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I'm looking to compute the global stable and unstable manifolds for the equilibrium points of the following autonomous vector field on the plane. Apologies for the formatting, I haven't figured out MathJax yet.

x'= -x

y'= y - y^4

For x and y in the Real numbers

So far I've computed the equilibrium points at (0,0) and (0,1) and their linear stability as a saddle point and a sink respectively.

I'm not sure how to continue with the question and would appreciate any pointers.

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    So, what's exactly your problem? Have you tried to sketch the phase portrait first? Do you know the relation between (un)stable manifolds and eigenspaces of Jacobi matrix? – 2017-01-07
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    Yes I've sketched the phase portrait and I think I understand which are the stable and unstable manifolds but I'm unsure of the notation on how to write them. So I've got that stable manifold for (0,0) is just the x axis and the unstable is y < 1, and the stable for (0,1) is y>0. Atleast I think. Could be very wrong. – 2017-01-09
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    I'm not sure I do know the relation between stable or unstable manifolds and the eigenspaces of the jacbian matrix. I understand by looking at the eigen vectors you can tell what kind of stability an equilibrium point has but nothing to do with manifolds. – 2017-01-09
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    To me it seems that you are on the right way. By the way, **eigenvalues** are enough to tell you about stability. Eigenvectors are for the following. When you have linear system with hyperbolic eigenvalues (i.e. no eigenvalues with zero real part) you have two linear subspaces: stable, which is spanned by eigenvectors (or root vectors) that correspond to eigenvalues with negative real part, and unstable one. In nonlinear case these subspaces still can be extracted from Jacobi matrix, but they are not invariant. Instead you have stable and unstable manifolds that are tangent to these subspaces. – 2017-01-09
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    By the way, you can refer to [this answer](http://math.stackexchange.com/questions/1241375/a-differentiable-manifold-of-class-mathcalcr-tangent-to-e-pm-and-re/1241421#1241421) – 2017-01-09

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