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

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.

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 ${\bf R}^n$.