How can I sketch the stable and unstable manifolds of the system $$u'=v$$ $$v' = 2u-u^3-v$$?
I know the definitions of the stable and unstable manifold, but I have no Idea how to even begin sketching it. Any help is appreciated.
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Find saddle equillibria. Find their eigenspaces: (un)stable manifold is tangent to (un)stable eigenspace. Just plot something that is more or less compatible with tangency condition and with vector field. It works as a crude approximation. – Evgeny Mar 31 '17 at 16:08
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Thankyou! I see how to do it now. But does that mean I can draw something that curves over and matches the spirals at $(\sqrt{2}, 0), (-\sqrt{2},0)$? Also, if you post that as an answer I will accept it. – Retired account Mar 31 '17 at 16:18
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Well, you can draw spirals just spiraling :) I'm a bit in a hurry now, so I prefer to try to give helpful directions in comments. I answered a similar question (IMO) a while ago, by the way. – Evgeny Mar 31 '17 at 16:55
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A small MATHEMATICA script to draw the unstable manifold which is located along a section of
$$ v \approx -2u +\frac{u^3}{7}+\frac{3u^5}{539} $$
n = 5;
V = Sum[a[k] U^k, {k, 1, n}];
res = D[V, U] V - 2 U + U^3 + V;
equs = CoefficientList[res, U];
sola = Solve[equs == 0, Table[a[k], {k, 1, n}]][[1]];
V0 = V /. sola
gr1 = StreamPlot[{v, 2 u - u^3 - v}, {u, -3, 3}, {v, -3, 3}];
gr2 = Plot[V0, {U, -1, 1}, PlotStyle -> {Thick, Red}];
Show[gr1, gr2]
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