D3 Dihedral Logistic Map of Fractional Order

In this paper, the D3 dihedral logistic map of fractional order is introduced. The map presents a dihedral symmetry D3. It is numerically shown that the construction and interpretation of the bifurcation diagram versus the fractional order requires special attention. The system stability is determined and the problem of hidden attractors is analyzed. Furthermore, analytical and numerical results show that the chaotic attractor of integer order, with D3 symmetries, looses its symmetry in the fractional-order variant.

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D3 Dihedral Logistic Map of Fractional Order

10.20944/preprints202112.0303.v1 ◽

2021 ◽

Author(s):

Marius-F. Danca ◽

Nikolay Kuznetsov

Keyword(s):

Bifurcation Diagram ◽

Fractional Order ◽

Chaotic Attractor ◽

Logistic Map ◽

System Stability ◽

Hidden Attractors ◽

Integer Order

In this paper the D 3 dihedral logistic map of fractional order is introduced. The map 1 presents a dihedral symmetry D 3 . It is numerically shown that the construction and interpretation 2 of the bifurcation diagram versus the fractional order require special attention. The system stability 3 is determined and the problem of hidden attractors is analyzed. Also, analytical and numerical 4 results show that the chaotic attractor of integer order, with D 3 symmetries, looses its symmetry 5 in the fractional-order variant.

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A New Fractional-Order Chaotic System with Different Families of Hidden and Self-Excited Attractors

Entropy ◽

10.3390/e20080564 ◽

2018 ◽

Vol 20 (8) ◽

pp. 564 ◽

Cited By ~ 36

Author(s):

Jesus Munoz-Pacheco ◽

Ernesto Zambrano-Serrano ◽

Christos Volos ◽

Sajad Jafari ◽

Jacques Kengne ◽

...

Keyword(s):

Fractional Order ◽

Chaotic System ◽

Chaotic Attractor ◽

Equilibrium Points ◽

Fractional Order System ◽

Order System ◽

Chaotic Attractors ◽

Equilibrium System ◽

Hidden Attractors ◽

Hidden Chaotic Attractor

In this work, a new fractional-order chaotic system with a single parameter and four nonlinearities is introduced. One striking feature is that by varying the system parameter, the fractional-order system generates several complex dynamics: self-excited attractors, hidden attractors, and the coexistence of hidden attractors. In the family of self-excited chaotic attractors, the system has four spiral-saddle-type equilibrium points, or two nonhyperbolic equilibria. Besides, for a certain value of the parameter, a fractional-order no-equilibrium system is obtained. This no-equilibrium system presents a hidden chaotic attractor with a `hurricane’-like shape in the phase space. Multistability is also observed, since a hidden chaotic attractor coexists with a periodic one. The chaos generation in the new fractional-order system is demonstrated by the Lyapunov exponents method and equilibrium stability. Moreover, the complexity of the self-excited and hidden chaotic attractors is analyzed by computing their spectral entropy and Brownian-like motions. Finally, a pseudo-random number generator is designed using the hidden dynamics.

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On Chaos in the Fractional-Order Discrete-Time Unified System and Its Control Synchronization

Entropy ◽

10.3390/e20070530 ◽

2018 ◽

Vol 20 (7) ◽

pp. 530 ◽

Cited By ~ 20

Author(s):

Amina-Aicha Khennaoui ◽

Adel Ouannas ◽

Samir Bendoukha ◽

Xiong Wang ◽

Viet-Thanh Pham

Keyword(s):

Fractional Order ◽

Discrete Time ◽

Chaotic Behavior ◽

Numerical Results ◽

Complete Synchronization ◽

Control Laws ◽

Integer Order

In this paper, we propose a fractional map based on the integer-order unified map. The chaotic behavior of the proposed map is analyzed by means of bifurcations plots, and experimental bounds are placed on the parameters and fractional order. Different control laws are proposed to force the states to zero asymptotically and to achieve the complete synchronization of a pair of fractional unified maps with identical or nonidentical parameters. Numerical results are used throughout the paper to illustrate the findings.

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A Class of Integer Order and Fractional Order Hyperchaotic Systems via the Chen System

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127416501091 ◽

2016 ◽

Vol 26 (06) ◽

pp. 1650109 ◽

Cited By ~ 3

Author(s):

Fei Xu

Keyword(s):

Differential Equation ◽

Fractional Order ◽

Chaotic System ◽

Chaotic Attractor ◽

Equation System ◽

Hyperchaotic System ◽

Differential Equation System ◽

Chen System ◽

Integer Order ◽

Hyperchaotic Systems

In this article, we investigate the generation of a class of hyperchaotic systems via the Chen chaotic system using both integer order and fractional order differential equation systems. Based on the Chen chaotic system, we designed a system with four nonlinear ordinary differential equations. For different parameter sets, the trajectory of the system may diverge or display a hyperchaotic attractor with double wings. By linearizing the ordinary differential equation system with divergent trajectory and designing proper switching controls, we obtain a chaotic attractor. Similar phenomenon has also been observed in linearizing the hyperchaotic system. The corresponding fractional order systems are also considered. Our investigation indicates that, switching control can be applied to either linearized chaotic or nonchaotic differential equation systems to create chaotic attractor.

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Coexisting Hidden and Self-Excited Attractors in an Economic Model of Integer or Fractional Order

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127421500620 ◽

2021 ◽

Vol 31 (04) ◽

pp. 2150062

Author(s):

Marius-F. Danca

Keyword(s):

Economic System ◽

Fractional Order ◽

Economic Model ◽

Foreign Capital ◽

Capital Inflow ◽

Hidden Attractors ◽

Coexisting Attractors ◽

Integer Order ◽

Coexistence Of Attractors ◽

System Variables

In this paper, the dynamics of an economic system with foreign financing, of integer or fractional order, are analyzed. The symmetry of the system determines the existence of two pairs of coexisting attractors. The integer-order version of the system proves to have several combinations of coexisting hidden attractors with self-excited attractors. Because one of the system variables represents the foreign capital inflow, the presence of hidden attractors could be of real interest in economic models. The fractional-order variant presents another interesting coexistence of attractors in the fractional-order space.

