Nongeneric bifurcations near heterodimensional cycles with inclination flip in \(\mathbb{R}^4\)
type: Item
instance of: scholarly article
title: Nongeneric bifurcations near heterodimensional cycles with inclination flip in \(\mathbb{R}^4\)
zbMATH Open document ID: 1257.34026
author: Shigui Ruan
author: Dan Liu
author: Deming Zhu
published in: Discrete and Continuous Dynamical Systems. Series S
publication date: 2011-06-10T00:00:00Z
review text: This paper deals with bifurcations from a heteroclinic cycle in families of vector fields in \(\mathbb{R}^4\). It is assumed that the cycle connects two hyperbolic equilibria \(p_1\) and \(p_2\) with different saddle indices (dimension of the unstable manifold). More precisely, it is assumed that all eigenvalues of the linearizations at the equilibria are real and simple; \(p_1\) has saddle index 3 and \(p_2\) has saddle index 2. The main nongeneric condition is that the (nonrobust) heteroclinic orbit connecting \(p_2\) to \(p_1\) undergoes an inclination flip at \(p_1\). The focus is on bifurcating single around dynamics, i.e., heteroclinic cycles, 1-homoclinic and 1-periodic orbits. To this end, the authors construct a first return map. This construction is based on considerations in [\textit{D. Zhu}, Acta Math. Sin., New Ser. 14, No. 3, 341--352 (1998; Zbl 0932.37032)].
reviewed by: Jürgen Knobloch
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 37G25
Mathematics Subject Classification ID: 34C37
Mathematics Subject Classification ID: 34C05
zbMATH DE Number: 5906837
zbMATH Keywords: bifurcation
zbMATH Keywords: heterodimensional cycle
zbMATH Keywords: inclination flip
zbMATH Keywords: Poincaré map
MaRDI profile type: Publication
full work available at URL: dcdss.2011.4.1511
OpenAlex ID: W2333430261
DOI: 10.3934/DCDSS.2011.4.1511
description: scientific article; zbMATH DE number 5906837