Forced symmetry breaking of homoclinic cycles in a PDE with O(2) symmetry
type: Item
instance of: scholarly article
title: Forced symmetry breaking of homoclinic cycles in a PDE with O(2) symmetry
zbMATH Open document ID: 0858.65123
DOI: 10.1016/0377-0427(95)00212-X
author: Peter Ashwin
author: Klaus Böhmer
author: Zhen Mei
published in: Journal of Computational and Applied Mathematics
publication date: 1997-03-25T00:00:00Z
review text: This paper concerns a numerical study of solutions near homoclinic orbits for forced symmetry breaking of a partial differential equation (PDE) with O(2) symmetry to one with SO(2) symmetry. The authors concentrate on the Kuramoto-Sivashinsky equation with spatially periodic boundary conditions. The breakup of structurally stable homoclinic cycles under perturbations breaking the symmetries is investigated via the introduction of flux term that breaks the reflectional symmetry while retaining the translational symmetry. As \textit{P. Chossat} [Nonlinearity 6, No. 5, 723-731 (1993; Zbl 0784.58052)] has shown if the reflectional symmetry is broken then such a homoclinic orbit generically bifurcates to a quasiperiodic flow on a 2-torus. Comparisons of these results with numerical simulations of a Galerkin approximation of the equations are presented.
Mathematics Subject Classification ID: 65Z05
Mathematics Subject Classification ID: 35B10
Mathematics Subject Classification ID: 35Q40
Mathematics Subject Classification ID: 65M60
Mathematics Subject Classification ID: 37C55
zbMATH DE Number: 908446
zbMATH Keywords: Galerkin method
zbMATH Keywords: homoclinic orbits
zbMATH Keywords: symmetry breaking
zbMATH Keywords: Kuramoto-Sivashinsky equation
zbMATH Keywords: quasiperiodic flow
reviewed by: Maria Gousidou-Koutita
describes a project that uses: DSTool
MaRDI profile type: Publication
Wikidata QID: Q57928527
cites work: Heteroclinic cycles and modulated travelling waves in systems with 0(2) symmetry
cites work: Kuramoto–Sivashinsky Dynamics on the Center–Unstable Manifold
cites work: The Numerical Computation of Connecting Orbits in Dynamical Systems
cites work: NUMERICAL DETECTION AND CONTINUATION OF CODIMENSION-TWO HOMOCLINIC BIFURCATIONS
cites work: Forced reflectional symmetry breaking of an O(2)-symmetric homoclinic cycle
cites work: Q4849841
cites work: Inertial manifolds for the Kuramoto-Sivashinsky equation and an estimate of their lowest dimension
cites work: Q5526485
cites work: Back in the Saddle Again: A Computer Assisted Study of the Kuramoto–Sivashinsky Equation
cites work: Robust heteroclinic cycles
cites work: Using Melnikov's method to solve Silnikov's problems
cites work: An Adaptive Newton--Picard Algorithm with Subspace Iteration for Computing Periodic Solutions
cites work: Forced symmetry breaking of homoclinic cycles
cites work: Convection in a rotating fluid layer
cites work: Homoclinic period blow-up in reversible and conservative systems
Recommended article: Forced reflectional symmetry breaking of an O(2)-symmetric homoclinic cycle
Recommended article: Forced symmetry breaking of homoclinic cycles
Recommended article: Heteroclinic cycles for reaction diffusion systems by forced symmetry-breaking
Recommended article: Forced symmetry breaking perturbations for periodic solutions
Recommended article: Q4849841
Recommended article: Symmetry-breaking homoclinic bifurcations in diffusively coupled equations
Recommended article: Q4734523
Recommended article: On degenerate Hopf bifurcation with broken O(2) symmetry
Recommended article: Symmetry breaking Hopf bifurcations in equation with O(2) symmetry with application to the Kuramoto-Sivashinsky equation
Recommended article: Heteroclinic cycles involving periodic solutions in mode interactions with O(2) symmetry
description: scientific article; zbMATH DE number 908446