Image:Heteroclinic orbit in pendulum phaseportrait.png

[edit] Summary

Phaseportrait for the pendulum equation with the heteroclinic orbit highlighted. Created by Jitse Niesen using Matlab.

[edit] Discussion

How come the orbit isn't called homoclinic? The domain is periodic: starting and ending point are the same.

That depends on what you consider as the domain. If the domain is a circle (and hence periodic), which is the most natural choice, then you're right and the orbit is homoclinic. If the domain is R, the set of real numbers, then the starting and ending point are not the same. But you certainly have a point that this is a confusing example; thanks for that. -- Jitse Niesen 06:45, 2 February 2007 (UTC)

[edit] Licensing

I, the copyright holder of this work, hereby release it into the public domain. This applies worldwide. In case this is not legally possible: I grant anyone the right to use this work for any purpose, without any conditions, unless such conditions are required by law. Afrikaans | Alemannisch | Aragonés | العربية‎ | Български | Català | Česky | Cymraeg | Dansk | Deutsch | Ελληνικά | English | Español | Esperanto | Euskara | فارسی‎ | Français | Galego | 한국어 | हिन्दी | Hrvatski | Ido | Bahasa Indonesia | Íslenska | Italiano | עברית‎ | Kurdî / كوردي | Latina | Lietuvių | Magyar | Bahasa Melayu | Nederlands | Norsk (bokmål) | Norsk (nynorsk) | 日本語 | Polski | Português | Ripoarish | Română | Русский | Shqip | Slovenčina | Slovenščina | Српски | Svenska | ไทย | Türkçe | Українська | Tiếng Việt | Walon | 简体中文 | 繁體中文 | 粵語 | +/-

[edit] Matlab source

clf; 
axis([-2*pi 2*pi -3 3]);
daspect([1 1 1]);
hold on;

% Draw constant energy contours
qs = linspace(-2*pi, 2*pi, 101);
[Q,P] = meshgrid(qs, linspace(-3,3));
H = P.*P/2 - cos(Q);
contour(Q,P,H, [-0.95 -0.5 0.3  2 4], 'k'); 

% Draw energy = 0 contour
ps = sqrt(2+2*cos(qs));
plot(qs,ps, 'k');
plot(qs,-ps, 'k');

% Draw heteroclinic connection
qs = linspace(-pi, pi, 101);
ps = sqrt(2+2*cos(qs));
plot(qs,ps, 'r', 'LineWidth', 3);
plot([-pi pi], [0 0], 'r.', 'MarkerSize', 25);

% Arrows
plot(-pi+[-0.10 0.05], sqrt(6)+[0.05 0], 'k');
plot(-pi+[-0.10 0.05], sqrt(6)+[-0.05 0], 'k');
plot(pi+[-0.10 0.05], sqrt(2)+[0.05 0], 'k');
plot(pi+[-0.10 0.05], sqrt(2)+[-0.05 0], 'k');
plot([-0.10 0.05], [1.05 1], 'k');
plot([-0.10 0.05], [0.95 1], 'k');
plot([0.10 -0.05], -sqrt(2.6)+[0.05 0], 'k');
plot([0.10 -0.05], -sqrt(2.6)+[-0.05 0], 'k');
plot(-pi+[0.10 -0.05], -sqrt(2)+[0.05 0], 'k');
plot(-pi+[0.10 -0.05], -sqrt(2)+[-0.05 0], 'k');
plot(pi+[0.10 -0.05], -sqrt(6)+[0.05 0], 'k');
plot(pi+[0.10 -0.05], -sqrt(6)+[-0.05 0], 'k');
plot([-0.2 0.2], [2.1 2], 'r', 'LineWidth', 3);
plot([-0.2 0.2], [1.9 2], 'r', 'LineWidth', 3);

% Axes
xlabel('\it{x}');
ylabel('\it{x}''');
set(gca, 'XTick', [-2*pi -pi 0 pi 2*pi]);
set(gca, 'XTickLabel', {'-2pi' '-pi' '0' 'pi' '2pi'});

% Print
print -dpng 'heteroclinic_tmp.png';
system('convert -trim -bordercolor white -border 10 +repage heteroclinic_tmp.png heteroclinic.png');