A New 7D Hyperchaotic System with Five Positive Lyapunov Exponents Coined

2018 ◽  
Vol 28 (05) ◽  
pp. 1850057 ◽  
Author(s):  
Qigui Yang ◽  
Daoyu Zhu ◽  
Lingbing Yang
Keyword(s):  
Lyapunov Exponents ◽  
Lorenz System ◽  
Linear Feedback ◽  
Hyperchaotic System ◽  
Nonlinear Feedback ◽  
Feedback Controllers ◽  
Dynamical Behaviors ◽  
The Lorenz System ◽  
Hyperbolic Equilibrium ◽  
Good Agreement

This paper reports the finding of a new seven-dimensional (7D) autonomous hyperchaotic system, which is obtained by coupling a 1D linear system and a 6D hyperchaotic system that is constructed by adding two linear feedback controllers and a nonlinear feedback controller to the Lorenz system. This hyperchaotic system has very simple algebraic structure but can exhibit complex dynamical behaviors. Of particular interest is that it has a hyperchaotic attractor with five positive Lyapunov exponents and a unique equilibrium in a large range of parameters. Numerical analysis of phase trajectories, Lyapunov exponents, bifurcation, power spectrum and Poincaré projections verifies the existence of hyperchaotic and chaotic attractors. Moreover, stability of the hyperbolic equilibrium is analyzed and a complete mathematical characterization for 7D Hopf bifurcation is given. Finally, circuit experiment implements the hyperchaotic attractor of the 7D system, showing very good agreement with the simulation results.

A New 6D Hyperchaotic System with Four Positive Lyapunov Exponents Coined

2015 ◽  
Vol 25 (04) ◽  
pp. 1550060 ◽  
Author(s):  
Qigui Yang ◽  
Waleed Mahgoub Osman ◽  
Chuntao Chen
Keyword(s):  
Lyapunov Exponents ◽  
Lorenz System ◽  
Feedback Controller ◽  
Linear Feedback ◽  
Hyperchaotic System ◽  
Nonlinear Feedback ◽  
Dynamical Behaviors ◽  
Linear Feedback Controller ◽  
The Lorenz System ◽  
Hyperbolic Equilibrium

This paper reports the finding of a new six-dimensional (6D) autonomous hyperchaotic system, which is obtained by coupling a 1D linear system and a 5D hyperchaotic system that is constructed by adding a linear feedback controller and a nonlinear feedback controller to the Lorenz system. This hyperchaotic system has very simple algebraic structure but can exhibit complex dynamical behaviors. Of particular interest is that it has a hyperchaotic attractor with four positive Lyapunov exponents and a unique equilibrium in a large range of parameters. Numerical analysis of phase trajectories, Lyapunov exponents, bifurcation, power spectrum and Poincaré projections verifies the existence of the hyperchaotic and chaotic attractors. In addition, stability of the hyperbolic equilibrium is analyzed and two complete mathematical characterizations for 6D Hopf bifurcation are given.

scholarly journals Hyperchaos Numerical Simulation and Control in a 4D Hyperchaotic System

2013 ◽  
Vol 2013 ◽  
pp. 1-16 ◽  
Author(s):  
Junhai Ma ◽  
Yujing Yang
Keyword(s):  
Lyapunov Exponents ◽  
Equilibrium Points ◽  
Linear Feedback ◽  
Hyperchaotic System ◽  
Bifurcation Diagrams ◽  
Theory Analysis ◽  
Feedback Controllers ◽  
Dynamic Simulations ◽  
Wide Range ◽  
And Control

A hyperchaotic system is introduced, and the complex dynamical behaviors of such system are investigated by means of numerical simulations. The bifurcation diagrams, Lyapunov exponents, hyperchaotic attractors, the power spectrums, and time charts are mapped out through the theory analysis and dynamic simulations. The chaotic and hyper-chaotic attractors exist and alter over a wide range of parameters according to the variety of Lyapunov exponents and bifurcation diagrams. Furthermore, linear feedback controllers are designed for stabilizing the hyperchaos to the unstable equilibrium points; thus, we achieve the goal of a second control which is more useful in application.

A Note on Hidden Transient Chaos in the Lorenz System

2017 ◽  
Vol 18 (5) ◽  
pp. 427-434 ◽  
Author(s):  
Quan Yuan ◽  
Fang-Yan Yang ◽  
Lei Wang
Keyword(s):  
Rayleigh Number ◽  
Lorenz System ◽  
Critical Value ◽  
Transient Chaos ◽  
Topological Horseshoe ◽  
One Dimensional ◽  
Dynamical Behaviors ◽  
The Lorenz System ◽  
The One

AbstractIn this paper, the classic Lorenz system is revisited. Some dynamical behaviors are shown with the Rayleigh number $\rho $ somewhat smaller than the critical value 24.06 by studying the basins characterization of attraction of attractors and tracing the one-dimensional unstable manifold of the origin, indicating some interesting clues for detecting the existence of hidden transient chaos. In addition, horseshoes chaos is verified in the famous system for some parameters corresponding to the hidden transient chaos by the topological horseshoe theory.

