# Thread: Classification of fixed points of N-dimensional linear dynamical system?

1. ## Classification of fixed points of N-dimensional linear dynamical system?

I'm familiar with the classification of fixed points of linear dynamical systems in two dimensions; it's readily available in many a book, as well as good ol' Wiki (Linear dynamical system - Wikipedia, the free encyclopedia).

However, what happens with higher-order systems, say, three-dimensional? In that case, you'll end up having three eigenvalues -- presumably, different combinations of their signs give rise to different fixed point types. Has this been investigated? I've looked at numerous books, and all I ever seem to find is classification for two dimensions.

Any help with finding a book/paper/URL dealing with this would be much appreciated!

2. ## Re: Classification of fixed points of N-dimensional linear dynamical system?

Yes, it has been extensively investigated and depends upon the eigenvalues. If all eigenvalues are positive, you have an unstable point. If all eigenvalues are negative, you have a stable point. If some eigenvalues are positive and some negative, you have a kind of "saddlepoint" which you can "approach" along directions corresponding to eigenvectors with negative eigenvalues and move away from along directions corresponding to eigenvectors with positive eigenvalues.

3. ## Re: Classification of fixed points of N-dimensional linear dynamical system?

Thanks for the reply, HallsofIvy! I suspected it might be something like that; I also presume that complex eigenvalues will lead to other types like stable/unstable foci, as with the 2-D case.

Would you happen to know of some book or paper where this is explained? I'd like to have a source to refer to in future, but also, I'd like to cite it in a paper -- it's not by any means a requirement, but it would be extra "polish". I've had a look through quite a few books, but I haven't found anything beyond 2-D, as mentioned in the original post. (I'll happily provide a list of said books if you'd like to make sure I'm not just leeching off the forum and getting others to do my work for me )