The Method for Solving Homoclinic∕Heteroclinic Orbits- The Undetermined Coefficient Method and its applications for a New Chaotic System

2010 ◽  
Author(s):  
Yushu Chen ◽  
Liangqiang Zhou ◽  
Fangqi Chen ◽  
Jane W. Z. Lu ◽  
Andrew Y. T. Leung ◽  
...  
Keyword(s):  
Chaotic System ◽  
Heteroclinic Orbits ◽  
New Chaotic System ◽  
Undetermined Coefficient ◽  
Coefficient Method ◽  
Undetermined Coefficient Method

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.

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.

scholarly journals Evaluation of the Trapped Surface Wave of a Vertical Electric Dipole Based on Undetermined Coefficient Method in the Presence of N-Layered Region: A Graphical Approach

2019 ◽  
Vol 2019 ◽  
pp. 1-12 ◽  
Author(s):  
Hong Lei Xu ◽  
Ting Ting Gu ◽  
Yong Zhu ◽  
Xiao Wei ◽  
Liang Sheng Li ◽  
...  
Keyword(s):  
Surface Wave ◽  
Electric Dipole ◽  
Observation Point ◽  
Dipole Source ◽  
Graphical Approach ◽  
Vertical Electric Dipole ◽  
Undetermined Coefficient ◽  
Trapped Surface ◽  
Coefficient Method ◽  
Undetermined Coefficient Method

In previous studies, the trapped surface wave, which is defined by the residue sums, has been addressed in the evaluation of the Sommerfeld integrals describing electromagnetic field of a vertical dipole in the presence of three-layered or four-layered region. But unfortunately, the existing computational scheme cannot provide analytical solution of the field in the presence of the N-layered region when N > 4. The scope of this paper is to overcome the limitations in root finding algorithm implied by the previous approach and provide solution of poles in stratified media. A set of pole equations following with explicit expressions are derived based on the undetermined coefficient method, which enable a graphical approach to obtain initial values of real roots. Accordingly, the generated trapped surface wave components are computed when both the observation point and the electric dipole source are on or near the surface of a dielectric-coated conductor. Validity, efficiency, and accuracy of the proposed method are illustrated by numerical examples.

scholarly journals SOLUSI DARI PERSAMAAN CAUCHY–EULER NONHOMOGEN KASUS LOGARITMIK

2020 ◽  
Vol 9 (2) ◽  
pp. 125
Author(s):  
I GEDE PUTU MIKI SUKADANA ◽  
I NYOMAN WIDANA ◽  
KETUT JAYANEGARA
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Parameter Variation ◽  
Undetermined Coefficient ◽  
Constant Coefficients ◽  
Particular Solution ◽  
Ordinary Solution ◽  
The Right ◽  
Coefficient Method ◽  
Undetermined Coefficient Method

Ordinary differential equation is one form of differential equations that are often found in everyday life. One form of ordinary differential equations which has non–constant coefficients is the Cauchy–Euler differential equation. In the nonhomogeneous Cauchy–Euler differential equations, the undetermined coefficient and the parameter variation were the most method that often used to find the particular solution. This paper aimed to show a new solution that was shorter than the previous methods for nonhomogeneous Cauchy–Euler differential equations with the right side was a logarithmic form. The new solution had been proven to produce the same solution as the ordinary solution sought using the undetermined coefficient method.

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