scholarly journals Ghost Attractor in Fractional Order Blinking System and its Application

2021 ◽  
Author(s):  
Fatma Mohamed Kamal ◽  
Ahmed Elsaid ◽  
Amr Refaat Elsonbaty
Keyword(s):  
Fractional Order ◽  
Initial Conditions ◽  
Security Analysis ◽  
Period Doubling ◽  
Fast Switching ◽  
Dynamical Behaviors ◽  
Short Memory ◽  
Image Cryptosystem ◽  
Differential Attacks ◽  
First Time

Abstract In this paper, the occurrence of ghost attractor is verified in three cases of a proposed fractional order Rössler blinking system. Firstly, the dynamical behaviors of the short memory fractional order prototype-4 Rössler system with Chua’s diode are explored via bifurcation diagrams and Lyapunov exponents. It is depicted that this system exhibits a variety of dynamics including limit cycles, period doubling and chaos. Then, a proposed non-autonomous fractional order Rössler blinking system is introduced. Numerical simulations are employed to confirm the existence of ghost attractors at specific cases which involve very fast switching time between two composing autonomous fractional subsystems. It is found that the presented fractional order blinking system is very sensitive to system parameters, initial conditions and stochastic process parameters. Thus, the induced chaotic ghost attractor is utilized in a suggested ghost attractor-based chaotic image encryption scheme for first time. Finally, a detailed security analysis is carried out and reveals that the proposed image cryptosystem is immune against different types of attacks such as differential attacks, brute force attacks, cropping and statistical attacks.

Periodic-Doubling Bifurcation of a Circuit With a Fractional-Order Memristor

2019 ◽  
Author(s):  
Yajuan Yu ◽  
Yangquan Chen
Keyword(s):  
Fractional Order ◽  
Memory Loss ◽  
Period Doubling ◽  
Hysteresis Loops ◽  
State Variables ◽  
Discontinuous Change ◽  
Dynamical Behaviors ◽  
Sinusoidal Current ◽  
Memristive Circuit ◽  
Theoretical Analyses

Abstract A new fractional-order current-controlled memristor is proposed by the fact of the memory loss. Excited by sinusoidal current, the generalized hysteresis loops of the new fractional-order memristor are no longer symmetrical to the origin and the time to reach the steady state is longer than the integer-order memristor’s. The dynamical behaviors of a new fractional-order memristive circuit system whose state variables have different derivation orders are investigated by theoretical analyses and simulated numerically. It is shown that the new fractional-order memristive circuit system goes into chaos by period-doubling bifurcation; the periodic windows are induced by the discontinuous change of derivative order between variables.

A fractional-order ship power system: chaos and its dynamical properties

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Karthikeyan Rajagopal ◽  
Prakash Duraisamy ◽  
Goitom Tadesse ◽  
Christos Volos ◽  
Fahimeh Nazarimehr ◽  
...  
Keyword(s):  
Power System ◽  
Fractional Order ◽  
Chaotic Dynamics ◽  
Period Doubling ◽  
Order Form ◽  
Dynamical Behaviors ◽  
Fractional Order Controller ◽  
Parameter Values ◽  
Ship Power System ◽  
D Model

Abstract In this research, the ship power system is studied with a fractional-order approach. A 2-D model of a two-generator parallel-connected is considered. A chaotic attractor is observed for particular parameter values. The fractional-order form is calculated with the Adam–Bashforth–Moulton method. The chaotic response is identified even for the order 0.99. Phase portrait is generated using the Caputo derivative approach. Wolf’s algorithm is used to calculate Lyapunov exponents. For the considered values of parameters, one positive Lyapunov exponent confirms the existence of chaos. Bifurcation diagrams are presented to analyze the various dynamical behaviors and bifurcation points. Interestingly, the considered system is multistable. Also, antimonotonicity, period-doubling, and period halving are observed in the bifurcation diagram. As the last step, a fractional-order controller is designed to remove chaotic dynamics. Time plots are simulated to show the effectiveness of the controller.

On the Dynamics and Control of Fractional Chaotic Maps with Sine Terms

2020 ◽  
Vol 21 (6) ◽  
pp. 589-601 ◽  
Author(s):  
Ahlem Gasri ◽  
Adel Ouannas ◽  
Amina-Aicha Khennaoui ◽  
Samir Bendoukha ◽  
Viet-Thanh Pham
Keyword(s):  
Fractional Order ◽  
Chaotic Maps ◽  
Initial Conditions ◽  
Bifurcation Diagrams ◽  
Coexisting Attractors ◽  
Dynamical Behaviors ◽  
Control Schemes ◽  
Dynamics And Control ◽  
Numerical Tools ◽  
And Control

