Slow divergence integral on a Möbius band (Q1731850)

The study of limit cycles in slow-fast vector fields on orientable two-dimensional manifolds such as $\mathbb{R}^2$ typically relies on the notion of a slow divergence integral that is associated with the first iterate of the Poincaré map, or $1$-return map [\textit{F. Dumortier} and \textit{R. Roussarie}, Mem. Am. Math. Soc. 577, 100 p. (1996; Zbl 0851.34057)]. \par Here, the author studies slow-fast cycles -- and, specifically, bifurcations of so-called $1$-canard and $2$-canard cycles -- on non-orientable two-dimensional manifolds. For definiteness, he considers a smooth two-parameter family of slow-fast vector fields on the Möbius band which realises a Hopf breaking mechanism due to the presence of a generic turning point. He introduces a slow divergence integral that is related to the second iterate of the Poincaré map, or $2$-return map; he then gives a sufficient condition for the existence of a period-doubling bifurcation near a $1$-canard cycle in his family of vector fields which is expressed in terms of that integral. Moreover, the author proves finite cyclicity properties of ``singular'' $1$-homoclinic and $2$-homoclinic loops, thereby extending previous results on ``regular'' $1$-homoclinic loops of finite codimension [\textit{L.-S. Guimond}, Nonlinearity 12, No. 1, 59--78 (1999; Zbl 0917.58026)]. Finally, he discusses the generalisation of his results to jump breaking mechanisms and non-generic turning points [\textit{P. De Maesschalck} and \textit{F. Dumortier}, Proc. R. Soc. Edinb., Sect. A, Math. 138, No. 2, 265--299 (2008; Zbl 1263.37034)].