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!

[DrRocket]

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...._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!

 

There is nothing magic about dimension 2. The general theory applies in n dimensions in a straightforward manner -- the real part of the eigenvalues of the "A" matrix determines stability. Any good book on ordinary differential equations should present this. Roger Brockett's Finite Dimensional Linear Systems is an excellent text on linear synamical systems.

 

The non-linear case is much more difficult. for that see Lyapunov Theory.

Edited July 28, 2011 by DrRocket

[TokenMonkey]

There is nothing magic about dimension 2. The general theory applies in n dimensions in a straightforward manner -- the real part of the eigenvalues of the "A" matrix determines stability. Any good book on ordinary differential equations should present this. Roger Brockett's Finite Dimensional Linear Systems is an excellent text on linear synamical systems.

 

The non-linear case is much more difficult. for that see Lyapunov Theory.

 

Thanks for the reply.

 

The system I'm considering can definitely be linearized at the fixed points, so that will work fine.

 

I haven't got access to the book you mentioned, but I did find a nice summary (again, for 2-D systems) in James Robinson's "An Introduction to Ordinary Differential Equations" (Cambridge University Press).

 

To clarify my question further, what if, for example, you look at the eigenvalues in a 4-D system, two of which are real and > 0, and two of which are complex conjugates of each other, with real parts < 0. The real parts would then dictate that it is a saddle, but that combination (mixture of real and complex) isn't covered by any 2-D case. Is it then a special type of saddle?

 

I hope that makes sense! If not, I'll try to clarify even more.

Edited July 29, 2011 by TokenMonkey

[Schrödinger's hat]

Just look at your eigenvalues and think about the behavior if you perturbed your system slightly from your fixed point.

If one or more of your eigenvalues is has a positive real part, there exists the possibility that you'd perturb your system in such a way that the distance from your fixed point will increase. (i suppose this is roughly equivalent to a saddle)

If they are all negative then no (small) perturbation will cause the system to leave that point

If they are all positive then the point is unstable

If some of your eigenvalues are the same you will get degeneracies

and so on