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I'm looking for a reference to a proof of the theorem of stable / unstable manifolds , using the Hadamard method, ie, characterizing the stable / unstable manifolds locally as the graph of a differentiable function.

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I am not sure about the way they were proved, but in Perko's "Dynamical systems" these theorems are nicely given, and the book is quite well-known – Ilya Mar 29 '12 at 17:20
For what kind of situations ? Hyperbolic fixed point ? Invariant hyperbolic compact set ? Anyway, you should take a look at the book by Katok and Hasselblatt. – Ahriman Sep 27 '12 at 15:28

I believe you can find this proof in chapter $5$ of "Introduction to Dynamical Systems" by Michael Brin and Garrett Stuck. I highly recommend it.

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