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Consider a planar dynamical system described in polar coordinates as $$ \left\{ \begin{array}{ll} \dot{\theta}=\Delta - r \sin \theta,\\ \dot{r} = - r + 1 + \cos \theta, \end{array} \right. $$ where $\Delta$ is a constant such that $|\Delta|< \max_{0\le x\le 2\pi}(1+\cos x)\sin x$. I need to show that the system does not have any limit cycle.

Remark: The system has a stable node and a saddle node.

Maybe a better question is that what are the ideas, methods and tricks to prove of nonexistence of limit cycle, especially for planar systems. For example, Dulac- Bendixson is one of them. Is there any other?

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  • $\begingroup$ could you please write your original system(In x-y coordinate)? $\endgroup$ – Ali Taghavi Mar 23 '14 at 3:36
  • $\begingroup$ Do you allow polycycles, or do you really mean «limit cycle» as a periodic trajectory (i.e. without singularity) ? $\endgroup$ – Loïc Teyssier Mar 23 '14 at 10:39
  • $\begingroup$ @AliTaghavi You can use a change of variable to convert it to x-y coordinates but in fact its original form is the mentioned above. $\endgroup$ – Mohammad Khosravi Mar 24 '14 at 0:42
  • $\begingroup$ @LoïcTeyssier No I just wanna prove that there exist no limit cycle, absorbing(stable) limit cycle. If you know any paper, methods,... all are appreciated. $\endgroup$ – Mohammad Khosravi Mar 24 '14 at 0:43
  • $\begingroup$ What does numerical experimentation suggest ? If there is a limit cycle then it must occur in the closed disk $\{|r|\leq 2\}$. The point is that there is no generic/systematic way to deal with limit cycles, so I'm afraid you're not going to get any definite answer in that respect... $\endgroup$ – Loïc Teyssier Mar 24 '14 at 6:14

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