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From the figures of page 478 and 479 of this paper one find that the author probably means that we have a (canard) limit cycle for the system

$$\begin{cases} x'=y-x^2\\ y'=\epsilon(a-x) \end{cases}$$

On the other hand it can be easily shown that the system under consideration

has no limit cycle when $a\neq 0$. Here is a proof:

Proof:

Putting $x:=x+a,\;y:=y-a^2$ we would obtain the following system:

$$\begin{cases} x'=y-x^2-2ax\\ y'=-\epsilon x\end{cases}$$

When $a\neq 0$ the system has no closed orbit by the following lemma which is a restatement of a lemma in the paper: A. Lins Neto, W. de Melo and C.C. Pugh, On Li´enard Equations, Proc. Symp.Geom. and Topol., Lectures Notes in Math. 597, Springer–Verlag, 1977 pp. 335–357

Lemma:

Assume that $f$ is an even polynomial and $g$ is an odd polynomial with $xg(x) \neq 0$ for all $x\neq 0$ then the lienard equation $$(1)\;\;\begin{cases} x'=y-(f(x)+g(x))\\ y'=-x\end{cases}$$ does not have any closed orbit.

The idea of the proof of lemma, which I learned from the above reference, is based on the following:

The system $(1)$ is transve to $$(2)\;\;\begin{cases} x'=y-f(x)\\ y'=-x\end{cases}$$ but $(2)$ has a center at origin. This obviously implies that $(1)$ can not have a closed orbit.

According to this situation we ask:

Does the figure in pages 478 and 479 really claim that the above system has a limit cycle? So is not a contradictory situation here?

I am interested in this question since 1999 when I was trying to learn the concept "Canard Limit Cycle".I presented this question, among other questions on other interpretations of canard, in my PHD thesis.

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  • $\begingroup$ @YCor Thank you for adding "ODE" tag. But was not the tag "Limit cycle" an appropriate tag? $\endgroup$ – Ali Taghavi Feb 3 '18 at 10:44
  • $\begingroup$ Because it's somewhat narrow: it's a tag with 0 follower at the moment, so I tend to erase such tags (a tag with too few followers makes little sense). Listing questions tagged "limitcycle" now yields 26 questions (mostly by you). A search on "limit cycle" yields 36 answers, mostly the same. Anyway I don't mind if you put this tag again. $\endgroup$ – YCor Feb 3 '18 at 12:58
  • $\begingroup$ @YCor I am sorry for this second message. Now I realize the meaning of "The number of follower". I think that the number of followers of a given tag is usually very less than the real number of participant who are interested in that subject. But you are right that in "Limit cycle" tag almost all questions are mine. Any way thank you for your attention to this question and adding ODE tag which was really a relevant tag. $\endgroup$ – Ali Taghavi Feb 4 '18 at 6:47
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    $\begingroup$ I agree. As a tag, "limitcycles" seems a reasonably focussed tag, and I should probably have kept it. At least you should follow it;) $\endgroup$ – YCor Feb 4 '18 at 13:37
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    $\begingroup$ Not substantial, but isn't the new system $$\begin{cases} x'=(y-a^2)-(x+a)^2=y-x^2{\color{red} -}2ax\\ y'=\epsilon (a-x-a)=-\epsilon x\end{cases}$$ $\endgroup$ – Pietro Majer Aug 5 at 20:26

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