On maximum bifurcation delay in real planar singularly perturbed vector fields

Abstract

In singularly perturbed vector fields, where the unperturbed vector field has a curve of singularities (a “critical curve”), orbits tend to be attracted towards or repelled away from this curve, depending on the sign of the divergence of the vector field at the curve. When at some point, this sign bifurcates from negative to positive, orbits will typically be repelled away immediately after passing the bifurcation point (“turning point”). Atypical behaviour is nevertheless observed as well, when orbits follow the critical curve for some distance after the turning point, before they repel away from it: a delay in the bifurcation is present. Interesting are systems that have a maximum bifurcation delay, i.e. there is a point on the critical curve beyond which orbits cannot stay close to the critical curve. This behaviour is known to appear in some systems in dimension 3 (see [E. Benoît (Ed.), Dynamic Bifurcations, in: Lecture Notes in Mathematics, vol. 1493, Springer-Verlag, Berlin, 1991]), and it is commonly believed that it is not an issue in (real) planar systems. Beside making the observation that it does appear in non-analytic planar systems, it is shown that whenever bifurcation delay appears, it has no non-trivial maximum for analytic planar vector fields. The proof is based on the notion of family blow-up at the turning point, on formal power series in terms of blow-up variables, the study of their Gevrey properties and analytic continuation of their Borel transform. These results complement existing results concerning the equivalence of local and global canard solutions in [A. Fruchard, R. Schäfke, Overstability and resonance, Ann. Inst. Fourier (Grenoble) 53 (1) (2003) 227–264].