scholarly journals A new type of adiabatic invariant for fractional order non-conservative Lagrangian systems

2020 ◽  
Vol 69 (22) ◽  
pp. 220401
Author(s):  
Xin-Xin Xu ◽  
Yi Zhang
Keyword(s):  
Fractional Order ◽  
Adiabatic Invariant ◽  
Lagrangian Systems ◽  
New Type

scholarly journals Synchronization of Fractional-Order Complex Chaotic Systems Based on Observers

Entropy ◽  
2019 ◽  
Vol 21 (5) ◽  
pp. 481 ◽  
Author(s):  
Zhonghui Li ◽  
Tongshui Xia ◽  
Cuimei Jiang
Keyword(s):  
Fractional Order ◽  
Chaotic Systems ◽  
Projective Synchronization ◽  
Pole Placement ◽  
Order Complex ◽  
Order Error ◽  
New Type ◽  
Modified Projective Synchronization ◽  
The Stability ◽  
Complex Chaotic Systems

By designing a state observer, a new type of synchronization named complex modified projective synchronization is investigated in a class of nonlinear fractional-order complex chaotic systems. Combining stability results of the fractional-order systems and the pole placement method, this paper proves the stability of fractional-order error systems and realizes complex modified projective synchronization. This method is so effective that it can be applied in engineering. Additionally, the proposed synchronization strategy is suitable for all fractional-order chaotic systems, including fractional-order hyper-chaotic systems. Finally, two numerical examples are studied to show the correctness of this new synchronization strategy.

Sixth-Kind Chebyshev Spectral Approach for Solving Fractional Differential Equations

2019 ◽  
Vol 20 (2) ◽  
pp. 191-203 ◽  
Author(s):  
W. M. Abd-Elhameed ◽  
Y. H. Youssri
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Chebyshev Polynomials ◽  
Symmetric Functions ◽  
Numerical Algorithms ◽  
Initial Boundary ◽  
Algebraic Equations ◽  
Fractional Order Differential Equations ◽  
New Type ◽  
Connection Formulae

AbstractThe basic aim of this paper is to develop new numerical algorithms for solving some linear and nonlinear fractional-order differential equations. We have developed a new type of Chebyshev polynomials, namely, Chebyshev polynomials of sixth kind. This type of polynomials is a special class of symmetric orthogonal polynomials, involving four parameters that were constructed with the aid of the extended Sturm–Liouville theorem for symmetric functions. The proposed algorithms are basically built on reducing the fractional-order differential equations with their initial/boundary conditions to systems of algebraic equations which can be efficiently solved. The new proposed algorithms are supported by a detailed study of the convergence and error analysis of the sixth-kind Chebyshev expansion. New connection formulae between Chebyshev polynomials of the second and sixth kinds were established for this study. Some examples were displayed to illustrate the efficiency of the proposed algorithms compared to other methods in literature. The proposed algorithms have provided accurate results, even using few terms of the proposed expansion.

A General Method to Study the Co-Existence of Different Hybrid Synchronizations in Fractional-Order Chaotic Systems

2019 ◽  
Vol 20 (3-4) ◽  
pp. 351-359
Author(s):  
Adel Ouannas ◽  
Samir Bendoukha ◽  
Abdulrahman Karouma ◽  
Salem Abdelmalek
Keyword(s):  
Fractional Order ◽  
Stability Theory ◽  
Projective Synchronization ◽  
The Novel ◽  
Fractional Order Systems ◽  
Function Projective Synchronization ◽  
Novel Approach ◽  
Full State ◽  
Hybrid Function ◽  
New Type

AbstractReferring to incommensurate fractional-order systems, this paper proposes a new type of chaos synchronization by combining full state hybrid function projective synchronization (FSHFPS) and inverse full state hybrid function projective synchronization (IFSHFPS). In particular, based on stability theory of linear integer-order systems and stability theory of linear fractional-order systems, the co-existence of FSHFPS and IFSHFPS between incommensurate fractional chaotic (hyperchaotic) systems is proved. To illustrate the capabilities of the novel approach proposed herein, numerical and simulation results are given.

