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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.

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2 Answers 2

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In the last two cases I am having trouble visualing what this would even look like.

Have a look at Fig. 2.4 (b) of this book.

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 ${\bf R}^n$.

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  • $\begingroup$ Is there a reason why people don't focus on classification in higher dimensions (other than the fact that there are way more cases to classify)? $\endgroup$ Apr 24, 2013 at 5:22
  • $\begingroup$ @tacos_tacos_tacos Because this classification does not add much to our understanding of the behavior of the orbits. $\endgroup$
    – Artem
    Apr 24, 2013 at 11:49
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In last two cases, the positive (negative) eigenvalue will correspond to a 1-D unstable(stable) manifold, normal to a plane in which you will see the stable focus.

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