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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 $\Delta$?

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1 Answer 1

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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.
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