scholarly journals The impact of memory effect on space fractional strong quantum couplers with tunable decay behavior and its numerical simulation

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Ahmed S. Hendy ◽  
Mahmoud A. Zaky ◽  
Ramy M. Hafez ◽  
Rob H. De Staelen
Keyword(s):  
Fractional Derivative ◽  
Numerical Simulations ◽  
Fractional Order ◽  
Numerical Algorithm ◽  
Time And Space ◽  
Fractional Quantum ◽  
Nonlinear Schrödinger ◽  
Time Space ◽  
New Type ◽  
Decay Behavior

AbstractThe nontrivial behavior of wave packets in the space fractional coupled nonlinear Schrödinger equation has received considerable theoretical attention. The difficulty comes from the fact that the Riesz fractional derivative is inherently a prehistorical operator. In contrast, nonlinear Schrödinger equation with both time and space nonlocal operators, which is the cornerstone in the modeling of a new type of fractional quantum couplers, is still in high demand of attention. This paper is devoted to numerically study the propagation of solitons through a new type of quantum couplers which can be called time-space fractional quantum couplers. The numerical methodology is based on the finite-difference/Galerkin Legendre spectral method with an easy to implement numerical algorithm. The time-fractional derivative is considered to describe the decay behavior and the nonlocal memory of the model. We conduct numerical simulations to observe the performance of the tunable decay and the sharpness behavior of the time-space fractional strongly coupled nonlinear Schrödinger model as well as the performance of the numerical algorithm. Numerical simulations show that the time and space fractional-order operators control the decay behavior or the memory and the sharpness of the interface and undergo a seamless transition of the fractional-order parameters.

scholarly journals Dynamic Behavior Analysis and Robust Synchronization of a Novel Fractional-Order Chaotic System with Multiwing Attractors

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Chenhui Wang
Keyword(s):  
Fractional Order ◽  
Chaotic Systems ◽  
Chaotic Behavior ◽  
Chaotic Attractors ◽  
Stability Criteria ◽  
Minimum Order ◽  
3 Dimensional ◽  
Robust Synchronization ◽  
New Type ◽  
The Stability

To enrich the types of multiwing chaotic attractors in fractional-order chaotic systems (FOCSs), a new type of 3-dimensional FOCSs is designed in this study. The most important contribution of this FOCS consists in the coexistence of multiple multiwing chaotic attractors, including 2-wing, 3-wing, and 4-wing attractors. It is also indicated that the minimum order that the system can exhibit chaotic behavior is 0.84. Then, based on certain fractional stability criteria, a robust synchronization controller is derived for this kind of FOCSs with multiwing chaotic attractors and parametric uncertainties, and the stability of the synchronization error is proven strictly. Meanwhile, the theoretical analysis is tested by simulation results.

Numerical Simulation of Anomalous Vortex Beams on Different Fractional Fourier Transform Planes

2014 ◽  
Vol 556-562 ◽  
pp. 3745-3748 ◽  
Author(s):  
Zhi Ping Dai ◽  
Zhen Feng Yang ◽  
Zhen Jun Yang ◽  
Zhao Guang Pang ◽  
Shu Min Zhang
Keyword(s):  
Numerical Simulation ◽  
Fourier Transform ◽  
Fractional Order ◽  
Topological Charge ◽  
Fractional Fourier Transform ◽  
Analytical Expression ◽  
New Type ◽  
Vortex Beams ◽  
Beam Parameters

The properties of fractional Fourier transform of anomalous vortex beams are studied. A new type of analytical expression of fractional Fourier transform for anomolous vortex beams is obtained. The properties of anomolous vortex beams on different fractional Fourier transform planes with different parameters are illustrated. The results show that the anomolous vortex beams always has a doughnut profile, the distribution of intensity on different fractional Fourier transform planes highly depends on the fractional order and the beam parameters, such as the beam order and the topological charge.

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