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:


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 Liénard Equations, Proc. Symp. Geom. and Topol., Lectures Notes in Math. 597, Springer–Verlag, 1977 pp. 335–357.


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 Liénard 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 transversal 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". In my PhD thesis, I presented this question, among other questions about two other interpretations of "Canard". My questions were result of my doubt or misunderstanding about the three methods of interpretations of "Canard": Nonstandard analysis asymptotic analysis and blow up.

I wrote in my thesis: "However the author of this thesis did not understand these three interpretations of Canard concept completely but he feels that in each interpretation there is a kind of quasi contradictory situation". Then I wrote in my thesis the reasons for such guess about such possible contradictory situations.

  • $\begingroup$ @YCor Thank you for adding "ODE" tag. But was not the tag "Limit cycle" an appropriate tag? $\endgroup$ 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. $\endgroup$ 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 $$\begin{cases} x'=(y-a^2)-(x+a)^2=y-x^2{\color{red} -}2ax\\ y'=\epsilon (a-x-a)=-\epsilon x\end{cases}$$ $\endgroup$ Aug 5 '19 at 20:26

The figures 3.2 and 3.3 in Eckhaus's 1983 paper refer to the degenerate case that the function $f(x)$ in the differential equations $$\begin{cases} x'=y-f(x)\\ y'=\epsilon(a-x) \end{cases} $$ is quadratic at the origin, but the later analysis and demonstration of limit cycles assumes that $f(x)$ is cubic at the origin (as it is in the Van der Pol equation 3.3.2 in Eckhaus's paper). This is indeed required, see a later paper by Braaksma (page 487).
Note that equation 3.5.5 of Eckhaus gives the radius squared of the periodic solution as proportional to $1/g'(0)$, where $f'(x)=xg(x)$. This diverges for a quadratic $f(x)$, when $g'(0)=0$, consistent with the expectation that a cubic $f(x)$ is needed for a limit cycle.

So if one restricts oneself to the explicit calculations (assuming a cubic $f$) rather than the initial figures (for a quadratic $f$) there is not really "a contradictory situation" in Eckhaus's 1983 paper.

  • $\begingroup$ Thanks again for your answer. did you already experienced working on canard limit cycles based on either interpretations mentioned in the question, Non standard, blow up and asymptotic analysis? Any way your current answwer is very help full to me I will read the paper of Eckhaus again and your answer help me to follow his paper easier. My main problem with the non standard approach by Zvonkin and shubin was the following For $\epsilon=0$ all horizontal line are solution curves(without any RETURN. But for small $\epsilon$ they frequently assume that we have a Poincare return map $\endgroup$ Jan 20 '20 at 11:10
  • $\begingroup$ so what is the domain of the poincare map they are working in?why is it legal to assume a global domain for for Poincare map. I remember when i was a PHD student I realy tried to understand these approaches(3 approaches) but such kind of questions were my main obstructions for continuation. I am still curious about "CANARD". So i am curious if you already experience researching on these subject. $\endgroup$ Jan 20 '20 at 11:14
  • $\begingroup$ Yes it is understandable. i will try to understand it. The first thing i need is that i have to forget my mentality and feeling of "quasi-contradictions" which i was suffering during my phd study about this subject. If i orget that i can follow the subject more clear and more relax. believe me that I really appreciate your great help by your answer. The reason i consider another bounty is that perhaps I would receive some other points which probably help me to convince myself.After all the canard is a beautiful and mysterious concept. The mysterious point is that it is observed by Non $\endgroup$ Jan 20 '20 at 12:09
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    $\begingroup$ a suggestion: I think the bounty will attract more response if you can formulate a somewhat more precise question, so that readers have a feeling what they should focus on; that is not quite clear at present. $\endgroup$ Jan 20 '20 at 12:31
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    $\begingroup$ that particular item (justification of Poincaré return map) could be a good focused question, to which I have no answer at present (but quite possibly other MO users might have) $\endgroup$ Jan 20 '20 at 13:05

The updated question asks for the validity of an analysis of the limit cycle based on the Poincaré return map, for which the limit $\epsilon\rightarrow 0$ is singular. This has been studied in Asymptotic analysis of the peeling-off point of a French duck (1994), for the specific case of the Van der Pol oscillator $$ \begin{cases} \epsilon x'=y-f(x),\;\;f(x)=\tfrac{1}{3}x^3+\tfrac{1}{2}x^2,\\ y'=-(x+\alpha). \end{cases} $$

The Poincaré map gives a power series in $\epsilon$ for the values of $\alpha$ where a "canard" = "duck" limit cycle exists. The singularity at $\epsilon=0$ gives an additional contribution $\propto e^{-k/\epsilon}$, for a full expression of the form $$\alpha=\alpha_{\rm Poincaré}+C\epsilon^{-1/2}\exp[-k/\epsilon],$$ $$\alpha_{\rm Poincaré}=\epsilon+12\epsilon^2+346\epsilon^3+15186\epsilon^4+o(\epsilon^4)$$ (explicit expressions for the coefficients $C$ and $k$ are given as well).


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