Solutions and memory effect of fractional-order chaotic system: A review (Interdisciplinary Physics)

2021 ◽  
Author(s):  
Shaobo He ◽  
Huihai Wang ◽  
Kehui Sun
Keyword(s):  
Memory Effect ◽  
Fractional Order ◽  
Chaotic Systems ◽  
Reference Value ◽  
Order Derivative ◽  
Systems Modeling ◽  
Fractional Order Calculus ◽  
Solution Algorithms ◽  
System A ◽  
Short Memory

Abstract Fractional calculus is a 300 years topic, which has been introduced to real physics systems modeling and engineering applications. In the last few decades, fractional-order nonlinear chaotic systems have been widely investigated. Firstly, the most used methods to solve fractional-order chaotic systems are reviewed. Characteristics and memory effect in those method are summarized. Then we discussed the memory effect in the fractional-order chaotic systems through the fractional-order calculus and numerical solution algorithms. It shows that the integer-order derivative has full memory effect, while the fractional-order derivative has nonideal memory effect due to the kernel function. Memory lose and short memory are discussed. Finally, applications of the fractional-order chaotic systems regarding the memory effects are investigated. The work summarized in this manuscript provides reference value for the applied scientists and engineering community of fractional-order nonlinear chaotic systems.

scholarly journals Active Sliding Mode for Synchronization of a Wide Class of Four-Dimensional Fractional-Order Chaotic Systems

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Bin Wang ◽  
Yuangui Zhou ◽  
Jianyi Xue ◽  
Delan Zhu
Keyword(s):  
Sliding Mode Control ◽  
Fractional Order ◽  
Wide Class ◽  
Stability Theory ◽  
Chaotic Systems ◽  
Sliding Mode ◽  
Order System ◽  
Fractional Order Calculus ◽  
Simulation Results ◽  
The Stability

We focus on the synchronization of a wide class of four-dimensional (4-D) chaotic systems. Firstly, based on the stability theory in fractional-order calculus and sliding mode control, a new method is derived to make the synchronization of a wide class of fractional-order chaotic systems. Furthermore, the method guarantees the synchronization between an integer-order system and a fraction-order system and the synchronization between two fractional-order chaotic systems with different orders. Finally, three examples are presented to illustrate the effectiveness of the proposed scheme and simulation results are given to demonstrate the effectiveness of the proposed method.

Novel Cryptography Technique via Chaos Synchronization of Fractional-Order Derivative Systems

2018 ◽  
pp. 404-437
Author(s):  
Alain Giresse Tene ◽  
Timoleon Crépin Kofane
Keyword(s):  
Fractional Order ◽  
Chaos Synchronization ◽  
Chaotic Systems ◽  
Order Derivative ◽  
Random Numbers ◽  
Generation Process ◽  
Initial Condition ◽  
Fractional Order Derivative ◽  
Q Matrix ◽  
Fractional System

Synchronization of fractional-order-derivative systems for cryptography purpose is still exploratory and despite an increase in cryptography research, several challenges remain in designing a powerful cryptosystem. This chapter addresses the problem of synchronization of fractional-order-derivative chaotic systems using random numbers generator for a novel technique to key distribution in cryptography. However, there is evidence that researchers have approached the problem using integer order derivative chaotic systems. Consequently, the aim of the chapter lies in coding and decoding a text via chaos synchronization of fractional-order derivative, the performance analysis and the key establishment scheme following an application on a text encryption using the chaotic Mathieu-Van Der Pol fractional system. In order to improve the level of the key security, the Fibonacci Q-matrix is used in the key generation process and the initial condition; the order of the derivative of the responder system secretly shared between the responder and the receiver are also involved. It followed from this study that compared to the existing cryptography techniques, this proposed method is found to be very efficient due to the fact that, it improves the key security.

scholarly journals Anti-Synchronization of a Class of Chaotic Systems with Application to Lorenz System: A Unified Analysis of the Integer Order and Fractional Order

Mathematics ◽  
2019 ◽  
Vol 7 (6) ◽  
pp. 559 ◽  
Author(s):  
Liang Chen ◽  
Chengdai Huang ◽  
Haidong Liu ◽  
Yonghui Xia
Keyword(s):  
Fractional Order ◽  
Chaotic System ◽  
Finite Time ◽  
Chaotic Systems ◽  
Lorenz System ◽  
Unknown Parameters ◽  
Integer Order ◽  
System A ◽  
Control Scheme ◽  
Lorenz Chaotic System

The paper proves a unified analysis for finite-time anti-synchronization of a class of integer-order and fractional-order chaotic systems. We establish an effective controller to ensure that the chaotic system with unknown parameters achieves anti-synchronization in finite time under our controller. Then, we apply our results to the integer-order and fractional-order Lorenz system, respectively. Finally, numerical simulations are presented to show the feasibility of the proposed control scheme. At the same time, through the numerical simulation results, it is show that for the Lorenz chaotic system, when the order is greater, the more quickly is anti-synchronization achieved.

Introducing Robust Evolutionary Optimization in Noisy Fractional-Order Systems

2020 ◽  
Vol 30 (08) ◽  
pp. 2050119
Author(s):  
Wei Du ◽  
Le Tong
Keyword(s):  
Fractional Order ◽  
Chaotic Systems ◽  
Initial Point ◽  
Evolutionary Optimization ◽  
Experimental Results ◽  
Fractional Order Systems ◽  
Initial Values ◽  
System A ◽  
Series Of Experiments

This paper investigates the problem of searching for the robust initial values of noisy fractional-order chaotic systems when the desired output is given. The problem is addressed under the framework of robust evolutionary optimization. Two different ways of adding noise are considered: (1) the noise is added to the initial point; (2) the noise is added to the orbit of the system. A series of experiments are conducted to validate the effectiveness of robust evolutionary optimization. The experimental results reveal that robust initial values of noisy fractional-order chaotic systems can be obtained in an efficient way by introducing robust evolutionary optimization.