AbstractThis paper studies the dynamics of two fractional-order chaotic maps based on two standard chaotic maps with sine terms. The dynamic behavior of this map is analyzed using numerical tools such as phase plots, bifurcation diagrams, Lyapunov exponents and 0–1 test. With the change of fractional-order, it is shown that the proposed fractional maps exhibit a range of different dynamical behaviors including coexisting attractors. The existence of coexistence attractors is depicted by plotting bifurcation diagram for two symmetrical initial conditions. In addition, three control schemes are introduced. The first two controllers stabilize the states of the proposed maps and ensure their convergence to zero asymptotically whereas the last synchronizes a pair of non-identical fractional maps. Numerical results are used to verify the findings.

scholarly journals Complex Dynamical Behaviors of a Fractional-Order System Based on a Locally Active Memristor

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-13
Author(s):  
Yajuan Yu ◽  
Han Bao ◽  
Min Shi ◽  
Bocheng Bao ◽  
Yangquan Chen ◽  
...  
Keyword(s):  
Hopf Bifurcation ◽  
Hysteresis Loop ◽  
Fractional Order ◽  
Period Doubling ◽  
Order System ◽  
Periodic Signal ◽  
Dynamical Behaviors ◽  
Memristive System ◽  
Attraction Basins ◽  
The Stability

A fractional-order locally active memristor is proposed in this paper. When driven by a bipolar periodic signal, the generated hysteresis loop with two intersections is pinched at the origin. The area of the hysteresis loop changes with the fractional order. Based on the fractional-order locally active memristor, a fractional-order memristive system is constructed. The stability analysis is carried out and the stability conditions for three equilibria are listed. The expression of the fractional order related to Hopf bifurcation is given. The complex dynamical behaviors of Hopf bifurcation, period-doubling bifurcation, bistability and chaos are shown numerically. Furthermore, the bistability behaviors of the different fractional order are validated by the attraction basins in the initial value plane. As an alternative to validating our results, the fractional-order memristive system is implemented by utilizing Simulink of MATLAB. The research results clarify that the complex dynamical behaviors are attributed to two facts: one is the fractional order that affects the stability of the equilibria, and the other is the local activeness of the fractional-order memristor.

scholarly journals Chaos in a Financial System with Fractional Order and Its Control via Sliding Mode

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Paul Yaovi Dousseh ◽  
Cyrille Ainamon ◽  
Clément Hodévèwan Miwadinou ◽  
Adjimon Vincent Monwanou ◽  
Jean Bio Chabi Orou
Keyword(s):  
Fractional Order ◽  
Chaos Control ◽  
Sliding Mode ◽  
Financial System ◽  
Period Doubling ◽  
Fractional Order Systems ◽  
Uncertain Dynamics ◽  
Dynamical Behaviors ◽  
Order Case ◽  
Route To Chaos

In this paper, the dynamical behaviors and chaos control of a fractional-order financial system are discussed. The lowest fractional order found from which the system generates chaos is 2.49 for the commensurate order case and 2.13 for the incommensurate order case. Also, period-doubling route to chaos was found in this system. The results of this study were validated by the existence of a positive Lyapunov exponent. Besides, in order to control chaos in this fractional-order financial system with uncertain dynamics, a sliding mode controller is derived. The proposed controller stabilizes the commensurate and incommensurate fractional-order systems. Numerical simulations are carried out to verify the analytical results.

scholarly journals Corrigendum to “Chaos in a Financial System with Fractional Order and Its Control via Sliding Mode”

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-16
Author(s):  
Paul Yaovi Dousseh ◽  
Cyrille Ainamon ◽  
Clément Hodévèwan Miwadinou ◽  
Adjimon Vincent Monwanou ◽  
Jean Bio Chabi Orou
Keyword(s):  
Fractional Order ◽  
Chaos Control ◽  
Sliding Mode ◽  
Financial System ◽  
Period Doubling ◽  
Fractional Order Systems ◽  
Uncertain Dynamics ◽  
Dynamical Behaviors ◽  
Order Case ◽  
Route To Chaos

In this paper, the dynamical behaviors and chaos control of a fractional-order financial system are discussed. The lowest fractional order found from which the system generates chaos is 2.49 for the commensurate order case and 2.57 for the incommensurate order case. Also, the period-doubling route to chaos was found in this system. The results of this study were validated by the existence of a positive Lyapunov exponent. Besides, in order to control chaos in this fractional-order financial system with uncertain dynamics, a sliding mode controller is derived. The proposed controller stabilizes the commensurate and incommensurate fractional-order systems. Numerical simulations are carried out to verify the analytical results.

scholarly journals Quantum Security Analysis of AES

2019 ◽  
pp. 55-93 ◽  
Author(s):  
Xavier Bonnetain ◽  
María Naya-Plasencia ◽  
André Schrottenloher
Keyword(s):  
Security Analysis ◽  
Building Blocks ◽  
Quantum Cost ◽  
Secret Key ◽  
Trade Offs ◽  
Speed Up ◽  
Generic Attacks ◽  
Differential Attacks ◽  
First Time ◽  
New Framework

