From what I can find, the proof of the Centre Manifold Theorem uses the Contracting Mapping Theorem which gives a unique solution. So my question is, where does the non-uniqueness come from in the proof? I am very well aware of the examples in which the Centre Manifold can be non-unique, but I don't see where it comes from.

Furthermore, where in the proof of the Hartman-Grobman Theorem does the fact that the equilibrium is hyperbolic enter the picture. More precisely, how would the Hartman-Grobman Theorem fail if the equilibrium is non-hyperbolic.

Thank you in advance. 

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