Singular perturbations and vanishing passage through a turning point

Abstract

The paper deals with planar slow-fast cycles containing a unique generic turning point. We address the question on how to study canard cycles when the slow dynamics call be singular at the turning point. We more precisely accept a generic saddle-node bifurcation to pass through the turning point. It reveals that in this case the slow divergence integral is no longer the good tool to use, but its derivative with respect to the layer Variable still is. We provide general results as Well as a number of applications. We show how to treat the open problems presented in Artes et al. (2009) [1] and Dumortier and Rousseau (2009) [13], dealing respectively with the graphics DI2a and DF1a from Dumortier et al. (1994) [14]. (C) 2009 Elsevier Inc. All rights reserved.