CHAOTIC ATTRACTORS OF THE CONJUGATE LORENZ-TYPE SYSTEM

2007 ◽  
Vol 17 (11) ◽  
pp. 3929-3949 ◽  
Author(s):  
QIGUI YANG ◽  
GUANRONG CHEN ◽  
KUIFEI HUANG
Keyword(s):  
Numerical Simulations ◽  
Chaotic Dynamics ◽  
Lorenz System ◽  
Type System ◽  
Chaotic Attractors ◽  
Compound Structure ◽  
Chen System ◽  
Numerical Examples ◽  
Special Cases ◽  
Dynamical Properties

A new conjugate Lorenz-type system is introduced in this paper. The system contains as special cases the conjugate Lorenz system, conjugate Chen system and conjugate Lü system. Chaotic dynamics of the system in the parametric space is numerically and thoroughly investigated. Meanwhile, a set of conditions for possible existence of chaos are derived, which provide some useful guidelines for searching chaos in numerical simulations. Furthermore, some basic dynamical properties such as Lyapunov exponents, bifurcations, routes to chaos, periodic windows, possible chaotic and periodic-window parameter regions and the compound structure of the system are demonstrated with various numerical examples.

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 MODIFIED GENERALIZED LORENZ-TYPE SYSTEM AND ITS CANONICAL FORM

2009 ◽  
Vol 19 (06) ◽  
pp. 1931-1949 ◽  
Author(s):  
QIGUI YANG ◽  
KANGMING ZHANG ◽  
GUANRONG CHEN
Keyword(s):  
Special Form ◽  
Topological Structure ◽  
Complex Dynamics ◽  
Type System ◽  
Chaotic Attractors ◽  
Hopf Bifurcations ◽  
Compound Structure ◽  
Two Parameters ◽  
First Time ◽  
New System

In this paper, a modified generalized Lorenz-type system is introduced, which is state-equivalent to a simple and special form, and is parameterized by two parameters useful for chaos turning and system classification. More importantly, based on the parameterized form, two classes of new chaotic attractors are found for the first time in the literature, which are similar but nonequivalent in topological structure. To further understand the complex dynamics of the new system, some basic properties such as Lyapunov exponents, Hopf bifurcations and compound structure of the attractors are analyzed and demonstrated with careful numerical simulations.

More Dynamical Properties Revealed from a 3D Lorenz-like System

2014 ◽  
Vol 24 (10) ◽  
pp. 1450133 ◽  
Author(s):  
Haijun Wang ◽  
Xianyi Li
Keyword(s):  
Numerical Simulations ◽  
Local Stability ◽  
Equilibrium Points ◽  
Heteroclinic Orbits ◽  
Mathematical Work ◽  
Chaotic Attractors ◽  
Heteroclinic Cycles ◽  
Homoclinic And Heteroclinic Orbits ◽  
Dynamical Properties ◽  
Theoretical Results

After a 3D Lorenz-like system has been revisited, more rich hidden dynamics that was not found previously is clearly revealed. Some more precise mathematical work, such as for the complete distribution and the local stability and bifurcation of its equilibrium points, the existence of singularly degenerate heteroclinic cycles as well as homoclinic and heteroclinic orbits, and the dynamics at infinity, is carried out in this paper. In particular, another possible new mechanism behind the creation of chaotic attractors is presented. Based on this mechanism, some different structure types of chaotic attractors are numerically found in the case of small b > 0. All theoretical results obtained are further illustrated by numerical simulations. What we formulate in this paper is to not only show those dynamical properties hiding in this system, but also (more mainly) present a kind of way and means — both "locally" and "globally" and both "finitely" and "infinitely" — to comprehensively explore a given system.

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.

Generation of Periodic Orbits from Chaotic Attractors of Weakly Coupled Diode Resonators

1997 ◽  
Vol 07 (04) ◽  
pp. 897-902
Author(s):  
Jong Cheol Shin ◽  
Sook-Il Kwun ◽  
Youngtae Kim
Keyword(s):  
Periodic Orbits ◽  
Chaotic Dynamics ◽  
Chaotic Attractors ◽  
Unstable Periodic Orbits ◽  
Chaotic Motions ◽  
Small Perturbations ◽  
Controlling Chaos ◽  
Weakly Coupled ◽  
Dynamical Properties

We have designed coupled diode resonators to study the effect of small perturbations due to weak symmetric coupling on chaotic dynamics. Our experiment clearly demonstrated that chaos of the diode resonators was suppressed so that chaotic motions were converted into periodic ones with small modifications to the attractor when an appropriate coupling signal perturbed the diode resonators. Many unstable periodic orbits were stabilized and they were very stable depending on the dynamical properties of the coupling signals. Our results suggest that coupling of signals belonging to the same class is effective in controlling chaos.

