I am trying to understand the classification of fixed points in a dynamical systems context (fixed points of a system of two linear differential equations are places where both $x_1' = x_2' = 0$).
Representing 2D systems as a matrix equation $\vec x'=\matrix A\vec x$, Strogatz classifies fixed points based on $\tau$, the trace of the matrix and $\Delta$, the determinant of the matrix:
- unstable node: $\tau>\sqrt{4\Delta}$, $\Delta>0$
- unstable spiral (spiral source): $\sqrt{4\Delta}>\tau>0$, $\Delta>0$
- neutrally stable centers: $\tau=0$, $\Delta>0$
- stable spiral (spiral sink): $-\sqrt{4\Delta}<\tau<0$, $\Delta>0$
- stable node: $\tau<-\sqrt{4\Delta}$, $\Delta>0$
- saddle point: $\Delta<0$
However, he is very vague about the boundary cases. Specifically, what happens on the parabola $\tau^2-4\Delta=0$ and the line $\Delta=0$?
Strogatz mentions that these include star nodes (decoupled systems), degenerate nodes (one unique eigendirection), and nonisolating fixed points. However, in a later problem, he mentions "isolating fixed points". What is the difference?
How are all of these nonstandard edge cases classified in terms of $\tau$ and $\Delta$?