CHEN'S ATTRACTOR EXISTS

2004 ◽  
Vol 14 (09) ◽  
pp. 3167-3177 ◽  
Author(s):  
TIANSHOU ZHOU ◽  
YUN TANG ◽  
GUANRONG CHEN
Keyword(s):  
Homoclinic Orbit ◽  
Heteroclinic Orbit ◽  
Homoclinic Orbits ◽  
Heteroclinic Orbits ◽  
Series Expansions ◽  
Chen System ◽  
Homoclinic And Heteroclinic Orbits ◽  
Undetermined Coefficient ◽  
Coefficient Method ◽  
Undetermined Coefficient Method

By applying the undetermined coefficient method, this paper finds homoclinic and heteroclinic orbits in the Chen system. It analytically demonstrates that the Chen system has one heteroclinic orbit of Ši'lnikov type that connects two nontrivial singular points. The Ši'lnikov criterion guarantees that the Chen system has Smale horseshoes and the horseshoe chaos. In addition, there also exists one homoclinic orbit joined to the origin. The uniform convergence of the series expansions of these two types of orbits are proved in this paper. It is shown that the heteroclinic and homoclinic orbits together determine the geometric structure of Chen's attractor.

On Shil’nikov Analysis of Homoclinic and Heteroclinic Orbits of the T System

2012 ◽  
Vol 8 (2) ◽  
Author(s):  
Antonio Algaba ◽  
Fernando Fernández-Sánchez ◽  
Manuel Merino ◽  
Alejandro J. Rodríguez-Luis
Keyword(s):  
Heteroclinic Orbit ◽  
Heteroclinic Orbits ◽  
Homoclinic And Heteroclinic Orbits ◽  
Undetermined Coefficient ◽  
Global Connections ◽  
Coefficient Method ◽  
T System ◽  
Undetermined Coefficient Method

In the referenced paper, the authors use the undetermined coefficient method to analytically construct homoclinic and heteroclinic orbits in the T system. Unfortunately their method is not valid because they assume odd functions for the first component of the homoclinic and the heteroclinic orbit whereas these Shil'nikov global connections do not exhibit symmetry.

The Stability and Chaotic Motions of a Four-Wing Chaotic Attractor

2011 ◽  
Vol 48-49 ◽  
pp. 1315-1318 ◽  
Author(s):  
Xia Wang ◽  
Jian Ping Li ◽  
Jian Yin Fang
Keyword(s):  
Chaotic Attractor ◽  
Autonomous System ◽  
Equilibrium Points ◽  
Heteroclinic Orbits ◽  
Series Expansions ◽  
Chaotic Motions ◽  
Undetermined Coefficient ◽  
The Stability ◽  
Coefficient Method ◽  
Undetermined Coefficient Method

The stability and chaotic motions of a 3-D quadratic autonomous system with a four-wing chaotic attractor are investigated in this paper. Base on the linearization analysis, the stability of the equilibrium points is studied. By using the undetermined coefficient method, the heteroclinic orbits are found and the convergence of the series expansions of this type of orbits is proved. It analytically demonstrates that there exist heteroclinic orbits of Silnikov type connecting the equilibrium points. Therefore, Smale horseshoes and the horseshoe chaos occur for this system via the Silnikov criterion.

SHIL'NIKOV HETEROCLINIC ORBITS IN A CHAOTIC SYSTEM

2007 ◽  
Vol 21 (25) ◽  
pp. 4429-4436 ◽  
Author(s):  
FENG-YUN SUN
Keyword(s):  
Chaotic System ◽  
Chaotic Attractor ◽  
Heteroclinic Orbits ◽  
Geometric Structures ◽  
Undetermined Coefficient ◽  
Coefficient Method ◽  
Undetermined Coefficient Method

In this paper, a chaotic system which exhibits a chaotic attractor with only three equilibria for some parameters is considered. The existence of heteroclinic orbits of the Shil'nikov type in a chaotic system has been proved using the undetermined coefficient method. As a result, the Shil'nikov criterion guarantees that the system has Smale horseshoes. Moreover, the geometric structures of the attractor are determined by these heteroclinic orbits.

scholarly journals Heteroclinic, Homoclinic and Closed Orbits in the Chen System

2016 ◽  
Vol 26 (04) ◽  
pp. 1650072 ◽  
Author(s):  
G. Tigan ◽  
J. Llibre
Keyword(s):  
Nonlinear System ◽  
Homoclinic Orbits ◽  
Heteroclinic Orbits ◽  
Chen System ◽  
Homoclinic And Heteroclinic Orbits ◽  
Closed Orbits

Bounded orbits such as closed, homoclinic and heteroclinic orbits are discussed in this work for a Lorenz-like 3D nonlinear system. For a large spectrum of the parameters, the system has neither closed nor homoclinic orbits but has exactly two heteroclinic orbits, while under other constraints the system has symmetrical homoclinic orbits.

CLASSIFICATION OF CHAOS IN 3-D AUTONOMOUS QUADRATIC SYSTEMS-I: BASIC FRAMEWORK AND METHODS

2006 ◽  
Vol 16 (09) ◽  
pp. 2459-2479 ◽  
Author(s):  
TIANSHOU ZHOU ◽  
GUANGRONG CHEN
Keyword(s):  
Homoclinic Orbit ◽  
Chaotic Attractor ◽  
Three Dimensional ◽  
Heteroclinic Orbit ◽  
Homoclinic Orbits ◽  
Classification Problem ◽  
Heteroclinic Orbits ◽  
Orbit Type ◽  
Quadratic Systems ◽  
Hybrid Type

This paper is part I of a series of contributions on the classification problem of chaos in three-dimensional autonomous quadratic systems. We try to classify chaos, based on the Ši'lnikov criteria, in such a large class of systems into the following four types: (1) chaos of the Ši'lnikov homoclinic orbit type; (2) chaos of the Ši'lnikov heteroclinic orbit type; (3) chaos of the hybrid type; i.e. those with both Ši'lnikov homoclinic and homoclinic orbits; (4) chaos of other types. We are especially interested in finding out all the simplest possible forms of chaotic systems for each type of chaos. Our main contributions are to develop some effective classification methods and to provide a basic classification framework under which each of the four types of chaos can be justified by some examples that are useful for describing the feasibility and procedure of the classification. In particular, we show several novel chaotic attractors, e.g. one hybrid-type chaotic attractor with three equilibria, one heteroclinic orbit and one homoclinic orbit, and one 4-scroll chaotic attractor with five equilibria and two heteroclinic orbits.

New Heteroclinic Orbits Coined

2016 ◽  
Vol 26 (12) ◽  
pp. 1650194 ◽  
Author(s):  
Haijun Wang ◽  
Chang Li ◽  
Xianyi Li
Keyword(s):  
Homoclinic Orbit ◽  
Algebraic Surface ◽  
Type System ◽  
Homoclinic Orbits ◽  
Heteroclinic Orbits ◽  
Homoclinic And Heteroclinic Orbits ◽  
Invariant Algebraic Surface

We devote to studying the problem for the existence of homoclinic and heteroclinic orbits of Unified Lorenz-Type System (ULTS). Other than the known results that the ULTS has two homoclinic orbits to [Formula: see text] for [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and two heteroclinic orbits to [Formula: see text] for [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] on its invariant algebraic surface [Formula: see text], formulated in the literature by Yang and Chen [2014], we seize two new heteroclinic orbits of this Unified Lorenz-Type System. Namely, we rigorously prove that this system has another two heteroclinic orbits to [Formula: see text] and [Formula: see text] while no homoclinic orbit when [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text].

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.

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