scholarly journals Construction of a Class of High-Dimensional Discrete Chaotic Systems

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 365
Author(s):  
Hongyan Zang ◽  
Jianying Liu ◽  
Jiu Li
Keyword(s):  
Lyapunov Exponent ◽  
Numerical Simulations ◽  
Chaotic Systems ◽  
Dynamic Properties ◽  
Sufficient Conditions ◽  
High Dimensional ◽  
Chaotic Mapping ◽  
Lyapunov Exponent Spectrum

In this paper, a class of n-dimensional discrete chaotic systems with modular operations is studied. Sufficient conditions for transforming this kind of discrete mapping into a chaotic mapping are given, and they are proven by the Marotto theorem. Furthermore, several special systems satisfying the criterion are given, the basic dynamic properties of the solution, such as the trace diagram and Lyapunov exponent spectrum, are analyzed, and the correctness of the chaos criterion is verified by numerical simulations.

ON LORENZ-LIKE DYNAMIC SYSTEMS WITH STRENGTHENED NONLINEARITY AND NEW PARAMETERS

2010 ◽  
Vol 08 (02) ◽  
pp. 293-311 ◽  
Author(s):  
BO LIAO ◽  
YUAN YAN TANG ◽  
LU AN
Keyword(s):  
Lyapunov Exponent ◽  
Numerical Simulations ◽  
Dynamic Systems ◽  
Chaotic Systems ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Maximum Lyapunov Exponent ◽  
Preliminary Results ◽  
Quadratic Equations

This paper introduces two types of Lorenz-like three-dimensional quadratic autonomous chaotic systems with 7 and 8 new parameters free of choice, respectively. Both systems are investigated at the equilibriums to study their chaotic characteristics. We focus our attention on the second type of the introduced system which consists of three nonlinear quadratic equations. Predictably, coordinates of the equilibriums are prohibitively complex. Therefore, instead of directly analyzing their stability, we prove the asymptotical characterization of equilibriums by utilizing our preliminary results derived for the first type of system. Our result shows that, though the coordinates of equilibriums satisfy a ternary quadratic, the system still contains only three equilibriums in circumstances of chaos. Sufficient conditions for the chaotic appearance of systems are derived. Our results are further verified by numerical simulations and the maximum Lyapunov exponent for several examples. Our research takes a first step in investigating chaos in Lorenz-like dynamic systems with strengthened nonlinearity and general forms of parameters.

SYNCHRONIZING HIGH DIMENSIONAL CHAOTIC SYSTEMS VIA EIGENVALUE PLACEMENT WITH APPLICATION TO CELLULAR NEURAL NETWORKS

1999 ◽  
Vol 09 (04) ◽  
pp. 705-711 ◽  
Author(s):  
GIUSEPPE GRASSI ◽  
SAVERIO MASCOLO
Keyword(s):  
Neural Networks ◽  
Numerical Simulations ◽  
Linear Systems ◽  
Chaotic Systems ◽  
Cellular Neural Networks ◽  
High Dimensional ◽  
Suggested Approach ◽  
Error Dynamics ◽  
Controllability Property

In this paper a method for synchronizing high dimensional chaotic systems is developed. The objective is to generate a linear error dynamics between the master and the slave systems, so that synchronization is achievable by exploiting the controllability property of linear systems. The suggested approach is applied to Cellular Neural Networks (CNNs), which can be considered as a tool for generating complex hyperchaotic behaviors. Numerical simulations are carried out for synchronizing CNNs constituted by Chua's circuits.

HYPERCHAOS Qi SYSTEM

2010 ◽  
Vol 24 (24) ◽  
pp. 4771-4778 ◽  
Author(s):  
XING-YUAN WANG ◽  
YONG-FENG GAO ◽  
YAO-XIAN ZHANG
Keyword(s):  
Phase Diagram ◽  
Lyapunov Exponent ◽  
Numerical Simulations ◽  
Bifurcation Diagram ◽  
Nonlinear Term ◽  
Linear Term ◽  
Nonlinear Controller ◽  
Chaos System ◽  
Lyapunov Exponent Spectrum

This paper presents a four-dimensional hyperchaos Qi system, obtained by adding linear term and nonlinear term of nonlinear controller to Qi chaos system. The hyperchaos Qi system is studied by bifurcation diagram, Lyapunov exponent spectrum and phase diagram. Numerical simulations show that the new system's behavior can be periodic, chaotic and hyperchaotic as the parameter varies.

