scholarly journals On the Caputo-Fabrizio fractal fractional representation for the Lorenz chaotic system

2021 ◽  
Vol 6 (11) ◽  
pp. 12395-12421
Author(s):  
Anastacia Dlamini ◽  
◽  
Emile F. Doungmo Goufo ◽  
Melusi Khumalo
Keyword(s):  
Chaotic System ◽  
Fractional Derivatives ◽  
Chaotic Systems ◽  
Chaotic Behavior ◽  
Dynamical Behavior ◽  
Integral Operators ◽  
Fractional Dimension ◽  
Science And Engineering ◽  
Widespread Application ◽  
Lorenz Chaotic System

<abstract><p>The widespread application of chaotic dynamical systems in different fields of science and engineering has attracted the attention of many researchers. Hence, understanding and capturing the complexities and the dynamical behavior of these chaotic systems is essential. The newly proposed fractal-fractional derivative and integral operators have been used in literature to predict the chaotic behavior of some of the attractors. It is argued that putting together the concept of fractional and fractal derivatives can help us understand the existing complexities better since fractional derivatives capture a limited number of problems and on the other side fractal derivatives also capture different kinds of complexities. In this study, we use the newly proposed Caputo-Fabrizio fractal-fractional derivatives and integral operators to capture and predict the behavior of the Lorenz chaotic system for different values of the fractional dimension $ q $ and the fractal dimension $ k $. We will look at the well-posedness of the solution. For the effect of the Caputo-Fabrizio fractal-fractional derivatives operator on the behavior, we present the numerical scheme to study the graphical numerical solution for different values of $ q $ and $ k $.</p></abstract>

Dynamics, Synchronization and Electronic Implementations of a New Lorenz-like Chaotic System with Nonhyperbolic Equilibria

2019 ◽  
Vol 29 (14) ◽  
pp. 1950197 ◽  
Author(s):  
P. D. Kamdem Kuate ◽  
Qiang Lai ◽  
Hilaire Fotsin
Keyword(s):  
Chaotic System ◽  
Chaotic Systems ◽  
Lorenz System ◽  
Chaotic Behavior ◽  
Piecewise Linear ◽  
Period Doubling ◽  
Adaptive Synchronization ◽  
The Novel ◽  
Algebraic Equations ◽  
The Lorenz System

The Lorenz system has attracted increasing attention on the issue of its simplification in order to produce the simplest three-dimensional chaotic systems suitable for secure information processing. Meanwhile, Sprott’s work on elegant chaos has revealed a set of 19 chaotic systems all described by simple algebraic equations. This paper presents a new piecewise-linear chaotic system emerging from the simplification of the Lorenz system combined with the elegance of Sprott systems. Unlike the majority, the new system is a non-Shilnikov chaotic system with two nonhyperbolic equilibria. It is multiplier-free, variable-boostable and exclusively based on absolute value and signum nonlinearities. The use of familiar tools such as Lyapunov exponents spectra, bifurcation diagrams, frequency power spectra as well as Poincaré map help to demonstrate its chaotic behavior. The novel system exhibits inverse period doubling bifurcations and multistability. It has only five terms, one bifurcation parameter and a total amplitude controller. These features allow a simple and low cost electronic implementation. The adaptive synchronization of the novel system is investigated and the corresponding electronic circuit is presented to confirm its feasibility.

scholarly journals S-Box Based Image Encryption Application Using a Chaotic System without Equilibrium

2019 ◽  
Vol 9 (4) ◽  
pp. 781 ◽  
Author(s):  
Xiong Wang ◽  
Ünal Çavuşoğlu ◽  
Sezgin Kacar ◽  
Akif Akgul ◽  
Viet-Thanh Pham ◽  
...  
Keyword(s):  
Image Encryption ◽  
Chaotic System ◽  
Chaotic Systems ◽  
Electronic Circuit ◽  
Chaotic Behavior ◽  
Encryption Algorithm ◽  
Phase Portraits ◽  
Hidden Attractors ◽  
New Chaotic System ◽  
Image Encryption Algorithm

Chaotic systems without equilibrium are of interest because they are the systems with hidden attractors. A nonequilibrium system with chaos is introduced in this work. Chaotic behavior of the system is verified by phase portraits, Lyapunov exponents, and entropy. We have implemented a real electronic circuit of the system and reported experimental results. By using this new chaotic system, we have constructed S-boxes which are applied to propose a novel image encryption algorithm. In the designed encryption algorithm, three S-boxes with strong cryptographic properties are used for the sub-byte operation. Particularly, the S-box for the sub-byte process is selected randomly. In addition, performance analyses of S-boxes and security analyses of the encryption processes have been presented.

