A Switched Hyperchaotic System and Analysis of Dynamical Properties

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.

HYPERCHAOS Qi SYSTEM

International Journal of Modern Physics B ◽

10.1142/s0217979210055895 ◽

2010 ◽

Vol 24 (24) ◽

pp. 4771-4778 ◽

Cited By ~ 1

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 ◽

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.

Constructing Keyed Strong S-Box Using an Enhanced Quadratic Map

10.1142/s0218127421501467 ◽

2021 ◽

Vol 31 (10) ◽

pp. 2150146

Author(s):

Yuanyuan Si ◽

Hongjun Liu ◽

Yuehui Chen

Keyword(s):

Phase Diagram ◽

Fixed Point ◽

Lyapunov Exponent ◽

Uniform Distribution ◽

Bifurcation Diagram ◽

Experimental Results ◽

Kolmogorov Entropy ◽

Symmetric Cryptography ◽

Quadratic Map ◽

Nonlinear Component

As the only nonlinear component for symmetric cryptography, S-Box plays an important role. An S-Box may be vulnerable because of the existence of fixed point, reverse fixed point or short iteration cycles. To construct a keyed strong S-Box, first, a 2D enhanced quadratic map (EQM) was constructed, and its dynamic behaviors were analyzed through phase diagram, Lyapunov exponent, Kolmogorov entropy, bifurcation diagram and randomness testing. The results demonstrated that the state points of EQM have uniform distribution, ergodicity and better randomness. Then a keyed strong S-Box construction algorithm was designed based on EQM, and the fixed point, reverse fixed point, and short cycles were eliminated. Experimental results verified the algorithm’s feasibility and effectiveness.

Realization of a Novel Logarithmic Chaotic System and Its Characteristic Analysis

10.1142/s0218127419300040 ◽

2019 ◽

Vol 29 (02) ◽

pp. 1930004 ◽

Cited By ~ 3

Author(s):

Xiaoyuan Wang ◽

Xiaotao Min ◽

Jun Yu ◽

Yiran Shen ◽

Guangyi Wang ◽

...

Keyword(s):

Phase Diagram ◽

Lyapunov Exponent ◽

Theoretical Analysis ◽

Chaotic System ◽

Nonlinear Term ◽

Experimental Results ◽

Characteristic Analysis ◽

Chaotic Parameter ◽

Hardware Circuit ◽

To further improve the complexity of the chaotic system and broaden the chaotic parameter range, a novel logarithmic chaotic system was constructed by adding a nonlinear term of logarithm. The dynamic characteristics of the chaotic system were analyzed by chaotic phase diagram, bifurcation diagram, Lyapunov exponent spectrum, Poincaré mapping and dynamical map, etc. The system was digitized by DSP simulation, and the corresponding experimental results are completely consistent with the theoretical analysis. Furthermore, the equivalent hardware circuit was designed and theoretical analysis was verified by its experimental results.

Chaos in a Meminductor-Based Circuit

10.1142/s0218127416501303 ◽

2016 ◽

Vol 26 (08) ◽

pp. 1650130 ◽

Cited By ~ 18

Author(s):

Fang Yuan ◽

Guangyi Wang ◽

Peipei Jin ◽

Xiaoyuan Wang ◽

Guojin Ma

Keyword(s):

Lyapunov Exponent ◽

Equivalent Circuit ◽

Smooth Curve ◽

Experimental Results ◽

Chaotic Oscillator ◽

Coexisting Attractors ◽

Dynamical Behaviors ◽

Equilibrium Set ◽

Curve Model ◽

A smooth curve model of meminductor and its equivalent circuit have been designed, on the condition that the meminductor is commonly unavailable. The equivalent circuit can be used for breadboard experiments for various application circuit designs of meminductor. Based on the meminductor, a new chaotic oscillator is proposed. The dynamical behaviors of the oscillator are investigated, including equilibrium set, Lyapunov exponent spectrum, bifurcations and dynamical map of the system. Particularly, the amplitude controlling is analyzed and coexisting attractors are found for conditions of different parameters. Furthermore, the experimental results are given to confirm the correction of the proposed meminductor model and the meminductor-based oscillator.

3D Grid Multi-Wing Chaotic Attractors

10.1142/s0218127418500451 ◽

2018 ◽

Vol 28 (04) ◽

pp. 1850045 ◽

Cited By ~ 5

Author(s):

Nan Yu ◽

Yan-Wu Wang ◽

Xiao-Kang Liu ◽

Jiang-Wen Xiao

Keyword(s):

Numerical Simulations ◽

Bifurcation Diagram ◽

Mirror Symmetry ◽

Electrical Circuit ◽

Circuit Diagram ◽

Chaotic Attractors ◽

Switching Functions ◽

Rotation Transformation ◽

Index 2 ◽

Dynamical Properties

As reported in the existing literature, wing attractors are confined to 1D [Formula: see text]-wing attractors, 2D [Formula: see text]-grid wing attractors. In this paper, we break this limitation and generate 3D [Formula: see text]-grid multi-wing chaotic attractors (GMWCAs). The 3D GMWCAs are produced via the following three steps: (1) applying rotation transformation to a double-wing Lorenz-like system to ensure that its saddle-focus equilibria with index 2 are located on the plane [Formula: see text]; (2) extending the wing attractors of the transformed Lorenz-like system along the [Formula: see text]-axis to have mirror symmetry; (3) introducing stair switching functions to increase the number of saddle-focus equilibria with index 2 along the [Formula: see text]-axis and [Formula: see text]-axis. Furthermore, some basic dynamical properties of the 3D chaotic system, including equilibria, symmetry, dissipativity, Lyapunov exponents and bifurcation diagram, are investigated and a module-based unified circuit diagram is designed. The effectiveness of this approach is confirmed by both numerical simulations and electrical circuit experiment.

