scholarly journals Energy Conservation, Singular Orbits, and FPGA Implementation of Two New Hamiltonian Chaotic Systems

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-15
Author(s):  
Enzeng Dong ◽  
Guanghan Liu ◽  
Zenghui Wang ◽  
Zengqiang Chen
Keyword(s):  
Secure Communication ◽  
Chaotic Systems ◽  
Equilibrium Points ◽  
Dynamical Evolution ◽  
Casimir Energy ◽  
Chaotic Attractors ◽  
Chaotic Flows ◽  
Field Programmable ◽  
Singular Orbits ◽  
Nist Tests

Since the conservative chaotic system (CCS) has no general attractors, conservative chaotic flows are more suitable for the chaos-based secure communication than the chaotic attractors. In this paper, two Hamiltonian conservative chaotic systems (HCCSs) are constructed based on the 4D Euler equations and a proposed construction method. These two new systems are investigated by equilibrium points, dynamical evolution map, Hamilton energy, and Casimir energy. They look similar, but it is found that one can be explained using Casimir power and another cannot be explained in terms of the mechanism of chaos. Furthermore, a pseudorandom signal generator is developed based on these proposed systems, which are tested based on NIST tests and implemented by using the field programmable gate array (FPGA) technique.

A Useful Chaotic Family with Single Linearity and Circuit Implementation Based on FPGA

2016 ◽  
Vol 26 (01) ◽  
pp. 1750017 ◽  
Author(s):  
Zeshi Yuan ◽  
Hongtao Li ◽  
Xiaohua Zhu
Keyword(s):  
Field Programmable Gate Array ◽  
Chaotic Systems ◽  
Equilibrium Points ◽  
Three Dimensional ◽  
Register Transfer Level ◽  
Chaotic Circuit ◽  
Chaotic Flows ◽  
Chaotic Model ◽  
Field Programmable ◽  
Simulation Results

Recently, a series of typical three-dimensional dissipative chaotic flows where all but one of the nonlinearities are quadratic are studied. Based on this research, a novel chaotic model with only one single linearity is proposed by introducing cubic terms and four new chaotic systems with various characteristics are found. Besides, a chaotic family with a single linearity is constructed with those four chaotic systems and 12 existing systems SL1–SL[Formula: see text] of the chaotic flows. Exploiting the new systems, basic dynamic behaviors are analyzed, including the strange attractors, equilibrium points, Lyapunov exponents as well as the property of multistability. In addition, the corresponding simulation results are illustrated to show those properties expressly. In realizing the chaotic circuit, we utilize the field programmable gate array (FPGA), which is of considerable flexibility, good programmability and stability, instead of analog devices that are easily affected by surroundings. More importantly, the circuit of the proposed chaotic family is realized on a single FPGA over register transfer level (RTL) using 32-bit fixed-point operation. Finally, an experimental FPGA-based circuit is constructed, and the output results are shown on oscilloscope, which agree well with the numerical simulations.

scholarly journals A Multiscroll Chaotic Attractors with Arrangement of Saddle-Shapes and Its Field Programmable Gate Array (FPGA) Implementation

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Faqiang Wang ◽  
Yufang Xiao
Keyword(s):  
Chaotic System ◽  
Field Programmable Gate Array ◽  
Equilibrium Points ◽  
Research Result ◽  
Step Function ◽  
Phase Locked Loops ◽  
Chaotic Attractors ◽  
Digital To Analog Converter ◽  
Field Programmable ◽  
Gate Array

Based on the step function and signum function, a chaotic system which can generate multiscroll chaotic attractors with arrangement of saddle-shapes is proposed and the stability of its equilibrium points is analyzed. The under mechanism for the generation of multiscroll chaotic attractors and the reason for the arrangement of saddle shapes and being symmetric about y-axis are presented, and the rule for controlling the number of scroll chaotic attractors with saddle shapes is designed. Based on the core chips including Altera Cyclone IV EP4CE10F17C8 Field Programmable Gate Array and Digital to Analog Converter chip AD9767, the peripheral circuit and the Verilog Hardware Description Language program for realization of the proposed multiscroll chaotic system is constructed and some experimental results are presented for confirmation. The research result shows that the occupation of multipliers and Phase-Locked Loops in Field Programmable Gate Array is zero.

CHAOTIC MODELING AND SIMULATION IN ROTATION–TRANSLATION MODELS

2011 ◽  
Vol 21 (10) ◽  
pp. 3023-3031 ◽  
Author(s):  
CHRISTOS H. SKIADAS ◽  
CHARILAOS SKIADAS
Keyword(s):  
Modeling And Simulation ◽  
Difference Equations ◽  
Simulation Analysis ◽  
Equilibrium Points ◽  
Chaotic Attractors ◽  
Chaotic Flows ◽  
Fast Growing ◽  
Special Cases ◽  
The World ◽  
Discrete Methods

Chaotic modeling and simulation is a fast growing field that influences the world around us, and consequently also influences our ways of approaching, analyzing and solving problems. In this paper the rotation–translation case is presented based on the theory analyzed in the book — Chaotic Modeling and Simulation: Analysis of Chaotic Models, Attractors and Forms published by the above authors [Skiadas & Skiadas, 2008]. An overview of the chaotic flows in rotation–translation is given. Special cases and illustrations of chaotic attractors and forms are presented and a method of comparative presentation and analysis of the flows by using both continuous and discrete methods is applied by deriving the appropriate differential and difference equations' analogue. Furthermore, an analysis of chaotic attractors resulting in the models proposed is given along with an exploration of the characteristic or equilibrium points.

