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I want to make sure I understand the definition of these terms. If someone could correct me or let me know if I am right I would appreciate it.

The stable manifold of a fixed point is the set of points along the flow that are sent toward the fixed point (as time moves forward).

The unstable manifold of a fixed point is the set of points along the flow that are sent away from the fixed point (as time moves forward).

Alternatively, we could swap toward and away above if we take time to move backward.

Thanks.

Zachary
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Yes and no. The stable manifold consists of points that approach the fixed point in the limit $t \to +\infty$. The unstable manifold consists of points that approach the fixed point in the limit $t \to -\infty$.

You can't generally tell that a point is in the unstable manifold by looking at what happens as time goes forward, and you can't generally tell that a point is in the stable manifold by looking at what happens as time goes backward.

Robert Israel
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    So let's say we had a fixed point in $\mathbb R^2$ that was an unstable spiral and no other fixed points. That means the stable manifold would be empty and the unstable manifold would be all of $\mathbb R^2$? – Zachary Dec 04 '14 at 00:21
  • The stable manifold would be empty. The unstable manifold might not be all of ${\mathbb R}^2$ though: there are other things that could happen, e.g. limit cycles. – Robert Israel Dec 04 '14 at 00:57
  • This was very helpful. My thanks. – Zachary Dec 04 '14 at 01:02