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

Question

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.

Answer 1

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.

Answer 2

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).