Design and FPGA Implementation of New Multidimensional Chaotic Map for Secure Communication

2021 ◽  
pp. 2150280
Author(s):  
Belqassim Bouteghrine ◽  
Camel Tanougast ◽  
Said Sadoudi
Keyword(s):  
Secure Communication ◽  
Chaotic Systems ◽  
Chaotic Maps ◽  
Chaotic Behavior ◽  
Chaotic Map ◽  
Two Phase ◽  
Multiple Parameters ◽  
Random Keys ◽  
Field Programmable ◽  
Constrained Devices

Due to their structure and complexity, chaotic systems have been introduced in several domains such as electronic circuits, commerce domain, encryption and network security. In this paper, we propose a novel multidimensional chaotic system with multiple parameters and nonlinear terms. Then, a two-phase algorithm is presented for investigating the chaotic behavior using bifurcation and Lyapunov exponent (LE) theories. Finally, we illustrate the performances of our proposal by constructing three (03) chaotic maps (3-D, 4-D and 5-D) and implementing the 3-D map on Field-Programmable-Gate-Array (FPGA) boards to generate random keys for securing a client–server communication purpose. Based on the achieved results, the proposed scheme is considered an ideal candidate for numerous resource-constrained devices and internet of the things (IoT) applications.

scholarly journals Modeling, Synchronization, and FPGA Implementation of Hamiltonian Conservative Hyperchaos

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Enzeng Dong ◽  
Xiaodong Jiao ◽  
Shengzhi Du ◽  
Zengqiang Chen ◽  
Guoyuan Qi
Keyword(s):  
Euler Equations ◽  
Engineering Application ◽  
Casimir Energy ◽  
Dissipative Systems ◽  
Adaptive Synchronization ◽  
Higher Dimensional ◽  
Field Programmable ◽  
Five Dimension ◽  
Hyperchaotic Systems ◽  
Nist Tests

Conservative chaotic systems have potentials in engineering application because of their superiority over the dissipative systems in terms of ergodicity and integer dimension. In this paper, five-dimension Euler equations are constructed by integrating two of sub-Euler equations, which are contributory to the exploration of higher-dimensional systems. These Euler equations compose the conservative parts from their antisymmetric structure, which have been proved to be both Hamiltonian and Casimir energy conservative. Furthermore, a family of Hamiltonian conservative hyperchaotic systems are proposed by breaking the conservation of Casimir energy. The numerical analysis shows that the system displays some interesting behaviors, such as the coexistence of quasi-periodic, chaotic, and hyperchaotic behaviors. Adaptive synchronization method is used to realize the hyperchaos synchronization. Finally, the system passed the NIST tests successfully. Field programmable gate array (FPGA) platform is used to implement the proposed Hamiltonian conservative hyperchaos.

A Novel Class of Chaotic Flows with Infinite Equilibriums and Their Application in Chaos-Based Communication Design Using DCSK

2018 ◽  
Vol 73 (7) ◽  
pp. 609-617 ◽  
Author(s):  
Karthikeyan Rajagopal ◽  
Serdar Çiçek ◽  
Abdul Jalil M. Khalaf ◽  
Viet-Thanh Pham ◽  
Sajad Jafari ◽  
...  
Keyword(s):  
Chaotic System ◽  
Communication Systems ◽  
Chaotic Systems ◽  
Equilibrium Points ◽  
Engineering Application ◽  
Chaotic Flows ◽  
Fundamental Properties ◽  
New Chaotic System ◽  
Shift Keying ◽  
Chaos Shift Keying

AbstractDiscovering chaotic systems with interesting features has been of interest in the recent years. One such important and interesting feature is the type and shape of equilibrium points. We introduce a class of chaotic systems which could show different types of infinite equilibrium points. The fundamental properties of the proposed systems like bifurcation diagram and Lyapunov exponents are investigated. An electronic circuit of the presented chaotic systems is implemented. In addition, a chaos-based communication application by the differential chaos shift keying method with the new chaotic system is designed and tested for engineering application. According to the design test results, the proposed chaos-based communication system is successful. Therefore, the new chaotic system can be used in chaos-based communication systems.

scholarly journals CHAOS ENTANGLEMENT: A NEW APPROACH TO GENERATE CHAOS

2013 ◽  
Vol 23 (05) ◽  
pp. 1330014 ◽  
Author(s):  
HONGTAO ZHANG ◽  
XINZHI LIU ◽  
XUEMIN SHEN ◽  
JUN LIU
Keyword(s):  
Secure Communication ◽  
Chaotic Systems ◽  
Complex Dynamics ◽  
Saddle Points ◽  
Dynamical Analysis ◽  
Chaotic Attractors ◽  
New Approach ◽  
One Dimensional ◽  
Linear Subsystem ◽  
Stable Linear