HOMOCLINIC AND HETEROCLINIC ORBITS AND BIFURCATIONS OF A NEW LORENZ-TYPE SYSTEM

2011 ◽  
Vol 21 (09) ◽  
pp. 2695-2712 ◽  
Author(s):  
XIANYI LI ◽  
HAIJUN WANG
Keyword(s):  
Chaotic Attractor ◽  
Lorenz System ◽  
Type System ◽  
Pitchfork Bifurcation ◽  
Heteroclinic Orbits ◽  
Chen System ◽  
Homoclinic And Heteroclinic Orbits ◽  
Dynamical Behaviors ◽  
The Lorenz System ◽  
The Stability

In this paper, a new Lorenz-type system with chaotic attractor is formulated. The structure of the chaotic attractor in this new system is found to be completely different from that in the Lorenz system or the Chen system or the Lü system, etc., which motivates us to further study in detail its complicated dynamical behaviors, such as the number of its equilibrium, the stability of the hyperbolic and nonhyperbolic equilibrium, the degenerate pitchfork bifurcation, the Hopf bifurcation and the local manifold character, etc., when its parameters vary in their space. The existence or nonexistence of homoclinic and heteroclinic orbits of this system is also rigorously proved. Numerical simulation evidences are also presented to examine the corresponding theoretical analytical results.

GENERATING HYPERCHAOTIC ATTRACTORS WITH THREE POSITIVE LYAPUNOV EXPONENTS VIA STATE FEEDBACK CONTROL

2009 ◽  
Vol 19 (02) ◽  
pp. 651-660 ◽  
Author(s):  
GUOSI HU
Keyword(s):  
Lyapunov Exponents ◽  
State Feedback ◽  
Electronic Circuit ◽  
Three Dimensional ◽  
State Feedback Control ◽  
Hyperchaotic System ◽  
Lorenz Chaotic System ◽  
Simulation Results ◽  
New Hyperchaotic System ◽  
Good Agreement

This letter presents a new hyperchaotic system, which was obtained by adding a nonlinear quadratic controller to the first equation and a linear controller to the second equation of the three-dimensional autonomous modified Lorenz chaotic system. This system uses only two multipliers but can generate very complex strange attractors with three positive Lyapunov exponents. The system is not only demonstrated by numerical simulations but also implemented via an electronic circuit, showing very good agreement with the simulation results.

THE COMPUTATION OF LYAPUNOV EXPONENTS FOR PERIODIC TRAJECTORIES

2005 ◽  
Vol 15 (12) ◽  
pp. 4075-4080 ◽  
Author(s):  
JIALIN HONG
Keyword(s):  
Lyapunov Exponents ◽  
Lorenz System ◽  
Periodic Trajectory ◽  
Computational Time ◽  
Periodic Trajectories ◽  
Linear Differential Systems ◽  
Periodic Linear ◽  
Group Methods ◽  
Linear Differential ◽  
The Lorenz System

We present a new method for the numerical computation of Lyapunov exponents of periodic trajectories which is crucial in the investigation of dynamics. The computational time of this method is merely a period of the periodic trajectory considered, when Lyapunov exponents can be approximated as precise as one wants. Our method stems from a combination between Floquet theory on periodic linear differential systems and Lie group methods in structure preserving algorithms on manifolds. The Lyapunov exponents of a periodic trajectory of the Lorenz system and a periodic solitary wave of the nonlinear Schrödinger equation with periodic coefficients are investigated by using the method.

Application of Indicators for Chaos in Chaotic Circuit Systems

2016 ◽  
Vol 26 (11) ◽  
pp. 1650182 ◽  
Author(s):  
Da-Zhu Ma ◽  
Zhi-Chao Long ◽  
Yu Zhu
Keyword(s):  
Lyapunov Exponent ◽  
Numerical Experiments ◽  
Lorenz System ◽  
Hyperchaotic System ◽  
Threshold Values ◽  
Chaotic Circuit ◽  
Qualitative Properties ◽  
Regular Motion ◽  
Fast Lyapunov Indicator ◽  
The Lorenz System

