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] Licensing

I, the author 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 | فارسی | Français | Galego | 한국어 | हिन्दी | Hrvatski | Ido | Bahasa Indonesia | Íslenska | Italiano | עברית | Latina | Lietuvių | Magyar | Bahasa Melayu | Nederlands | Norsk (bokmål) | Norsk (nynorsk) | 日本語 | Polski | Português | Ripoarish | Română | Русский | 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');