BIFURCATION ANALYSIS OF CHEN'S EQUATION

2000 ◽  
Vol 10 (08) ◽  
pp. 1917-1931 ◽  
Author(s):  
TETSUSHI UETA ◽  
GUANRONG CHEN
Keyword(s):  
Bifurcation Analysis ◽  
Continuous Time ◽  
Chaotic Systems ◽  
Lorenz System ◽  
Three Dimensional ◽  
Dynamical Properties ◽  
The Lorenz System

Anticontrol of chaos by making a nonchaotic system chaotic has led to the discovery of some new chaotic systems, particularly the continuous-time three-dimensional autonomous Chen's equation with only two quadratic terms. This paper further investigates some basic dynamical properties and various bifurcations of Chen's equation, thereby revealing its different features from some other chaotic models such as its origin, the Lorenz system.

scholarly journals Numerical detection of unstable periodic orbits in continuous-time dynamical systems with chaotic behaviors

2007 ◽  
Vol 14 (5) ◽  
pp. 615-620 ◽  
Author(s):  
Y. Saiki
Keyword(s):  
Dynamical Systems ◽  
Periodic Orbits ◽  
Infinite Number ◽  
Continuous Time ◽  
Chaotic Systems ◽  
Lorenz System ◽  
Complex Phenomenon ◽  
Unstable Periodic Orbits ◽  
Newton Raphson ◽  
The Lorenz System

Abstract. An infinite number of unstable periodic orbits (UPOs) are embedded in a chaotic system which models some complex phenomenon. Several algorithms which extract UPOs numerically from continuous-time chaotic systems have been proposed. In this article the damped Newton-Raphson-Mees algorithm is reviewed, and some important techniques and remarks concerning the practical numerical computations are exemplified by employing the Lorenz system.

Generating the New Chaotic Attractors of the Lorenz System Family via Anti-Controlling Method

2012 ◽  
Vol 542-543 ◽  
pp. 1042-1046 ◽  
Author(s):  
Xin Deng
Keyword(s):  
Chaotic System ◽  
Chaotic Systems ◽  
Lorenz System ◽  
Chaotic Attractors ◽  
Chen System ◽  
Lü System ◽  
New Chaotic System ◽  
Dynamical Properties ◽  
The Lorenz System ◽  
Lu System

In this paper, the first new chaotic system is gained by anti-controlling Chen system,which belongs to the general Lorenz system; also, the second new chaotic system is gained by anti-controlling the first new chaotic system, which belongs to the general Lü system. Moreover,some basic dynamical properties of two new chaotic systems are studied, either numerically or analytically. The obtained results show clearly that Chen chaotic system and two new chaotic systems also can form another Lorenz system family and deserve further detailed investigation.

A GALLERY OF LORENZ-LIKE AND CHEN-LIKE ATTRACTORS

2013 ◽  
Vol 23 (04) ◽  
pp. 1330011 ◽  
Author(s):  
XIONG WANG ◽  
GUANRONG CHEN
Keyword(s):  
Chaotic Attractor ◽  
Chaotic Systems ◽  
Lorenz System ◽  
Three Dimensional ◽  
Chaotic Attractors ◽  
Lorenz Attractor ◽  
Chen System ◽  
The Lorenz System ◽  
Algebraic Forms ◽  
General Dynamic

In this article, three-dimensional autonomous chaotic systems with two quadratic terms, similar to the Lorenz system in their algebraic forms, are studied. An attractor with two clearly distinguishable scrolls similar to the Lorenz attractor is referred to as a Lorenz-like attractor, while an attractor with more intertwine between the two scrolls similar to the Chen attractor is referred to as a Chen-like attractor. A gallery of Lorenz-like attractors and Chen-like attractors are presented. For several different families of such systems, through tuning only one real parameter gradually, each of them can generate a spectrum of chaotic attractors continuously changing from a Lorenz-like attractor to a Chen-like attractor. Some intrinsic relationships between the Lorenz system and the Chen system are revealed and discussed. Some common patterns of the Lorenz-like and Chen-like attractors are found and analyzed, which suggest that the instability of the two saddle-foci of such a system somehow determines the shape of its chaotic attractor. These interesting observations on the general dynamic patterns hopefully could shed some light for a better understanding of the intrinsic relationships between the algebraic structures and the geometric attractors of these kinds of chaotic systems.

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.

A New Category of Three-Dimensional Chaotic Flows with Identical Eigenvalues

2020 ◽  
Vol 30 (02) ◽  
pp. 2050026 ◽  
Author(s):  
Zahra Faghani ◽  
Fahimeh Nazarimehr ◽  
Sajad Jafari ◽  
Julien C. Sprott
Keyword(s):  
Chaotic Systems ◽  
Autonomous Systems ◽  
Three Dimensional ◽  
Special Property ◽  
Computer Search ◽  
Chaotic Flows ◽  
Stable Equilibria ◽  
Dynamical Properties ◽  
Simple Systems ◽  
First Time

In this paper, some new three-dimensional chaotic systems are proposed. The special property of these autonomous systems is their identical eigenvalues. The systems are designed based on the general form of quadratic jerk systems with 10 terms, and some systems with stable equilibria. Using a systematic computer search, 12 simple chaotic systems with identical eigenvalues were found. We believe that systems with identical eigenvalues are described here for the first time. These simple systems are listed in this paper, and their dynamical properties are investigated.