scholarly journals A New 4-D Chaotic System with Hidden Attractor and its Circuit Implementation

2018 ◽  
Vol 7 (3) ◽  
pp. 1245 ◽  
Author(s):  
Aceng Sambas ◽  
Mustafa Mamat ◽  
Sundarapandian Vaidyanathan ◽  
Muhammad Mohamed ◽  
Mada Sanjaya
Keyword(s):  
Chaotic System ◽  
Chaotic Systems ◽  
Electronic Circuit ◽  
Chaotic Behavior ◽  
Phase Portraits ◽  
Hidden Attractor ◽  
Circuit Realization ◽  
New Chaotic System ◽  
Quadratic Nonlinearities ◽  
Work First

In the chaos literature, there is currently significant interest in the discovery of new chaotic systems with hidden chaotic attractors. A new 4-D chaotic system with only two quadratic nonlinearities is investigated in this work. First, we derive a no-equilibrium chaotic system and show that the new chaotic system exhibits hidden attractor. Properties of the new chaotic system are analyzed by means of phase portraits, Lyapunov chaos exponents, and Kaplan-Yorke dimension. Then an electronic circuit realization is shown to validate the chaotic behavior of the new 4-D chaotic system. Finally, the physical circuit experimental results of the 4-D chaotic system show agreement with numerical simulations.

scholarly journals Chaos Control and Synchronization via Switched Output Control Strategy

Complexity ◽  
2017 ◽  
Vol 2017 ◽  
pp. 1-11 ◽  
Author(s):  
Runzi Luo ◽  
Haipeng Su ◽  
Yanhui Zeng
Keyword(s):  
Chaotic System ◽  
Control Strategy ◽  
Chaos Control ◽  
Chaotic Systems ◽  
State Variables ◽  
Output Control ◽  
Lorenz Chaotic System ◽  
Control And Synchronization ◽  
Theoretical Results

This paper investigates the control and synchronization of a class of chaotic systems with switched output which is assumed to be switched between the first and the second state variables of chaotic system. Some novel and yet simple criteria for the control and synchronization of a class of chaotic systems are proposed via the switched output. The generalized Lorenz chaotic system is taken as an example to show the feasibility and efficiency of theoretical results.

scholarly journals Parallel multi-layer selector S-Box based on lorenz chaotic system with FPGA implementation

2020 ◽  
Vol 19 (2) ◽  
pp. 784
Author(s):  
Mohamed Saber ◽  
Esam Hagras
Keyword(s):  
Power Consumption ◽  
Chaotic System ◽  
High Speed ◽  
Computational Models ◽  
Chaotic Systems ◽  
Fpga Implementation ◽  
Differential Probability ◽  
Substitution Boxes ◽  
Lorenz Chaotic System ◽  
Hidden Data

<p><span>The substitution box (S-Box) is the main block in the encryption system, which replaces the non-encrypted data by dynamic secure and hidden data. S-Box can be designed based on complex nonlinear chaotic systems that presented in recent papers as a chaotic S-Box. The hardware implementation of these chaotic systems suffers from long processing time (low speed), and high-power consumption since it requires a large number of non-linear computational models. In this paper, we present a high-speed FPGA implementation of Parallel Multi-Layer Selector Substitution Boxes based on the Lorenz Chaotic System (PMLS S-Box). The proposed PMLS chaotic S-Box is modeled using Xilinx System Generator (XSG) in 32 bits fixed-point format, and the architecture implemented into Xilinx Spartan-6 X6SLX45 board. The maximum frequency of the proposed PMLS chaotic S-Box is 381.764 MHz, with dissipates of 77 mwatt. Compared to other S-Box chaotic systems, the proposed one achieves a higher frequency and lower power consumption. In addition, the proposed PMLS chaotic S-Box is analyzed based on S-Box standard tests such as; Bijectivity property, nonlinearity, strict avalanche criterion, differential probability, and bits independent criterion. The five different standard results for the proposed S-Box indicate that PMLSC can effectively resist crypto-analysis attacks, and is suitable for secure communications.</span></p>

scholarly journals A novel chaotic system and its topological horseshoe

2013 ◽  
Vol 18 (1) ◽  
pp. 66-77 ◽  
Author(s):  
Chunlai Li ◽  
Lei Wu ◽  
Hongmin Li ◽  
Yaonan Tong
Keyword(s):  
Lyapunov Exponent ◽  
Chaotic System ◽  
Chaotic Systems ◽  
Signal Amplitude ◽  
Topological Horseshoe ◽  
Lorenz Chaotic System ◽  
Construction Pattern ◽  
Lyapunov Exponent Spectrum

