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.

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.

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.

Jacobi analysis for an unusual 3D autonomous system

2020 ◽  
Vol 17 (04) ◽  
pp. 2050062 ◽  
Author(s):  
Chunsheng Feng ◽  
Qiujian Huang ◽  
Yongjian Liu
Keyword(s):  
Chaotic System ◽  
Stable Equilibrium ◽  
Equilibrium Points ◽  
Dynamical Behavior ◽  
Three Dimensional ◽  
Chen System ◽  
Parameter Region ◽  
Stable Equilibria ◽  
The Lorenz System ◽  
Deviation Vector

Little seems to be known about the study of the chaotic system with only Lyapunov stable equilibria from the perspective of differential geometry. Therefore, this paper presents Jacobi analysis of an unusual three-dimensional (3D) autonomous chaotic system. Under certain parameter conditions, this system has positive Lyapunov exponents and only two linear stable equilibrium points, which means that chaotic attractor and Lyapunov stable equilibria coexist. The dynamical behavior of the deviation vector near the whole trajectories (including all equilibrium points) is analyzed in detail. The results show that the value of the deviation curvature tensor at equilibrium points is only related to parameters; the two equilibrium points of the system are Jacobi stable if the parameters satisfy certain conditions. Particularly, for a specific set of parameters, the linear stable equilibrium points of the system are always Jacobi unstable. A periodic orbit that is Lyapunov stable is also proven to be always Jacobi unstable. Next, Jacobi-stable regions of the Lorenz system, the Chen system and the system under study are compared for specific parameters. It can be found that although these three chaotic systems are very similar, their regions of Jacobi stable parameters are much different. Finally, by comparing Jacobi stability with Lyapunov stability, the obtained results demonstrate that the Jacobi stable parameter region is basically symmetric with the Lyapunov stable parameter region.

Jacobi Stability Analysis of the Chen System

2019 ◽  
Vol 29 (10) ◽  
pp. 1950139 ◽  
Author(s):  
Qiujian Huang ◽  
Aimin Liu ◽  
Yongjian Liu
Keyword(s):  
Periodic Orbit ◽  
Lorenz System ◽  
Equilibrium Points ◽  
Dynamical Behavior ◽  
Chen System ◽  
Nonlinear Connection ◽  
Berwald Connection ◽  
The Lorenz System ◽  
Nonzero Equilibrium ◽  
Deviation Vector

In this paper, the research of the Jacobi stability of the Chen system is performed by using the KCC-theory. By associating a nonlinear connection and a Berwald connection, five geometrical invariants of the Chen system are obtained. The Jacobi stability of the Chen system at equilibrium points and a periodic orbit is investigated in terms of the eigenvalues of the deviation curvature tensor. The obtained results show that the origin is always Jacobi unstable, while the Jacobi stability of the other two nonzero equilibrium points depends on the values of the parameters. And a periodic orbit of the Chen system is proved to be also Jacobi unstable. Furthermore, Jacobi stability regions of the Chen system and the Lorenz system are compared. Finally, the dynamical behavior of the components of the deviation vector near the equilibrium points is also discussed.

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.

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.

Chen's attractor exists if Lorenz repulsor exists: The Chen system is a special case of the Lorenz system

2013 ◽  
Vol 23 (3) ◽  
pp. 033108 ◽  
Author(s):  
Antonio Algaba ◽  
Fernando Fernández-Sánchez ◽  
Manuel Merino ◽  
Alejandro J. Rodríguez-Luis
Keyword(s):  
Lorenz System ◽  
Chen System ◽  
The Lorenz System ◽  
Special Case

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.

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.

scholarly journals Chen System as a Controlled Weather Model — Physical Principle, Engineering Design and Real Applications

2018 ◽  
Vol 28 (04) ◽  
pp. 1830009 ◽  
Author(s):  
Pitikhate Sooraksa ◽  
Guanrong Chen
Keyword(s):  
Engineering Design ◽  
Time Reversal ◽  
Lorenz System ◽  
Physical Principle ◽  
Indoor Climate ◽  
Chen System ◽  
Weather Model ◽  
Weather Control ◽  
The Lorenz System ◽  
Control Application

This paper presents the Chen system as a controlled weather model. Mathematically, the Chen system is dual to the Lorenz system via time reversal. Physically, the Chen system can be viewed as a controlled weather model from the anti-control perspective. This paper illustrates the physical principle of this controlled weather model, and develops an engineering design of the model for real indoor climate (temperature-humidity) regulation, with a perspective on outdoor weather control application.

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