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On fractional Liénard--type systems

Revista Mexicana de Física ◽

10.31349/revmexfis.65.618 ◽

2019 ◽

Vol 65 (6 Nov-Dec) ◽

pp. 618 ◽

Cited By ~ 1

Author(s):

A. Fleitas ◽

J. A. Mendez-Bermudez ◽

J. E. Napoles Valdes ◽

J. M. Sigarreta Almira

Keyword(s):

Fractional Order ◽

Type Systems ◽

Numerical Results ◽

General Context ◽

Integer Order ◽

Global And Local

In this work we present numerical results of classical Li\'{e}nard--type systems in a very general context, since we consider several types of derivatives (integer order and fractional order, global and local). Additionally we made theoretical-methodological observations. En este trabajo presentamos resultados num´ericos de sistemas tipo Li´enard en un contexto muy general ya que consideramos varios tipos dederivadas (de orden entero y fraccionario, globales y locales). Adicionalmente hacemos observaciones te ´oricas y metodol´ogicas.

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Complexity and Chimera States in a Network of Fractional-Order Laser Systems

Symmetry ◽

10.3390/sym13020341 ◽

2021 ◽

Vol 13 (2) ◽

pp. 341

Author(s):

Shaobo He ◽

Hayder Natiq ◽

Santo Banerjee ◽

Kehui Sun

Keyword(s):

Numerical Simulation ◽

Numerical Solution ◽

Bifurcation Diagram ◽

Fractional Order ◽

Dynamical Model ◽

Chimera States ◽

Laser Systems ◽

Complexity Algorithm ◽

Derivative Order

By applying the Adams-Bashforth-Moulton method (ABM), this paper explores the complexity and synchronization of a fractional-order laser dynamical model. The dynamics under the variance of derivative order q and parameters of the system have examined using the multiscale complexity algorithm and the bifurcation diagram. Numerical simulation outcomes demonstrate that the system generates chaos with the decreasing of q. Moreover, this paper designs the coupled fractional-order network of laser systems and subsequently obtains its numerical solution using ABM. These solutions have demonstrated chimera states of the proposed fractional-order laser network.

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Invariant Image Representation Using Novel Fractional-Order Polar Harmonic Fourier Moments

Sensors ◽

10.3390/s21041544 ◽

2021 ◽

Vol 21 (4) ◽

pp. 1544

Author(s):

Chunpeng Wang ◽

Hongling Gao ◽

Meihong Yang ◽

Jian Li ◽

Bin Ma ◽

...

Keyword(s):

Image Reconstruction ◽

Fractional Order ◽

Continuous Functions ◽

Kernel Functions ◽

Superior Performance ◽

Image Description ◽

Orthogonal Moments ◽

Integer Order ◽

Geometric Invariance ◽

Order Continuous

Continuous orthogonal moments, for which continuous functions are used as kernel functions, are invariant to rotation and scaling, and they have been greatly developed over the recent years. Among continuous orthogonal moments, polar harmonic Fourier moments (PHFMs) have superior performance and strong image description ability. In order to improve the performance of PHFMs in noise resistance and image reconstruction, PHFMs, which can only take integer numbers, are extended to fractional-order polar harmonic Fourier moments (FrPHFMs) in this paper. Firstly, the radial polynomials of integer-order PHFMs are modified to obtain fractional-order radial polynomials, and FrPHFMs are constructed based on the fractional-order radial polynomials; subsequently, the strong reconstruction ability, orthogonality, and geometric invariance of the proposed FrPHFMs are proven; and, finally, the performance of the proposed FrPHFMs is compared with that of integer-order PHFMs, fractional-order radial harmonic Fourier moments (FrRHFMs), fractional-order polar harmonic transforms (FrPHTs), and fractional-order Zernike moments (FrZMs). The experimental results show that the FrPHFMs constructed in this paper are superior to integer-order PHFMs and other fractional-order continuous orthogonal moments in terms of performance in image reconstruction and object recognition, as well as that the proposed FrPHFMs have strong image description ability and good stability.

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Analysis of a New Class of Impulsive Implicit Sequential Fractional Differential Equations

International Journal of Nonlinear Sciences and Numerical Simulation ◽

10.1515/ijnsns-2019-0030 ◽

2020 ◽

Vol 21 (6) ◽

pp. 571-587 ◽

Cited By ~ 2

Author(s):

Akbar Zada ◽

Sartaj Ali ◽

Tongxing Li

Keyword(s):

Differential Equations ◽

Fractional Order ◽

Fractional Differential Equations ◽

Sufficient Conditions ◽

Uniqueness Of Solutions ◽

Integer Order ◽

New Class ◽

Proposed Model ◽

Fractional Differential ◽

Sequential Fractional Differential Equations

AbstractIn this paper, we study an implicit sequential fractional order differential equation with non-instantaneous impulses and multi-point boundary conditions. The article comprehensively elaborate four different types of Ulam’s stability in the lights of generalized Diaz Margolis’s fixed point theorem. Moreover, some sufficient conditions are constructed to observe the existence and uniqueness of solutions for the proposed model. The proposed model contains both the integer order and fractional order derivatives. Thus, the exponential function appearers in the solution of the proposed model which will lead researchers to study fractional differential equations with well known methods of integer order differential equations. In the last, few examples are provided to show the applicability of our main results.

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