Classifying local behavior of fixed points using eigenvalues from linear stability analysis of 3D system

Question

I've learned about classification of fixed points of 2D systems using linear stability analysis and I am wondering how if at all I can apply the same process to analyzing local behavior of fixed points in 3D systems.

I am specifically wondering about these cases:

• 1 positive, 2 distinct negative real eigenvalues
• 1 positive, 2 repeated negative real eigenvalues
• 1 positive, 2 complex eigenvalues with negative real part
• 1 negative, 2 complex eigenvalues with negative real part

In the first two cases I am guessing that it works something like 2D: the stable manifold is a 2D manifold and the unstable manifold is a curve. I am wondering what the difference would be between 2 distinct versus 2 repeated would be.

In the last two cases I am having trouble visualing what this would even look like.

Any help or resources are appreciated.

Answer 1

In the last two cases I am having trouble visualing what this would even look like.

Have a look at Fig. 2.4 (b) of this book.

In general, I do not know many sources that specifically distinguish between various cases as on the plane (note, e.g., that stable node is topologically equivalent to a stable focus, and so forth). However, in a very old book by Nemytskii and Stepanov you can find a very detailed classification of various types of equilibria in ${\bf R}^n$.

Answer 2

In last two cases, the positive (negative) eigenvalue will correspond to a 1-D unstable(stable) manifold, normal to a plane in which you will see the stable focus.