Global generic dynamics close to symmetry
type: Item
instance of: scholarly article
title: Global generic dynamics close to symmetry
zbMATH Open document ID: 1255.34045
author: Alexandre A. P. Rodrigues
author: Isabel S. Labouriau
published in: Journal of Differential Equations
publication date: 2012-09-12T00:00:00Z
review text: The paper studies the dynamics that arises in generic perturbations of an asymptotically stable heteroclinic cycle in \(S^3\). The cycle is assumed to involve two saddle-foci of different type and is structurally stable within the class of \((\mathbb Z_2\oplus\mathbb Z_2)\)-symmetric vector fields. It is also assumed that the cycle contains a two-dimensional connection that persists as a transverse intersection of invariant surfaces under symmetry-breaking perturbations. It is shown that breaking the symmetry in a two-parameter family leads to a wide range of dynamical behavior, i.e., an attracting periodic trajectory; other heteroclinic trajectories; homoclinic orbits; \(n\)-pulses; suspended horseshoes and cascades of bifurcations of periodic trajectories near an unstable homoclinic cycle of Shilnikov's type. It is also shown that, generically, the coexistence of linked homoclinic orbits at the two saddle-foci has codimension two and occurs arbitrarily close to the symmetric cycle.
reviewed by: Eugene Ershov
Mathematics Subject Classification ID: 34C37
Mathematics Subject Classification ID: 34C28
Mathematics Subject Classification ID: 34C29
Mathematics Subject Classification ID: 37D05
Mathematics Subject Classification ID: 37G35
Mathematics Subject Classification ID: 34C14
Mathematics Subject Classification ID: 34D10
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 34C05
zbMATH DE Number: 6081512
zbMATH Keywords: heteroclinic networks
zbMATH Keywords: symmetry breaking
zbMATH Keywords: Bykov cycles
zbMATH Keywords: Shilnikov's homoclinic cycles
MaRDI profile type: Publication
full work available at URL: j.jde.2012.06.009
OpenAlex ID: W1973887680
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DOI: 10.1016/J.JDE.2012.06.009
description: scientific article; zbMATH DE number 6081512