File:Example of application of Poincaré–Bendixson theorem.png

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Example_of_application_of_Poincaré–Bendixson_theorem.png ‎(415 × 580 pixels, file size: 64 KB, MIME type: image/png)

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English: Example of application of Poincaré–Bendixson theorem. The figure shows the vector field of autonomous two dimensional differential equations

There is a square space closed by 0 ≤ x ≤ 10, 0 ≤ y ≤ 101 (blue area), and any vector on the boundary flow inward or is tangent to the boundary. There also is an equilibrium point (2, 5) and it is repelling. According to Poincaré–Bendixson theorem, a closed orbit (thick black closed curve in the figure) exists inside the square space except for the equilibrium point. Five curves with colors in the figure are the orbits of this differential equations. They are attracted to the black closed curve.

Reference: 千葉 逸人 (2021) 解くための微分方程式と力学系理論, 現代数学社, p. 204 ISBN: 978-4-7687-0570-4.
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Source Own work
Author Yapparina

Scilab source

Scilab source
Scilab source
 This media was created with Scilab, a free open-source software.

Here is a listing of the Scilab source used to create this file. 

clear;
clf;

function dX = LC(t,X) //  X(1) = x,  X(2) = y
    dX=zeros(2,1);
  dX(1) = 10 - X(1) - 4 * X(1) * X(2) / (1 + X(1)^2) ; 
  dX(2) = X(1) * (1 - X(2) / (1 + X(1)^2)) ; 
endfunction

xc=-1:1:11;
yc=-10:10:110;

fchamp(LC,0,xc,yc,1)

T = linspace(0,100,100000);
x01 = [0 ; 0];
x1 = ode(x01,0,T,LC);
x02 = [1.0143 ; 4.06];
x2 = ode(x02,0,T,LC);
x03 = [10 ; 0];
x3 = ode(x03,0,T,LC);
x04 = [10 ; 101];
x4 = ode(x04,0,T,LC);
x05 = [10 ; 50];
x5 = ode(x05,0,T,LC);
x06 = [10 ; 20];
x6 = ode(x06,0,T,LC);
plot(x1(1,:),x1(2,:),x2(1,:),x2(2,:),x3(1,:),x3(2,:),x4(1,:),x4(2,:),x5(1,:),x5(2,:),x6(1,:),x6(2,:));

zoom_rect([-1,-10,11,110])

xlabel('x');
ylabel('y');
gh=gca();
gh.font_style=2;
gh.font_size=5;
gh.x_label.font_style=3;
gh.x_label.font_size=6;
gh.y_label.font_style=3;
gh.y_label.font_size=6;

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I, the copyright holder of this work, hereby publish it under the following license:
Creative Commons CC-Zero This file is made available under the Creative Commons CC0 1.0 Universal Public Domain Dedication.
The person who associated a work with this deed has dedicated the work to the public domain by waiving all of their rights to the work worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law. You can copy, modify, distribute and perform the work, even for commercial purposes, all without asking permission.

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current13:00, 17 May 2023Thumbnail for version as of 13:00, 17 May 2023415 × 580 (64 KB)Yapparina (talk | contribs)Uploaded own work with UploadWizard

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