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# Stable and unstable manifolds question watch

1. how would i start going about proving that if p belongs to the global manifold then the flow of p tends to the hyperbolic saddle point as t (time) tends to infinity?

intuitively the problem makes sense as p would also belong to the flow of the local stable manifold which goes towards the hyperbolic saddle point as t increases. Not sure how to prove that mathematically or is it acceptable to explain it like this.

got another one asking if the unstable and stable manifolds for a hyperbolic point can intersect at a point which is not a fixed point. Again intuitively im thinking no because it would mean there would be an intersection between the flow of local stable/unstable manifolds which I don't think is possible?

2. (Original post by Kim-Jong-Illest)
...
Sorry, can't help my self.

This isn't the best forum to post on. I suggest reposting on the maths forum - a subforum of this one. Mods can move threads, but it usually takes a few hours. May still take a while to get a useful response, even there.
3. BUMP - as this thread's now been moved, and is worthy of some knowledgeably person's attention.
4. (Original post by Kim-Jong-Illest)
how would i start going about proving that if p belongs to the global manifold then the flow of p tends to the hyperbolic saddle point as t (time) tends to infinity?

intuitively the problem makes sense as p would also belong to the flow of the local stable manifold which goes towards the hyperbolic saddle point as t increases. Not sure how to prove that mathematically or is it acceptable to explain it like this.
I'm very rusty on this material, but I'll attempt to help. First, are you able to clarify this question a bit? I'm unfamiliar with the term "global manifold" (do you mean "global invariant manifold"?) Second, what level of rigour is expected here? Is this an undergrad differential equations course or a graduate dynamical systems course?

got another one asking if the unstable and stable manifolds for a hyperbolic point can intersect at a point which is not a fixed point. Again intuitively im thinking no because it would mean there would be an intersection between the flow of local stable/unstable manifolds which I don't think is possible?

Isn't a homoclinic point one that lies on the intersection of the stabel and unstable manifolds of a given fixed point? I don't remember much about them, but I do recall that homoclinic orbits get very complicated very fast!

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