Slow divergence integral on a Möbius band

Abstract

The slow divergence integral has proved to be an important tool in the study of slow-fast cycles defined on an orientable two-dimensional manifold (e.g. R^2). The goal of our paper is to study 1-canard cycle and 2-canard cycle bifurcations on a non-orientable two-dimensional manifold (e.g. the Möbius band) by using similar techniques. Our focus is on smooth slow-fast models with a Hopf breaking mechanism. The same results can be proved for a jump breaking mechanism and non-generic turning points. The slow-fast bifurcation problems on the Möbius band require the study of the 2-return map attached to such 1- and 2-canard cycles. We give a simple sufficient condition, expressed in terms of the slow divergence integral, for the existence of a period-doubling bifurcation near the 1-canard cycle. We also prove the finite cyclicity property of “singular” 1- and 2-homoclinic loops (“regular” 1-homoclinic loops of finite codimension have been studied by Guimond).