Lyapunov exponent (LE), fast Lyapunov indicator (FLI), relative finite-time Lyapunov indicator (RLI), smaller alignment index (SALI), and generalized alignment index (GALI) are some of the available methods in most conservative systems. This study focuses on the effects of the above indicators on dissipative chaotic circuit systems such as the Lorenz system and a hyperchaotic model. Numerical experiments show that the performances of the chaos indicators in the hyperchaotic system are almost similar to those in the Lorenz system. These indicators clearly provide transition from chaotic to regular motion. However, FLI, RLI, SALI, and GALI cannot describe transition from chaos to hyperchaos. These indicators are also applied to study a new four-dimensional chaotic circuit system. The basic dynamic behaviors and structures are investigated analytically and numerically. The dynamic qualitative properties of individual orbits are observed using an oscilloscope. Moreover, the entire set of LE about the parameter is found to have three threshold values. Comparisons show that all chaos indicators are able to capture chaotic and periodic motion in chaotic circuit systems, but SALI displays significantly different behavior in several periodic orbits. SALI drops exponentially to zero for “morphologically regular” orbits that are actually unstable and sensitive to perturbation. This conclusion can also be confirmed by GALI.

scholarly journals Analysis of Multiple Structural Changes in Financial Contagion Based on the Largest Lyapunov Exponents

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Rui Wang ◽  
Xiaofeng Hui ◽  
Xuechao Zhang
Keyword(s):  
Time Series ◽  
Lyapunov Exponents ◽  
Structural Changes ◽  
Lorenz System ◽  
Stock Index ◽  
Strong Nonlinearity ◽  
New Model ◽  
Multiple Structural Changes ◽  
Largest Lyapunov Exponents ◽  
The Lorenz System

A modified multiple structural changes model is built to test structural breaks of the financial system based on calculating the largest Lyapunov exponents of the financial time series. Afterwards, the Lorenz system is used as a simulation example to inspect the new model. As the Lorenz system has strong nonlinearity, the verification results show that the new model has good capability in both finding the breakpoint and revealing the changes in nonlinear characteristics of the time series. The empirical study based on the model used daily data from the S&P 500 stock index during the global financial crisis from 2005 to 2012. The results provide four breakpoints of the period, which divide the contagion into four stages: stationary, local outbreak, global outbreak, and recovery period. An additional significant result is the obvious chaos characteristic difference in the largest Lyapunov exponents and the standard deviation at various stages, particularly at the local outbreak stage.

A 5D HYPERCHAOTIC SYSTEM WITH THREE POSITIVE LYAPUNOV EXPONENTS COINED

2013 ◽  
Vol 23 (06) ◽  
pp. 1350109 ◽  
Author(s):  
QIGUI YANG ◽  
CHUNTAO CHEN
Keyword(s):  
Hopf Bifurcation ◽  
Power Spectrum ◽  
Lyapunov Exponents ◽  
Hyperchaotic System ◽  
Unique Equilibrium ◽  
Nonlinear Controller ◽  
Dynamical Behaviors ◽  
The Stability ◽  
New Hyperchaotic System ◽  
New System

This paper reports the finding of a five-dimensional (5D) new hyperchaotic system with three positive Lyapunov exponents, which is obtained by adding a nonlinear controller to the first equation of a 4D hyperchaotic system. The algebraical form of the hyperchaotic system is very similar to the 5D controlled Lorenz-like systems but they are different and, in fact, nonequivalent in topological structures. Of particular interest is the fact that the hyperchaotic system has a hyperchaotic attractor with three positive Lyapunov exponents under unique equilibrium or three equilibria. To further analyze the new system, the corresponding hyperchaotic and chaotic attractor are firstly numerically verified through investigating phase trajectories, Lyapunov exponents, bifurcation, analysis of power spectrum and Poincaré projections. Moreover, some complex dynamical behaviors such as the stability of hyperbolic or nonhyperbolic equilibrium and two complete mathematical characterizations for 5D Hopf bifurcation are rigorously derived and studied.

scholarly journals Dynamical behavior and control of a new hyperchaotic Hamiltonian system

2021 ◽  
Vol 7 (4) ◽  
pp. 5117-5132
Author(s):  
Junhong Li ◽  
◽  
Ning Cui
Keyword(s):  
Hamiltonian System ◽  
Nonholonomic Constraint ◽  
Equilibrium Points ◽  
Dynamical Behavior ◽  
Linear Feedback ◽  
Hyperchaotic System ◽  
Linear Feedback Control ◽  
Dynamical Behaviors ◽  
Complex Phenomena ◽  
And Control

<abstract><p>In this paper, we firstly formulate a new hyperchaotic Hamiltonian system and demonstrate the existence of multi-equilibrium points in the system. The characteristics of equilibrium points, Lyapunov exponents and Poincaré sections are studied. Secondly, we investigate the complex dynamical behaviors of the system under holonomic constraint and nonholonomic constraint, respectively. The results show that the hyperchaotic system can generated by introducing constraint. Additionally, the hyperchaos control of the system is achieved by applying linear feedback control. The numerical simulations are carried out in order to analyze the complex phenomena of the systems.</p></abstract>

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