Asymptotic expansion of planar canard solutions near a non-generic turning point

TL;DR

This paper develops an asymptotic expansion method for planar canard solutions near degenerate turning points in singularly perturbed ODEs, extending previous approaches to non-generic cases.

Contribution

It introduces a new asymptotic analysis framework for canard solutions at non-generic turning points using a recent correspondence, including the definition of a specialized function family.

Findings

Derived asymptotic expansions in powers of the small parameter

Extended analysis to degenerate (non-generic) turning points

Provided a systematic method for studying canard solutions near complex turning points

Abstract

This paper deals with the asymptotic study of the so-called canard solutions, which arise in the study of real singularly perturbed ODEs. Starting near an attracting branch of the "slow curve", those solutions are crossing a turning point before following for a while a repelling branch of the "slow curve". Assuming that the turning point is degenerate (or non-generic), we apply a correspondence presented in a recent paper. This application needs the definition of a family of functions $\phi$ that is studied in a first part. Then, we use the correspondence is used to compute the asymptotic expansion in the powers of the small parameter for the canard solution.