Novel Cryptography Technique via Chaos Synchronization of Fractional-Order Derivative Systems

2021 ◽  
pp. 765-798
Author(s):  
Alain Giresse Tene ◽  
Timoleon Crépin Kofane
Keyword(s):  
Fractional Order ◽  
Chaos Synchronization ◽  
Chaotic Systems ◽  
Order Derivative ◽  
Random Numbers ◽  
Generation Process ◽  
Initial Condition ◽  
Fractional Order Derivative ◽  
Q Matrix ◽  
Fractional System

Synchronization of fractional-order-derivative systems for cryptography purpose is still exploratory and despite an increase in cryptography research, several challenges remain in designing a powerful cryptosystem. This chapter addresses the problem of synchronization of fractional-order-derivative chaotic systems using random numbers generator for a novel technique to key distribution in cryptography. However, there is evidence that researchers have approached the problem using integer order derivative chaotic systems. Consequently, the aim of the chapter lies in coding and decoding a text via chaos synchronization of fractional-order derivative, the performance analysis and the key establishment scheme following an application on a text encryption using the chaotic Mathieu-Van Der Pol fractional system. In order to improve the level of the key security, the Fibonacci Q-matrix is used in the key generation process and the initial condition; the order of the derivative of the responder system secretly shared between the responder and the receiver are also involved. It followed from this study that compared to the existing cryptography techniques, this proposed method is found to be very efficient due to the fact that, it improves the key security.

scholarly journals Adaptive Synchronization of Fractional-Order Complex Chaotic system with Unknown Complex Parameters

Entropy ◽  
2019 ◽  
Vol 21 (2) ◽  
pp. 207 ◽  
Author(s):  
Ruoxun Zhang ◽  
Yongli Liu ◽  
Shiping Yang
Keyword(s):  
Fractional Order ◽  
Complex Variable ◽  
Chaotic Systems ◽  
Adaptive Synchronization ◽  
Order Complex ◽  
System A ◽  
Synchronization Scheme ◽  
Complex Valued ◽  
Complex Chaotic System ◽  
Variable Inequality

This paper investigates the problem of synchronization of fractional-order complex-variable chaotic systems (FOCCS) with unknown complex parameters. Based on the complex-variable inequality and stability theory for fractional-order complex-valued system, a new scheme is presented for adaptive synchronization of FOCCS with unknown complex parameters. The proposed scheme not only provides a new method to analyze fractional-order complex-valued system but also significantly reduces the complexity of computation and analysis. Theoretical proof and simulation results substantiate the effectiveness of the presented synchronization scheme.

Novel Cryptography Technique via Chaos Synchronization of Fractional-Order Derivative Systems

2021 ◽  
pp. 2220-2253
Author(s):  
Alain Giresse Tene ◽  
Timoleon Crépin Kofane
Keyword(s):  
Fractional Order ◽  
Chaos Synchronization ◽  
Chaotic Systems ◽  
Order Derivative ◽  
Random Numbers ◽  
Generation Process ◽  
Initial Condition ◽  
Fractional Order Derivative ◽  
Q Matrix ◽  
Fractional System

Synchronization of fractional-order-derivative systems for cryptography purpose is still exploratory and despite an increase in cryptography research, several challenges remain in designing a powerful cryptosystem. This chapter addresses the problem of synchronization of fractional-order-derivative chaotic systems using random numbers generator for a novel technique to key distribution in cryptography. However, there is evidence that researchers have approached the problem using integer order derivative chaotic systems. Consequently, the aim of the chapter lies in coding and decoding a text via chaos synchronization of fractional-order derivative, the performance analysis and the key establishment scheme following an application on a text encryption using the chaotic Mathieu-Van Der Pol fractional system. In order to improve the level of the key security, the Fibonacci Q-matrix is used in the key generation process and the initial condition; the order of the derivative of the responder system secretly shared between the responder and the receiver are also involved. It followed from this study that compared to the existing cryptography techniques, this proposed method is found to be very efficient due to the fact that, it improves the key security.

GENERALIZED THEORY OF MAGNETO-THERMO-VISCOELASTIC SPHERICAL CAVITY PROBLEM UNDER FRACTIONAL ORDER DERIVATIVE: STATE SPACE APPROACH

2020 ◽  
Vol 9 (11) ◽  
pp. 9769-9780
Author(s):  
S.G. Khavale ◽  
K.R. Gaikwad
Keyword(s):  
State Space ◽  
Fractional Order ◽  
Order Derivative ◽  
Laplace Inversion ◽  
State Space Approach ◽  
Spherical Region ◽  
Fractional Order Derivative ◽  
Numerical Laplace Inversion ◽  
Mathcad Software ◽  
Space Approach

This paper is dealing the modified Ohm's law with the temperature gradient of generalized theory of magneto-thermo-viscoelastic for a thermally, isotropic and electrically infinite material with a spherical region using fractional order derivative. The general solution obtained from Laplace transform, numerical Laplace inversion and state space approach. The temperature, displacement and stresses are obtained and represented graphically with the help of Mathcad software.

A Review: Comparative Analysis of Chaotic Systems With Their Fractional Order Counterparts

2020 ◽  
Author(s):  
Niyojeet Shrirao ◽  
Alpana Pandey ◽  
Vijayshri Chaurasia ◽  
Ramji Gupta ◽  
Nancy Modi ◽  
...  
Keyword(s):  
Comparative Analysis ◽  
Fractional Order ◽  
Chaotic Systems
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