I've tried mathworld and wiki but I can't find the n-dimensional version of Taylor's Theorem. Is it formulated in terms of the Jacobian?(adsbygoogle = window.adsbygoogle || []).push({});

In my dynamics book, it states that a map f from R^2 to itself has an attracting fixed point p if f(p)=p and all eigenvalues of the jacobian lie inside the unit circle, a repelling fixed point p if all eigenvalues lie outside the unit circle, and a saddle point if one is inside and one is outside.

I'm going to try to justify that terminology for myself and I think I need a higher D analog of the mean value theorem; an n-D Taylor's theorem. I guess my ultimate goal is to prove the following:

Let f be a map from R^n to itself. If p is a fixed point of f and all eigenvalues of the Jacobian of f at p are inside the n-dimensional hypersphere, then p is an attracting fixed point. By this, I mean that there is a neighborhood of p for which all points in the neighborhood converge to p upon iteration of f.

Likewise, if all eigenvalues are outside the n-dimensional unit sphere, then there is a nieghborhood N around p such that f(N) contains N and for all x in N\{p}, (f^m)(x) is not in N for some m>0.

Finally, I want to show that for saddle points, there exist functions f such that f has an attracting fixed point and functions g such that f has a repelling fixed point.

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# N-dimensional Tayor's Theorem and Dynamics

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