In this paper we analyze for the first time the post-quantum security of AES. AES is the most popular and widely used block cipher, established as the encryption standard by the NIST in 2001. We consider the secret key setting and, in particular, AES-256, the recommended primitive and one of the few existing ones that aims at providing a post-quantum security of 128 bits. In order to determine the new security margin, i.e., the lowest number of non-attacked rounds in time less than 2128 encryptions, we first provide generalized and quantized versions of the best known cryptanalysis on reduced-round AES, as well as a discussion on attacks that don’t seem to benefit from a significant quantum speed-up. We propose a new framework for structured search that encompasses both the classical and quantum attacks we present, and allows to efficiently compute their complexity. We believe this framework will be useful for future analysis.Our best attack is a quantum Demirci-Selçuk meet-in-the-middle attack. Unexpectedly, using the ideas underlying its design principle also enables us to obtain new, counter-intuitive classical TMD trade-offs. In particular, we can reduce the memory in some attacks against AES-256 and AES-128.One of the building blocks of our attacks is solving efficiently the AES S-Box differential equation, with respect to the quantum cost of a reversible S-Box. We believe that this generic quantum tool will be useful for future quantum differential attacks. Judging by the results obtained so far, AES seems a resistant primitive in the post-quantum world as well as in the classical one, with a bigger security margin with respect to quantum generic attacks.

scholarly journals Fractional isospectral and non-isospectral AKNS hierarchies and their analytic methods for N-fractal solutions with Mittag-Leffler functions

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Bo Xu ◽  
Yufeng Zhang ◽  
Sheng Zhang
Keyword(s):  
Spectral Problem ◽  
Fractional Order ◽  
Point Of View ◽  
Adjoint Equations ◽  
Scattering Data ◽  
Bilinear Method ◽  
Soliton Equation ◽  
Scattering Transform ◽  
First Time ◽  
Hirota Bilinear

AbstractAblowitz–Kaup–Newell–Segur (AKNS) linear spectral problem gives birth to many important nonlinear mathematical physics equations including nonlocal ones. This paper derives two fractional order AKNS hierarchies which have not been reported in the literature by equipping the AKNS spectral problem and its adjoint equations with local fractional order partial derivative for the first time. One is the space-time fractional order isospectral AKNS (stfisAKNS) hierarchy, three reductions of which generate the fractional order local and nonlocal nonlinear Schrödinger (flnNLS) and modified Kortweg–de Vries (fmKdV) hierarchies as well as reverse-t NLS (frtNLS) hierarchy, and the other is the time-fractional order non-isospectral AKNS (tfnisAKNS) hierarchy. By transforming the stfisAKNS hierarchy into two fractional bilinear forms and reconstructing the potentials from fractional scattering data corresponding to the tfnisAKNS hierarchy, three pairs of uniform formulas of novel N-fractal solutions with Mittag-Leffler functions are obtained through the Hirota bilinear method (HBM) and the inverse scattering transform (IST). Restricted to the Cantor set, some obtained continuous everywhere but nondifferentiable one- and two-fractal solutions are shown by figures directly. More meaningfully, the problems worth exploring of constructing N-fractal solutions of soliton equation hierarchies by HBM and IST are solved, taking stfisAKNS and tfnisAKNS hierarchies as examples, from the point of view of local fractional order derivatives. Furthermore, this paper shows that HBM and IST can be used to construct some N-fractal solutions of other soliton equation hierarchies.

scholarly journals A New No Equilibrium Fractional Order Chaotic System, Dynamical Investigation, Synchronization, and Its Digital Implementation

Inventions ◽  
2021 ◽  
Vol 6 (3) ◽  
pp. 49
Author(s):  
Zain-Aldeen S. A. Rahman ◽  
Basil H. Jasim ◽  
Yasir I. A. Al-Yasir ◽  
Raed A. Abd-Alhameed ◽  
Bilal Naji Alhasnawi
Keyword(s):  
Adaptive Control ◽  
Fractional Order ◽  
Chaotic System ◽  
Real World ◽  
Experimental Results ◽  
Chaotic Attractors ◽  
State Variables ◽  
Digital Implementation ◽  
Dynamical Behaviors ◽  
Real World Applications

In this paper, a new fractional order chaotic system without equilibrium is proposed, analytically and numerically investigated, and numerically and experimentally tested. The analytical and numerical investigations were used to describe the system’s dynamical behaviors including the system equilibria, the chaotic attractors, the bifurcation diagrams, and the Lyapunov exponents. Based on the obtained dynamical behaviors, the system can excite hidden chaotic attractors since it has no equilibrium. Then, a synchronization mechanism based on the adaptive control theory was developed between two identical new systems (master and slave). The adaptive control laws are derived based on synchronization error dynamics of the state variables for the master and slave. Consequently, the update laws of the slave parameters are obtained, where the slave parameters are assumed to be uncertain and are estimated corresponding to the master parameters by the synchronization process. Furthermore, Arduino Due boards were used to implement the proposed system in order to demonstrate its practicality in real-world applications. The simulation experimental results were obtained by MATLAB and the Arduino Due boards, respectively, with a good consistency between the simulation results and the experimental results, indicating that the new fractional order chaotic system is capable of being employed in real-world applications.

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