PROJECTIVE SYNCHRONIZATION OF UNIDENTICAL CHAOTIC SYSTEMS BASED ON STABILITY CRITERION

2006 ◽  
Vol 16 (04) ◽  
pp. 1049-1056 ◽  
Author(s):  
HONGJIE YU ◽  
JIANHUA PENG ◽  
YANZHU LIU
Keyword(s):  
Linear Systems ◽  
Stability Criterion ◽  
Chaotic Systems ◽  
Three Dimensional ◽  
New Method ◽  
Projective Synchronization ◽  
Partially Linear ◽  
Higher Dimensional ◽  
The Lorenz System ◽  
The Stability

A new method of projective synchronization of unidentical chaotic systems is proposed in this letter. This method is based on the stability criterion of linear systems. The response of two unidentical chaotic systems can synchronize up to any desired scaling factor by a suitable separation of the systems. The new method of projective synchronization is suitable not only for the three-dimensional coupled partially linear systems, but also for higher dimensional even hyperchaotic systems. The simplicity and effectiveness of the proposed method are illustrated by the Lorenz system, the four-dimensional partially linear system, the four-dimensional hyperchaotic Rösser system and Chua's circuit system as four numerical examples.

CHAOS OR TURBULENCE?

1992 ◽  
Vol 02 (04) ◽  
pp. 1005-1009 ◽  
Author(s):  
RAY BROWN ◽  
LEON O. CHUA
Keyword(s):  
Dynamical System ◽  
Lorenz System ◽  
Three Dimensional ◽  
New Technique ◽  
Chaotic Dynamical System ◽  
The Lorenz System

We present a new three-dimensional autonomous chaotic dynamical system that appears to have a closer relationship to turbulence than the Lorenz system. We have developed this system using the new technique of dynamical synthesis.

CHAOS SYNCHRONIZATION VIA CONTROLLING PARTIAL STATE OF CHAOTIC SYSTEMS

2001 ◽  
Vol 11 (06) ◽  
pp. 1737-1741 ◽  
Author(s):  
XINGHUO YU ◽  
YANXING SONG
Keyword(s):  
Invariant Manifold ◽  
Fourth Order ◽  
Chaos Synchronization ◽  
Chaotic Systems ◽  
Lorenz System ◽  
Dynamic Properties ◽  
Rossler System ◽  
Rössler System ◽  
Partial State ◽  
The Lorenz System

An invariant manifold based chaos synchronization approach is proposed in this letter. A novel idea of using only a partial state of chaotic systems to synchronize the coupled chaotic systems is presented by taking into account the inherent dynamic properties of the chaotic systems. The effectiveness of the approach and idea is tested on the Lorenz system and the fourth-order Rossler system.

Distinguishing Lorenz and Chen Systems Based Upon Hamiltonian Energy Theory

2017 ◽  
Vol 27 (02) ◽  
pp. 1750024 ◽  
Author(s):  
Shijian Cang ◽  
Aiguo Wu ◽  
Zenghui Wang ◽  
Zengqiang Chen
Keyword(s):  
Energy Function ◽  
Lorenz System ◽  
Three Dimensional ◽  
Type System ◽  
Chen System ◽  
First Order ◽  
Dimensional Phase Space ◽  
Ellipsoidal Surface ◽  
Generalized Hamiltonian Realization ◽  
The Lorenz System

Solving the linear first-order Partial Differential Equations (PDEs) derived from the unified Lorenz system, it is found that there is a unified Hamiltonian (energy function) for the Lorenz and Chen systems, and the unified energy function shows a hyperboloid of one sheet for the Lorenz system and an ellipsoidal surface for the Chen system in three-dimensional phase space, which can be used to explain that the Lorenz system is not equivalent to the Chen system. Using the unified energy function, we obtain two generalized Hamiltonian realizations of these two chaotic systems, respectively. Moreover, the energy function and generalized Hamiltonian realization of the Lü system and a four-dimensional hyperchaotic Lorenz-type system are also discussed.

When Two Dual Chaotic Systems Shake Hands

2014 ◽  
Vol 24 (06) ◽  
pp. 1450086 ◽  
Author(s):  
J. C. Sprott ◽  
Xiong Wang ◽  
Guanrong Chen
Keyword(s):  
Parameter Space ◽  
Chaotic Systems ◽  
Lorenz System ◽  
Time Variable ◽  
Chen System ◽  
Common Parameter ◽  
The Lorenz System ◽  
The Relationship

This letter reports an interesting finding that the parametric Lorenz system and the parametric Chen system "shake hands" at a particular point of their common parameter space, as the time variable t → +∞ in the Lorenz system while t → -∞ in the Chen system. This helps better clarify and understand the relationship between these two closely related but topologically nonequivalent chaotic systems.

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