The Borel Transform of Canard Values and Its Singularities

Canards were discovered in the early 80s by É. Benoît, F.  and M. Diener, and J.-L. Callot in the study of the famous van der Pol equation (Benoît et al. [3]). Given a real differential equation, singularly perturbed by \(\varepsilon \) small, a canard solution — if it exists — has the particularity to follow partially or totally a slow curve from its attractive part to its repulsive part for a certain value of the control parameter a, named a canard value. A generalization to complex ODEs leads to overstable solutions, bounded in a neighbourhood of a turning point, i.e. a point where the slow curve presents an inversion of stability. It is known (Benoît et al. [5] and Canalis-Durand et al. [7]) that canard values admit a unique asymptotic expansion of Gevrey order 1 denoted by \(\hat{a}\), so that the Borel transform \(\tilde{a}(t)\) of \(\hat{a}(\varepsilon )\) is analytic near the origin. Using the recent theory of composite asymptotic expansions due to A. Fruchard and R. Schäfke (Fruchard and Schäfke [11]), we study and describe the first singularity of this Borel transform \(\tilde{a}\). This article focuses on a Riccati equation where \(x,y,\varepsilon ,a\in \mathbb {C}\). For this equation, the formal series \(\hat{a}\) is Borel summable in every direction except the real positive axis which constitutes a Stokes line. We first obtain an estimate of the difference of the canard values. This estimate contains an exponentially small term and a Gevrey asymptotic expansion. Then this result is translated into the Borel plane. It follows that the Borel transform \(\tilde{a}\) can be analytically continued to \(\mathbb {C}\setminus [1/3,+\infty [\) and has an isolated singularity at \(t=1/3\) on the first sheet.