N-dimensional Tayor's Theorem and Dynamics

[phoenixthoth]

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?

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.

 

[homology]

I've tried mathworld and wiki but I can't find the n-dimensional version of Taylor's Theorem.

http://mathworld.wolfram.com/TaylorSeries.html

Go about 4/5ths of the way down.

 

[phoenixthoth]

Oh I suppose I just need an n-dimensional Mean Value Theorem, not Taylor's Theorem.

 

[phoenixthoth]

http://mathworld.wolfram.com/TaylorSeries.html

Go about 4/5ths of the way down.

Thanks! I was looking under Taylor's Theorem, not Taylor Series. Doh!

 

[Hurkyl]

Bah, I was writing a nice post on the n-dimensional taylor series.

Do you really need a multidimensional mean value theorem? Can't you do the whole thing one dimension at a time?

 

[phoenixthoth]

Thanks anyway. But I'm not sure how I'd do it one dimension at a time. Can you please explain that in general terms? You don't mean induction do you?

After all this, I found that I had an analysis book in my little pathetic library anyway. I took one look at the n-m dimensional mean value theorem and knew that the proof would look roughly the same as it does for one dimension. Oh wait... I'm not sure how the eigenvalues being in the unit hypersphere relate. *ponders

 

[Hurkyl]

Well, in the target space, isn't the mapping attractive if and only if it is attractive in each dimension?

Analyzing the source space one dimension at a time may be possible, but it's not obvious, and I'm not sure it's necessary.

 

[Hurkyl]

Also, what about taking a simple differential approximation? (f, x, a are all vectors)

f(x) = f(a) + df (x-a) + R(x-a)

where R -> 0 as x -> a

 

[phoenixthoth]

Oh yeah, that's true. Hmm... That's good enough for me. Thanks, Hurkyl.

 

[phoenixthoth]

I'm curious to know what the eigenvalues of df have to do with it. Do you know?

 

[Hurkyl]

Well, if a is the fixed point, then the goal is:

|f(x) - a| < |x - a|

We can write the differential approximation as:

f(x) - a = (df + R) (x - a)

|f(x) - a| <= |df + R| |x - a| <= (|df| + |R|) |x - a|

Where |A| denotes the operator norm (matrix norm) of A.

That is,

$$|A| = \sup_{x \neq 0} \frac{|Ax|}{|x|}$$

If A has a complete set of eigenvectors (that is, n linearly independent eigenvectors), then it is a straightforward exercise to show that |A| is simply the largest absolute value of its eigenvalues. (I have a hunch this is true for all matrices, but I don't recall for sure)

Also, we have that |R| --> 0 as x --> a since R --> 0 as x --> a

So, if |df| < 1, we can pick a neighborhood of a such that |R| < 1 - |df|, and thus

|f(x) - a| <= |df + R| |x - a| <= (|df| + |R|) |x - a| < |x - a|

And, thus, a is an attractive fixed point.

 

[phoenixthoth]

Very cool. I had forgotten that |A| is simply the largest absolute value of its eigenvalues. Doh!

 

[Hurkyl]

It's easy to forget a lot of things until you start writing them down.