Classification of fixed points of N-dimensional linear dynamical system?

[TokenMonkey]

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 (http://en.wikipedia.org/wiki/Linear_dynamical_system#Classification_in_two_dimensions).

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!

 

[Octonion]

You're having trouble finding info on it because there isn't much classification to be done. For three linear equations the types of situations that can arise are much more plentiful than in two dimensions. Most work in three-dimensional systems of diff eqs is with NONLINEAR equations that give rise to chaos. Classifying the fixed points of all 3D systems linear diff eqs would be a hell of a waste of time when the classification itself does not provide much insight. The use of the Trace-determinant plane picture of fixed points in two dimensions is to apply them to the local behavior of corresponding fixed points of nonlinear systems. But in 3D nonlinear systems give rise to chaos and thus a classification of the linear fixed points would not have much of a use.

 

[TokenMonkey]

Well, the thing is that my system is actually nonlinear, but linearisable at the fixed points. From what I've gathered from other forums, textbooks, and an applied-maths-educated friend, while the exact classifications may not be straightforward, stability is. More specifically, the real parts of the eigenvalues: if they are all positive, unstable node; if they are all negative, we have a stable node; if there is a mixture of positive and negative, we're at a saddle point. In this sense, the 2-D case extends to the N-D case.

Would you say this is incorrect?

 

[Octonion]

If all of the eigenvalues are negative then the fixed point in n-dimensional space is asymptotically stable and if all three are positive then the fixed point is unstable but the term node, as far as I've heard it used in my courses, was reserved for 2D systems. When we did three dimensional systems our Professor, Steven Strogatz, focused in on his research which is in chaotic systems so in all reality I didn't see too much of any other type of three dimensional system. We worked with strange attractors and those don't have any linearisable fixed points.

 

[Bob-Shandy]

Hi! To this subject, I have some more details to give!

With higher-order systems, let’s say three-dimensional, you have about height possible cases:
Node if all the eigenvalues are real and negative; Repellor if all positive reals; Saddle point index 1 if all real with one positive and others negatives; Saddle point index 2 if all real with rather one negative; Spiral node if one real and two complex conjugate but all of them with negative real parts; Spiral repellor if one real and two complex conjugate but all with positive real parts; Spiral saddle index 1 if one positive real and the two others complex conjugate with negative real parts and finally, Spiral saddle index 2 in case one negative real and the two others complex conjugate with positive real parts.

For more information, see Book by Julien Clinton Sprott, Chaos and Time-Series Analysis, Oxford University Press Inc., New York 2003