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 $\Delta$?
Best Answer
An isolated fixed point means that one can construct a region around the fixed point such that no other fixed points lie within. A nonisolated fixed point is the converse (i.e. there are other fixed points arbitrarily close; in practice, these end up being lines or a plane of fixed points).
As far as classification:
For 2D linear systems (or for linearization predictions concerning 2D nonlinear systems):
Isolated fixed point
CASE #1: Saddle Point
Nonisolated fixed points
CASE #2: Line of Ляпуно́в (Lyapunov) stable fixed points
CASE #3: Plane of fixed points
CASE #4: Line of unstable fixed points
Isolated fixed point
CASE #5: Stable Node
CASE #6: Stable Star
CASE #7: Stable Degenerate Node
CASE #8: Stable Spiral
CASE #9: Stable Center
CASE #10: Unstable Spiral
CASE #11: Unstable Star
CASE #12: Unstable Degenerate Node
CASE #13: Unstable Node
General Notes:
For 2D linear systems, the above predictions are always accurate.
For 2D nonlinear systems, when the above are used as predictions: