I encountered this question in Strogatz's "nonlinear dynamics and chaos", which requires to draw a phase portrait with exactly three closed orbits and one fixed point. Due to the index theorem, the fixed point must be enclosed by all three orbits, which seems to me can only be achieved by a solar-system like structure (fixed point as the sun). But after that I couldn't find any portrait that doesn't break the typological constraints (e.g. no crossed lines, etc). Does any one has a solution for this? Thanks!


You can fill the space between the closed orbits with spiraling trajectories connecting the orbits. No conservation of energy is assumed, i.e., the system is not Hamiltionian, so this is possible.

For instance, make the fixed point and the second closed orbit stable, the first and third closed orbit unstable.

  • $\begingroup$ Hi, Lutzl. But in this case there will be points on the closed orbit with more than one direction. $\endgroup$ – Yuanzhao Mar 1 '14 at 10:43
  • $\begingroup$ With "connecting" I mean only the asymptotic behavior, the spirals never reach the closed orbits. $\endgroup$ – LutzL Mar 1 '14 at 10:53
  • $\begingroup$ Oh, it's my fault. I thought the Poincaré-Bendixson Thm had ruled out this possibility. Thank you, Lutzl! $\endgroup$ – Yuanzhao Mar 1 '14 at 11:11

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