LINEAR AND NONLINEAR GENERALIZED SYNCHRONIZATION OF CHAOTIC SYSTEMS

2010 ◽  
Vol 24 (31) ◽  
pp. 6143-6155
Author(s):  
XING-YUAN WANG ◽  
JING ZHANG
Keyword(s):  
Numerical Simulations ◽  
Chaotic Systems ◽  
Sufficient Conditions ◽  
State Observer ◽  
Drive System ◽  
Generalized Synchronization ◽  
Response System ◽  
Linear And Nonlinear ◽  
Synchronization Scheme ◽  
Application Scope

In this paper, the linear and nonlinear generalized synchronization of chaotic systems is investigated. Based on the modified state observer method, a new synchronization approach is proposed with more extensive application scope. The proposed synchronization scheme can realize the linear and nonlinear generalized synchronizations of same dimensional or different dimensional chaotic systems. Sufficient conditions of global asymptotic generalized synchronization between the drive system and the response system are gained on the basis of the state observer theory. Numerical simulations further illustrate the effectiveness of the proposed scheme.

scholarly journals The Related Extension and Application of the Ši'lnikov Theorem

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Baoying Chen
Keyword(s):  
Chaotic System ◽  
Chaotic Systems ◽  
Autonomous Systems ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
High Dimensional ◽  
Extended Version

The traditional Ši'lnikov theorems provide analytic criteria for proving the existence of chaos in high-dimensional autonomous systems. We have established one extended version of the Ši'lnikov homoclinic theorem and have given a set of sufficient conditions under which the system generates chaos in the sense of Smale horseshoes. In this paper, the extension questions of the Ši'lnikov homoclinic theorem and its applications are still discussed. We establish another extended version of the Ši'lnikov homoclinic theorem. In addition, we construct a new three-dimensional chaotic system which meets all the conditions in this extended Ši'lnikov homoclinic theorem. Finally, we list all well-known three-dimensional autonomous quadratic chaotic systems and classify them in the light of the Ši'lnikov theorems.

Dual Combination Synchronization of Six Chaotic Systems

2015 ◽  
Vol 11 (3) ◽  
Author(s):  
Junwei Sun ◽  
Suxia Jiang ◽  
Guangzhao Cui ◽  
Yanfeng Wang
Keyword(s):  
Adaptive Control ◽  
Numerical Simulations ◽  
Lyapunov Stability ◽  
Stability Theory ◽  
Chaotic Systems ◽  
Sufficient Conditions ◽  
Lyapunov Stability Theory ◽  
Combination Synchronization

Based on combination synchronization of three chaotic systems and combination–combination synchronization of four chaotic systems, a novel scheme of dual combination synchronization is investigated for six chaotic systems in the paper. Using combined adaptive control and Lyapunov stability theory of chaotic systems, some sufficient conditions are attained to realize dual combination synchronization of six chaotic systems. The corresponding theoretical proofs and numerical simulations are presented to demonstrate the effectiveness and correctness of the dual combination synchronization. Due to the complexity of dual combination synchronization, it will be more secure and interesting to transmit and receive signals in application of communication.

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.

A Switched Hyperchaotic System and Analysis of Dynamical Properties

2012 ◽  
Vol 452-453 ◽  
pp. 511-515
Author(s):  
Bian Li ◽  
Ai Xue Qi ◽  
Wei Bing Li
Keyword(s):  
Lyapunov Exponent ◽  
Numerical Simulations ◽  
Bifurcation Diagram ◽  
Experimental Results ◽  
Hyperchaotic System ◽  
Symbolic Function ◽  
Dynamical Properties ◽  
Lyapunov Exponent Spectrum

This paper introduces a new switched hyperchaotic system by utilizing symbolic function. The switched hyperchaotic system converts its states from one subsystem to another via symbolic functions. By the analysis of its dynamics, symmetry, dissipativity, Lyapunov exponent spectrum and bifurcation diagram of the system. The system is implemented by the FPGA hardware. It is shown that the experimental results are identical with numerical simulations, and the chaotic trajectories are much more complex.

scholarly journals Simultaneity of Synchronization and Antisynchronization in a Class of Chaotic Systems

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Zhi Liu ◽  
Rongwei Guo ◽  
Yi Qi ◽  
Cuimei Jiang
Keyword(s):  
Numerical Simulations ◽  
Chaotic System ◽  
Chaotic Systems ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Synchronization Phenomenon ◽  
All Solutions ◽  
Necessary And Sufficient ◽  
Theoretical Results

In this paper, a new synchronization phenomenon, that is, the simultaneity of synchronization and antisynchronization, is investigated for a class of chaotic systems. First, for a given chaotic system, necessary and sufficient conditions for the simultaneity of synchronization and antisynchronization are proved. Then, based on these conditions, all solutions of such synchronization phenomenon for a given chaotic system are derived. After that, physical controllers that are not only simple but also implementable are designed to realize the simultaneity of synchronization and antisynchronization in the above system. Finally, illustrative examples based on numerical simulations are used to verify the validity and effectiveness of the above theoretical results.

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