A new approach to generate chaotic phenomenon, called chaos entanglement, is proposed in this paper. The basic principle is to entangle two or multiple stable linear subsystems by entanglement functions to form an artificial chaotic system such that each of them evolves in a chaotic manner. Firstly, a new attractor, entangling a two-dimensional linear subsystem and a one-dimensional one by sine function, is presented as an example. Dynamical analysis shows that both entangled subsystems are bounded and all equilibra are unstable saddle points when chaos entanglement is achieved. Also, numerical computation shows that this system has one positive Lyapunov exponent, which implies chaos. Furthermore, two conditions are given to achieve chaos entanglement. Along this way, by different linear subsystems and different entanglement functions, a variety of novel chaotic attractors have been created and abundant complex dynamics are exhibited. Our discovery indicates that it is not difficult any more to construct new artificial chaotic systems/networks for engineering applications such as chaos-based secure communication. Finally, a possible circuit is given to realize these new chaotic attractors.

BIFURCATION IN A PIECEWISE LINEAR CIRCUIT WITH SWITCHING BOUNDARIES

2012 ◽  
Vol 22 (02) ◽  
pp. 1250034 ◽  
Author(s):  
ZHENGDI ZHANG ◽  
QINSHENG BI
Keyword(s):  
Periodic Orbits ◽  
Piecewise Linear ◽  
Equilibrium Points ◽  
Period Doubling ◽  
Dynamical Evolution ◽  
Phase Portraits ◽  
Chaotic Attractors ◽  
Generalized Jacobian ◽  
Linear Circuit ◽  
The Stability

By introducing time-dependent power source, a periodically excited piecewise linear circuit with double-scroll is established. In the absence of the excitation, all possible equilibrium points as well as the stability conditions are presented. Analyzing the corresponding characteristic equations with perturbation method, Hopf bifurcation conditions associated with the equilibria are derived, which can be demonstrated by the numerical simulations. The Hopf bifurcations of the two symmetric equilibrium points may cause two symmetric periodic orbits, which lead to single-scroll chaotic attractors via sequences of period-doubling bifurcations with the variation of the parameters. The two chaotic attractors expand to interact with each other to form an enlarged chaotic attractor with double-scroll. The behaviors on the switching boundaries are investigated by the generalized Jacobian matrix. When periodic excitation is applied to work on the circuit, three periodic orbits with the frequency of the excitation may exist, which can be called generalized equilibrium points (GEPs) with the same characteristic polynomials as those of the corresponding equilibrium points for the autonomous case. It is shown that when the trajectories do not pass across the switching boundaries, the solutions are the same as the GEPs. However, when the trajectories pass across the switching boundaries, complicated behaviors will take place. Three forms of chaotic attractors via different bifurcations can be observed and the influence of the switching boundaries on the phase portraits is discussed to explore the mechanism of the dynamical evolution.

scholarly journals A Chaotic System with an Infinite Number of Equilibrium Points: Dynamics, Horseshoe, and Synchronization

2016 ◽  
Vol 2016 ◽  
pp. 1-8 ◽  
Author(s):  
Viet-Thanh Pham ◽  
Christos Volos ◽  
Sundarapandian Vaidyanathan ◽  
Xiong Wang
Keyword(s):  
Chaotic System ◽  
Infinite Number ◽  
Chaotic Systems ◽  
Chaotic Behavior ◽  
Equilibrium Points ◽  
Equilibrium Analysis ◽  
Chaotic Attractors ◽  
Hidden Attractors ◽  
New Chaotic System ◽  
Topological Horseshoes

Discovering systems with hidden attractors is a challenging topic which has received considerable interest of the scientific community recently. This work introduces a new chaotic system having hidden chaotic attractors with an infinite number of equilibrium points. We have studied dynamical properties of such special system via equilibrium analysis, bifurcation diagram, and maximal Lyapunov exponents. In order to confirm the system’s chaotic behavior, the findings of topological horseshoes for the system are presented. In addition, the possibility of synchronization of two new chaotic systems with infinite equilibria is investigated by using adaptive control.

scholarly journals Controlling Chaotic Systems via Time-delayed Control

2021 ◽  
Vol 15 ◽  
pp. 44-49
Author(s):  
Ramy Farid ◽  
Abdul-Azim Ibrahim ◽  
Belal Abou-Zalam
Keyword(s):  
Numerical Simulation ◽  
Theoretical Analysis ◽  
Chaotic Systems ◽  
Equilibrium Points ◽  
Unstable Equilibrium ◽  
Chaotic Attractors ◽  
Chua’S Circuit ◽  
Chua's Circuit ◽  
Integral Time ◽  
Lyapunov Stabilization

Based on Lyapunov stabilization theory, this paper proposes a proportional plus integral time-delayed controller to stabilize unstable equilibrium points (UPOs) embedded in chaotic attractors. The criterion is successfully applied to the classic Chua's circuit. Theoretical analysis and numerical simulation show the effectiveness of this controller.

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