Canard limit cycle for certain singularly perturbed system (Is there a contradictory situation?) 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.
 A: 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.
A: 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).