Based on the construction pattern of Chen, Liu and Qi chaotic systems, a new threedimensional (3D) chaotic system is proposed by developing Lorenz chaotic system. It’s found that when parameter e varies, the Lyapunov exponent spectrum keeps invariable, and the signal amplitude can be controlled by adjusting e. Moreover, the horseshoe chaos in this system is investigated based on the topological horseshoe theory.

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.

scholarly journals Encryption in Chaotic Systems with Sinusoidal Excitations

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
G. Obregón-Pulido ◽  
A. Torres-González ◽  
R. Cárdenas-Rodríguez ◽  
G. Solís-Perales
Keyword(s):  
Chaotic System ◽  
Chaotic Systems ◽  
Chaotic Behavior ◽  
Sinusoidal Signal ◽  
Security Level ◽  
Chaotic Oscillator ◽  
Regulation Theory ◽  
Régulation Theory ◽  
Sinusoidal Signals ◽  
Encryption Method

In this contribution an encryption method using a chaotic oscillator, excited by “n” sinusoidal signals, is presented. The chaotic oscillator is excited by a sum of “n” sinusoidal signals and a message. The objective is to encrypt such a message using the chaotic behavior and transmit it, and, as the chaotic system is perturbed by the sinusoidal signal, the transmission security could be increased due to the effect of such a perturbation. The procedure is based on the regulation theory and consider that the receiver knows the frequencies of the perturbing signal, with this considerations the algorithm estimates the excitation in such a way that the receiver can cancel out the perturbation and all the undesirable dynamics in order to produce only the message. In this way we consider that the security level is increased.

scholarly journals Circuit Implementation Synchronization between Two Modified Fractional-Order Lorenz Chaotic Systems via a Linear Resistor and Fractional-Order Capacitor in Parallel Coupling

2021 ◽  
Vol 2021 ◽  
pp. 1-8 ◽  
Author(s):  
Juan Liu ◽  
Xuefeng Cheng ◽  
Ping Zhou
Keyword(s):  
Fractional Order ◽  
Chaotic System ◽  
Chaos Synchronization ◽  
Chaotic Systems ◽  
Electronic Circuit ◽  
Chaotic Attractors ◽  
Circuit Implementation ◽  
Lorenz Chaotic System ◽  
Simulation Results ◽  
Synchronization Scheme

In this study, a modified fractional-order Lorenz chaotic system is proposed, and the chaotic attractors are obtained. Meanwhile, we construct one electronic circuit to realize the modified fractional-order Lorenz chaotic system. Most importantly, using a linear resistor and a fractional-order capacitor in parallel coupling, we suggested one chaos synchronization scheme for this modified fractional-order Lorenz chaotic system. The electronic circuit of chaos synchronization for modified fractional-order Lorenz chaotic has been given. The simulation results verify that synchronization scheme is viable.

scholarly journals Simultaneous identification of the all parameters for the Lorenz chaotic system

2019 ◽  
Vol 6 (125) ◽  
pp. 68-75
Author(s):  
Anton Guda ◽  
Andrey Zimoglyad
Keyword(s):  
Dynamic System ◽  
Chaotic System ◽  
Measurement Errors ◽  
Chaotic Behavior ◽  
Moving Average ◽  
Regression Method ◽  
Linear Regression Method ◽  
New Approach ◽  
Lorenz Chaotic System ◽  
Simultaneous Identification

Drawbacks of the adaptive-searching methods, related with the problem of multi-parameter dynamic system identification are explored and highlighted. New approach, based on “moving regression” method is proposed. New approach is a hybrid method; it combines features of the “moving average” method, linear regression method and differential system representation. This combination allows to simultaneously determining complex dynamic system parameters, in spite of its chaotic behavior and measurement errors. New method possibilities are explored via identification process numerical simulation for the Lorenz chaotic system.

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