Slow divergence integral on a Möbius band
type: Item
instance of: scholarly article
title: Slow divergence integral on a Möbius band
zbMATH Open document ID: 1418.34114
author: Renato Huzak
published in: Journal of Differential Equations
publication date: 2019-03-14T00:00:00Z
review text: 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)].
Mathematics Subject Classification ID: 34E15
Mathematics Subject Classification ID: 34E17
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 34C40
Mathematics Subject Classification ID: 34C05
zbMATH DE Number: 7036303
zbMATH Keywords: slow-fast cycle
zbMATH Keywords: slow divergence integral
zbMATH Keywords: Poincaré map
zbMATH Keywords: non-orientable manifold
zbMATH Keywords: bifurcation
reviewed by: Nikola Popović
MaRDI profile type: Publication
full work available at URL: j.jde.2018.11.002
OpenAlex ID: W2900221698
cites work: Time analysis and entry-exit relation near planar turning points
cites work: Canard cycles in the presence of slow dynamics with singularities
cites work: Cyclicity of common slow-fast cycles
cites work: Canard cycles and center manifolds
cites work: Slow divergence integral and balanced canard solutions
cites work: Nonlinear oscillations, dynamical systems, and bifurcations of vector fields
cites work: Homoclinic loop bifurcations on a Möbius band
cites work: Limit cycles in slow-fast codimension 3 saddle and elliptic bifurcations
cites work: Q3711248
cites work: Relaxation oscillation and canard explosion
cites work: Canard cycles of finite codimension with two breaking parameters
DOI: 10.1016/J.JDE.2018.11.002
Recommended article: Slow divergence integrals in generalized Liénard equations near centers
Recommended article: Slow divergence integrals in classical Liénard equations near centers
Recommended article: Slowly divergent geodesics in moduli space
Recommended article: Asymptotic decomposition of slow integral manifolds
Recommended article: A note on the Duffin-Schaeffer conjecture with slow divergence
Recommended article: The Möbius function and distal flows
Recommended article: Pseudo-abelian integrals on slow-fast Darboux systems
Recommended article: Sobolev gradients for the Möbius energy
Recommended article: The Loewner differential equation and slit mappings
Recommended article: Asymptotics of a Slow Manifold
description: scientific article; zbMATH DE number 7036303