DYNAMICAL ANALYSIS OF A CHAOTIC SYSTEM WITH TWO DOUBLE-SCROLL CHAOTIC ATTRACTORS

2004 ◽  
Vol 14 (03) ◽  
pp. 971-998 ◽  
Author(s):  
WENBO LIU ◽  
GUANRONG CHEN
Keyword(s):  
Chaotic System ◽  
Three Dimensional ◽  
Dynamical Analysis ◽  
Chaotic Attractors ◽  
Compound Structure ◽  
Numerical Examples ◽  
Routes To Chaos ◽  
Basic Properties ◽  
Dynamical Behaviors

Dynamical behaviors of a three-dimensional autonomous chaotic system with two double-scroll attractors are studied. Some basic properties such as bifurcation, routes to chaos, periodic windows and compound structure are demonstrated with various numerical examples. System equilibria and their stabilities are discussed, and chaotic features of the attractors are justified numerically.

ON A GENERALIZED LORENZ CANONICAL FORM OF CHAOTIC SYSTEMS

2002 ◽  
Vol 12 (08) ◽  
pp. 1789-1812 ◽  
Author(s):  
SERGEJ ČELIKOVSKÝ ◽  
GUANRONG CHEN
Keyword(s):  
Large Class ◽  
Canonical Form ◽  
Chaotic Systems ◽  
Lorenz System ◽  
Chen System ◽  
Scalar Parameter ◽  
Generalized Lorenz System ◽  
Special Cases ◽  
The One ◽  
Generalized Lorenz Canonical Form

This paper shows that a large class of systems, introduced in [Čelikovský & Vaněček, 1994; Vaněček & Čelikovský, 1996] as the so-called generalized Lorenz system, are state-equivalent to a special canonical form that covers a broader class of chaotic systems. This canonical form, called generalized Lorenz canonical form hereafter, generalizes the one introduced and analyzed in [Čelikovský & Vaněček, 1994; Vaněček & Čelikovský, 1996], and also covers the so-called Chen system, recently introduced in [Chen & Ueta, 1999; Ueta & Chen, 2000].Thus, this new generalized Lorenz canonical form contains as special cases the original Lorenz system, the generalized Lorenz system, and the Chen system, so that a comparison of the structures between two essential types of chaotic systems becomes possible. The most important property of the new canonical form is the parametrization that has precisely a single scalar parameter useful for chaos tuning, which has promising potential in future engineering chaos design. Some other closely related topics are also studied and discussed in the paper.

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.

A UNIFIED LORENZ-TYPE SYSTEM AND ITS CANONICAL FORM

2006 ◽  
Vol 16 (10) ◽  
pp. 2855-2871 ◽  
Author(s):  
QIGUI YANG ◽  
GUANGRONG CHEN ◽  
TIANSHOU ZHOU
Keyword(s):  
Canonical Form ◽  
Special Form ◽  
Lorenz System ◽  
Type System ◽  
Chaotic Attractors ◽  
Lorenz Attractor ◽  
System A ◽  
Generalized Lorenz System ◽  
Two Parameters ◽  
First Time

Based on the generalized Lorenz system, a conjugate Lorenz-type system is introduced, and a new unified Lorenz-type system containing these two classes of systems is naturally constructed in the paper. Such a unified system is state-equivalent to a simple special form, which is parameterized by two parameters useful for chaos turning and system classification. More importantly, based on the parameterized form, three new chaotic attractors, called conjugate attractors, are found for the first time, which are conjugate to the Lorenz attractor, the Chen attractor, and the Lü attractor, respectively.

3D Grid Multi-Wing Chaotic Attractors

2018 ◽  
Vol 28 (04) ◽  
pp. 1850045 ◽  
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.

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