scholarly journals Quantifying local predictability of the Lorenz system using the nonlinear local Lyapunov exponent

2017 ◽  
Vol 10 (5) ◽  
pp. 372-378 ◽  
Author(s):  
Xiao-Wei HUAI ◽  
Jian-Ping LI ◽  
Rui-Qiang DING ◽  
Jie FENG ◽  
De-Qiang LIU
Keyword(s):  
Lyapunov Exponent ◽  
Lorenz System ◽  
Local Lyapunov Exponent ◽  
The Lorenz System

COLOR MAP OF LYAPUNOV EXPONENTS OF INVARIANT SETS

1999 ◽  
Vol 09 (07) ◽  
pp. 1459-1463 ◽  
Author(s):  
MARCO MONTI ◽  
WILLIAM B. PARDO ◽  
JONATHAN A. WALKENSTEIN ◽  
EPAMINONDAS ROSA ◽  
CELSO GREBOGI
Keyword(s):  
Lyapunov Exponent ◽  
Parameter Space ◽  
Lorenz System ◽  
Chaotic Behavior ◽  
Invariant Sets ◽  
Largest Lyapunov Exponent ◽  
Specific Behavior ◽  
Practical Applications ◽  
The Lorenz System ◽  
Different Levels

The largest Lyapunov exponent of the Lorenz system is used as a measure of chaotic behavior to construct parameter space color maps. Each color in these maps corresponds to different values of the Lyapunov exponent and indicates, in parameter space, the locations of different levels of chaos for the Lorenz system. Practical applications of these maps include moving in parameter space from place to place without leaving a region of specific behavior of the system.

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.

A Simple Method for Generating Mirror Symmetry Composite Multiscroll Chaotic Attractors

2020 ◽  
Vol 30 (11) ◽  
pp. 2050220
Author(s):  
Xuenan Peng ◽  
Yicheng Zeng
Keyword(s):  
Lyapunov Exponent ◽  
Mirror Symmetry ◽  
Chaotic Systems ◽  
Lorenz System ◽  
Chaotic Attractors ◽  
Simple Method ◽  
Dynamical Behaviors ◽  
The Lorenz System ◽  
Lyapunov Exponent Spectrum ◽  
New System

For further increasing the complexity of chaotic attractors, a new method for generating Mirror Symmetry Composite Multiscroll Chaotic Attractors (MSCMCA) is proposed. We take the Lorenz system as an example to explain the mechanism of the method. Firstly, by varying the signs and magnitudes of the nonlinear terms, the Lorenz system generates symmetrical attractors and different-magnitude attractors, respectively. Secondly, a modified Lorenz system is constructed by imposing several unified multilevel-logic pulse signals to the Lorenz system. The new system generates a novel chaotic attractor consisting of two pairs of different-magnitude symmetrical attractors. By adjusting the parameters of the pulse signals, the modified Lorenz system can also be controlled to generate novel grid multiscroll chaotic attractors, namely MSCMCA. Several dynamical behaviors of the new system are shown by equilibria analysis and Lyapunov exponent spectrum. Moreover, the method can be applied to other chaotic systems. Finally, a circuit of the modified Lorenz system is designed by Multisim software, and the simulation result proves the effectiveness of the method.

scholarly journals Dynamic Analysis of the Nonlinear Chaotic System with Multistochastic Disturbances

2014 ◽  
Vol 2014 ◽  
pp. 1-16
Author(s):  
Lingling Geng ◽  
Yongguang Yu ◽  
Shuo Zhang
Keyword(s):  
Lyapunov Exponent ◽  
Dynamic Analysis ◽  
Orthogonal Polynomial ◽  
Chaotic System ◽  
Lorenz System ◽  
Deterministic System ◽  
Maximum Lyapunov Exponent ◽  
Nonlinear Chaotic System ◽  
The Lorenz System ◽  
The Given

The nonlinear chaotic system with multistochastic disturbances is investigated. Based on the orthogonal polynomial approximation, the method of transforming the system into an equivalent deterministic system is given. Then dynamic analysis of the nonlinear chaotic system with multistochastic disturbances can be reduced into that of its equivalent deterministic system. Especially, the Lorenz system with multistochastic disturbances is studied to demonstrate the feasibility of the given method. And its dynamic behaviors are gained including the phase portrait, the bifurcation diagram, the Poincaré section, and the maximum Lyapunov exponent.

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.

CONTROL OF THE DIFFERENTIALLY FLAT LORENZ SYSTEM

2001 ◽  
Vol 11 (07) ◽  
pp. 1989-1996 ◽  
Author(s):  
JIN MAN JOO ◽  
JIN BAE PARK
Keyword(s):  
Equilibrium State ◽  
Lorenz System ◽  
Degree Of Freedom ◽  
Design Approach ◽  
State Trajectory ◽  
System A ◽  
Full State ◽  
Tracking Controller ◽  
The Lorenz System ◽  
Two Degree Of Freedom

This paper presents an approach for the control of the Lorenz system. We first show that the controlled Lorenz system is differentially flat and then compute the flat output of the Lorenz system. A two degree of freedom design approach is proposed such that the generation of full state feasible trajectory incorporates with the design of a tracking controller via the flat output. The stabilization of an equilibrium state and the tracking of a feasible state trajectory are illustrated.

3D Printing — The Basins of Tristability in the Lorenz System

2017 ◽  
Vol 27 (08) ◽  
pp. 1750128 ◽  
Author(s):  
Anda Xiong ◽  
Julien C. Sprott ◽  
Jingxuan Lyu ◽  
Xilu Wang
Keyword(s):  
Lorenz System ◽  
Initial Conditions ◽  
Equilibrium Points ◽  
Basins Of Attraction ◽  
Symmetric Pair ◽  
State Space Approach ◽  
3D Printers ◽  
The Lorenz System ◽  
Almost All ◽  
Space Approach

The famous Lorenz system is studied and analyzed for a particular set of parameters originally proposed by Lorenz. With those parameters, the system has a single globally attracting strange attractor, meaning that almost all initial conditions in its 3D state space approach the attractor as time advances. However, with a slight change in one of the parameters, the chaotic attractor coexists with a symmetric pair of stable equilibrium points, and the resulting tri-stable system has three intertwined basins of attraction. The advent of 3D printers now makes it possible to visualize the topology of such basins of attraction as the results presented here illustrate.

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