Resumo

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)

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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');