Hopf bifurcation on the hexagonal lattice with small frequency
type: Item
instance of: scholarly article
title: Hopf bifurcation on the hexagonal lattice with small frequency
zbMATH Open document ID: 0844.34030
author: Michael Renardy
published in: Advances in Differential Equations
publication date: 1996-08-13T00:00:00Z
review text: A system of six complex, cubic ODE's for \(z_i \in \mathbb{C}\), \(i = 1, 2, \dots, 6\), is considered which has the symmetry of a hexagonal lattice. The amplitudes \(z_i\) are related to six eigenmodes in the \((x,y)\)-plane, \[ z_{1,4} : e^{\pm i \alpha z}, \quad z_{2,3,5,6} : e^{i \alpha {\pm x \pm \sqrt 3 y \over 2}}. \] These equations arise by a center manifold reduction for the two-layer Bénard problem. Reduced systems for solutions of a prescribed symmetry for steady states and traveling waves (rolls, hexagons, triangles, rectangles, etc.) are discussed. The paper aims on Hopf bifurcation with a small frequency leading to periodic solutions with high period. In this case the Birkhoff normal form reduction is not possible.
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 35B32
Mathematics Subject Classification ID: 37G99
zbMATH DE Number: 844393
zbMATH Keywords: symmetry of a hexagonal lattice
zbMATH Keywords: center manifold reduction
zbMATH Keywords: two-layer Bénard problem
zbMATH Keywords: traveling waves
zbMATH Keywords: Hopf bifurcation
zbMATH Keywords: periodic solutions
reviewed by: Bodo Werner
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 844393