The existence of homoclinic orbits in the Lorenz system via the undetermined coefficient method

2019 ◽  
Vol 355 ◽  
pp. 497-515
Author(s):  
Juan Song ◽  
Yanmin Niu ◽  
Xiong Li
Keyword(s):  
Lorenz System ◽  
Homoclinic Orbits ◽  
Undetermined Coefficient ◽  
The Lorenz System ◽  
Coefficient Method ◽  
Undetermined Coefficient Method

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.

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 Analysis of a New Quadratic 3D Chaotic Attractor

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Shahed Vahedi ◽  
Mohd Salmi Md Noorani
Keyword(s):  
Lyapunov Exponents ◽  
Chaotic System ◽  
Chaotic Attractor ◽  
Three Dimensional ◽  
Bifurcation Diagrams ◽  
Basic Properties ◽  
Undetermined Coefficient ◽  
Coefficient Method ◽  
Undetermined Coefficient Method

A new three-dimensional chaotic system is introduced. Basic properties of this system show that its corresponding attractor is topologically different from some well-known systems. Next, detailed information on dynamic of this system is obtained numerically by means of Lyapunov exponents spectrum, bifurcation diagrams, and 0-1 chaos indicator test. We finally prove existence of this chaotic attractor theoretically using Shil’nikov theorem and undetermined coefficient method.

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.

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