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 Liénard 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 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.*

4more comments