# Isolated Versus Non-Isolated Fixed Point, 2D Dynamics

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 $\Delta$?

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):

• if $\Delta<0$:
Isolated fixed point
CASE #1: Saddle Point
• if $\Delta=0$:
Nonisolated fixed points
• if $\tau<0$:
CASE #2: Line of Ляпуно́в (Lyapunov) stable fixed points
• if $\tau=0$:
CASE #3: Plane of fixed points
• if $\tau>0$:
CASE #4: Line of unstable fixed points
• if $\Delta>0$:
Isolated fixed point
• if $\tau<-\sqrt{4\Delta}$:
CASE #5: Stable Node
• if $\tau=-\sqrt{4\Delta}$:
• if there are no uniquely determined eigenvectors (both can be anything):
CASE #6: Stable Star
• if there is one uniquely determined eigenvector (the other can be anything):
CASE #7: Stable Degenerate Node
• if $-\sqrt{4\Delta}<\tau<0$:
CASE #8: Stable Spiral
• if $\tau=0$:
CASE #9: Stable Center
• if $0<\tau<\sqrt{4\Delta}$:
CASE #10: Unstable Spiral
• if $\tau=\sqrt{4\Delta}$:
• if there are no uniquely determined eigenvectors (both can be anything):
CASE #11: Unstable Star
• if there is one uniquely determined eigenvector (the other can be anything):
CASE #12: Unstable Degenerate Node
• if $\sqrt{4\Delta}<\tau$:
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:

• The descriptions are always correct for cases #1, #5, #8, #10, and #13 but can be inaccurate for cases #2, #3, #4, #6, #7, #9, #11, and #12.
• Ambiguous cases #6, #7, #11, and #12 at least have their stability correctly determined.
• If the system is conservative, a prediction of case #9 is accurate.