A Novel Four-Order Chaotic System with N-Attractor Multi-Direction Multi-Scroll Attractors

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.678.81 ◽

2014 ◽

Vol 678 ◽

pp. 81-88

Author(s):

Wen Shuang Yin ◽

Dai Jun Wei ◽

Shi Qiang Chen

Keyword(s):

Lyapunov Exponent ◽

Bifurcation Diagram ◽

Chaotic System ◽

Operational Amplifier ◽

Experimental Results ◽

Order System ◽

Chaotic Attractors ◽

Maximum Lyapunov Exponent ◽

Nonlinear Functions ◽

New System

In this paper, a novel four-order system is proposed. It can generate N-attractor multi-direction multi-scroll attractor by adding simple nonlinear functions. We analyze the new system by using means of maximum Lyapunov exponent, bifurcation diagram and Poincaré maps of the system. Moreover, an minimum operational amplifier circuit is designed for realizing 2×(3×3 ×3) scroll chaotic attractors, and experimental results are also obtained, which verify chaos characteristics of the system.

Construction of a Class of High-Dimensional Discrete Chaotic Systems

Mathematics ◽

10.3390/math9040365 ◽

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 ◽

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.

Dynamics and Synchronization of a Novel Hyperchaotic System Without Equilibrium

10.1142/s0218127414500874 ◽

2014 ◽

Vol 24 (06) ◽

pp. 1450087 ◽

Cited By ~ 26

Author(s):

Viet-Thanh Pham ◽

Fadhil Rahma ◽

Mattia Frasca ◽

Luigi Fortuna

Keyword(s):

Continuous Time ◽

State Feedback ◽

Dimensional System ◽

Hyperchaotic System ◽

Third Order ◽

Circuit Realization ◽

Diffusive Coupling ◽

Dynamical Properties ◽

Time Autonomous ◽

A novel four-dimensional continuous-time autonomous hyperchaotic system which has no equilibrium is proposed in this paper. By starting from a third-order chaotic system and introducing a further variable performing state feedback, a four-dimensional system exhibiting hyperchaos is obtained. The basic dynamical properties of this system are investigated, such as equilibria and stability, Lyapunov exponent spectrum, and bifurcation diagrams. Furthermore, synchronization via diffusive coupling or control has been addressed. In the latter, parameter identification and synchronization are performed simultaneously. The circuit realization and experimental results are also presented.

Generation and Circuit Implementation of Grid Multi-Scroll Chaotic Attractors

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.336-338.1554 ◽

2013 ◽

Vol 336-338 ◽

pp. 1554-1557

Author(s):

Chao Xia Zhang

Keyword(s):

Lyapunov Exponent ◽

Numerical Simulations ◽

Experimental Results ◽

Chaotic Attractors ◽

Maximum Lyapunov Exponent ◽

Circuit Implementation

In this paper, grid multi-scroll chaotic attractors are firstly generated. The maximum Lyapunov exponent is further calculated to prove the existence of chaos in designed system. An improved module-based circuit is designed for realizing 5×3 grid scroll chaotic attractors, and the experimental results are also obtained, which is consistent with the numerical simulations.

Study on State Recognition of ASCE Benchmark Based on Lyapunov Exponent Spectrum Entropy

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.243-249.5435 ◽

2011 ◽

Vol 243-249 ◽

pp. 5435-5439 ◽

Cited By ~ 1

Author(s):

Jian Xi Yang ◽

Jian Ting Zhou ◽

Yue Chen

Keyword(s):

Lyapunov Exponent ◽

Time Sequence ◽

Maximum Lyapunov Exponent ◽

Entropy Analysis ◽

Structural System ◽

Time Duration ◽

State Recognition ◽

Chaotic Characteristic ◽

Lyapunov Exponent Spectrum ◽

Maximum Lyapunov Exponents

The paper has made a maximum Lyapunov exponent and Lyapunov exponent spectrum entropy analysis of ASCE Benchmark using non-linear theory and chaos time sequence. The maximum Lyapunov exponents in the two kinds of structural monitored data are both over zero, indicating that in the structural system chaos phenomenon has appeared. And, experiments have shown that the maximum Lyapunov exponent is sensitive of the amount of samples and the time delay. So, to compute the chaos index, the amount of samples and the time duration are of importance. Meanwhile, the Lyapunov exponent spectrum entropy is effective to measure the chaotic characteristic of the system, but ,the entropy is less sensitive to state recognition more than the max Lyapunov exponent.