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Consider a surface $S$ and a vector field on the surface which has a closed orbit. The vector field on both sides of the closed orbit spirals towards it, which gives us that the linearized Poincare return map has eigenvalue $\leq 1$. What additional information do I need to infer that the eigenvalue is strictly less than 1? I know that in a neighborhood of the closed orbit, the Gaussian curvature is negative. How are such conclusions generally made? Thanks a lot!

Edit after Ali Taghavi's comment: The integral curves of the vector field near the closed orbit have their accelerations ${\bf parallel}$ to the surface (this is kind of opposite of a geodesic).

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  • $\begingroup$ In your question, you did not assume any compatibility between your vector field and riemannian structute, so why you expect that the curvature effect on the dynamics? any way may be a paper By Romanovski "Limit cycle and complex geometry" could be indirectly related to your question. I was interested in such geometric compatibility in the following post:mathoverflow.net/questions/160945/… $\endgroup$
    – Ali Taghavi
    Commented Jan 11, 2015 at 12:51
  • $\begingroup$ please google "Limit cycles and complex geometry" the PDF of this paper is availlable in the webb. $\endgroup$
    – Ali Taghavi
    Commented Jan 11, 2015 at 12:57
  • $\begingroup$ @AliTaghavi Good point regarding the compatibility. I have edited the question accordingly. $\endgroup$
    – user65812
    Commented Jan 11, 2015 at 18:43
  • $\begingroup$ @AliTaghavi Unfortunately googling reveals a paper written in Russian. Where can I locate an English version? Thanks a lot! $\endgroup$
    – user65812
    Commented Jan 11, 2015 at 18:49
  • $\begingroup$ I have a PDF file of the english version. if you wish i can email you. $\endgroup$
    – Ali Taghavi
    Commented Jan 12, 2015 at 20:42

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