a fractional-order memristor is studied, and the influences of parameters are analyzed and compared. The results reflect that the resistance value of a fractional-order memristor can be affected by fraction-order, frequency, the switch resistor ratio, average mobility and so on. [...] Read more.
The properties of a fractional-order memristor is studied, and the influences of parameters are analyzed and compared. The results reflect that the resistance value of a fractional-order memristor can be affected by fraction-order, frequency, the switch resistor ratio, average mobility and so on. In addition, the circuit of a fractional-order memristor that is serially connected and connected in parallel with inductance and capacitance are studied. Then, the current–voltage characteristics of a simple series one-port circuits that are composed of a fractional-order memristor and a capacitor, or composed of a fractional-order memristor and a inductor are studied separately. The results demonstrate that at the periodic excitation, the memristor in the series circuits will have capacitive properties or inductive properties as the fractional order changes, the dynamical properties can be used in a memristive circuit. Full article
(This article belongs to the Special Issue Bifurcation and Chaos in Fractional-Order Systems)
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12 pages, 1614 KiB  
Article
Fractional Dynamics in Soccer Leagues
by António M. Lopes and Jose A. Tenreiro Machado
Symmetry 2020, 12(3), 356; https://doi.org/10.3390/sym12030356 - 1 Mar 2020
Cited by 3 | Viewed by 2446
Abstract
This paper addresses the dynamics of four European soccer teams over the season 2018–2019. The modeling perspective adopts the concepts of fractional calculus and power law. The proposed model embeds implicitly details such as the behavior of players and coaches, strategical and tactical [...] Read more.
This paper addresses the dynamics of four European soccer teams over the season 2018–2019. The modeling perspective adopts the concepts of fractional calculus and power law. The proposed model embeds implicitly details such as the behavior of players and coaches, strategical and tactical maneuvers during the matches, errors of referees and a multitude of other effects. The scale of observation focuses the teams’ behavior at each round. Two approaches are considered, namely the evaluation of the team progress along the league by a variety of heuristic models fitting real-world data, and the analysis of statistical information by means of entropy. The best models are also adopted for predicting the future results and their performance compared with the real outcome. The computational and mathematical modeling lead to results that are analyzed and interpreted in the light of fractional dynamics. The emergence of patterns both with the heuristic modeling and the entropy analysis highlight similarities in different national leagues and point towards some underlying complex dynamics. Full article
(This article belongs to the Special Issue Bifurcation and Chaos in Fractional-Order Systems)
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12 pages, 22008 KiB  
Article
Puu System of Fractional Order and Its Chaos Suppression
by Marius-F. Danca
Symmetry 2020, 12(3), 340; https://doi.org/10.3390/sym12030340 - 27 Feb 2020
Cited by 15 | Viewed by 3548
Abstract
In this paper, the fractional-order variant of Puu’s system is introduced, and, comparatively with its integer-order counterpart, some of its characteristics are presented. Next, an impulsive chaos control algorithm is applied to suppress the chaos. Because fractional-order continuous-time or discrete-time systems have not [...] Read more.
In this paper, the fractional-order variant of Puu’s system is introduced, and, comparatively with its integer-order counterpart, some of its characteristics are presented. Next, an impulsive chaos control algorithm is applied to suppress the chaos. Because fractional-order continuous-time or discrete-time systems have not had non-constant periodic solutions, chaos suppression is considered under some numerical assumptions. Full article
(This article belongs to the Special Issue Bifurcation and Chaos in Fractional-Order Systems)
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