Multiple slowly oscillating periodic solutions in coupled lossless transmission lines
type: Item
instance of: scholarly article
title: Multiple slowly oscillating periodic solutions in coupled lossless transmission lines
zbMATH Open document ID: 1144.34365
DOI: 10.1016/S1468-1218(03)00040-3
author: Shiwang Ma
author: Jianhong Wu
author: Wieslaw Krawcewicz
published in: Nonlinear Analysis. Real World Applications
publication date: 2004-06-22T00:00:00Z
review text: Using equivariant degree theory [cf. \textit{W. Krawcewicz} and \textit{J. Wu}, Theory of degrees with applications to bifurcations and differential equations (1997; Zbl 0882.58001)] the article studies symmetric periodic solutions in neutral functional differential equations with dihedral symmetry depending on one parameter. The authors first obtain symmetric Hopf bifurcation theorems in this general setting and then apply these theorems to a polygonal ring of diffusively coupled identical oscillators. The article studies existence, multiplicity and global continuations of branches of periodic solutions. In particular, it proves the existence of large amplitude synchronous (all cells have always equal value) and phase-locked (all cells oscillate identically but not with the same phase) periodic solutions in these ring-structured neutral systems. A concrete example illustrates the coexistence of a large number of slowly oscillating solutions. The problem of stability of these periodic solutions is, naturally, left open.
reviewed by: Jan Sieber
Mathematics Subject Classification ID: 34K13
Mathematics Subject Classification ID: 34K18
Mathematics Subject Classification ID: 34K40
Mathematics Subject Classification ID: 35B10
Mathematics Subject Classification ID: 37G40
Mathematics Subject Classification ID: 37L20
Mathematics Subject Classification ID: 47J15
zbMATH DE Number: 2076993
zbMATH Keywords: equivariant degree
zbMATH Keywords: Hopf bifurcation
zbMATH Keywords: neutral functional differential equation
zbMATH Keywords: dihedral symmetry
zbMATH Keywords: ring of identical oscillators
zbMATH Keywords: phase-locked oscillation
zbMATH Keywords: synchronous oscillation
zbMATH Keywords: slowly oscillatory periodic solution
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2076993