Theory and applications of equivariant normal forms and Hopf bifurcation for semilinear FDEs in Banach spaces
type: Item
instance of: scholarly article
title: Theory and applications of equivariant normal forms and Hopf bifurcation for semilinear FDEs in Banach spaces
zbMATH Open document ID: 1496.34112
author: Shangjiang Guo
published in: Journal of Differential Equations
publication date: 2022-03-04T00:00:00Z
review text: The paper extends existing methods for the analysis of autonomous delay differential equations on the existence of invariant manifolds to semilinear functional differential equations. The author summarises the paper in a succinct and comprehensive way: `We show that in the neighborhood of trivial solutions, variables can be chosen so that the form of the reduced vector field relies not only on the information of the linearized system at the critical point but also on the inherent symmetry. We observe that the normal forms give critical information about dynamical properties, such as generic local branching spatiotemporal patterns of equilibria and periodic solutions. As an important application of equivariant normal forms, we not only establish equivariant Hopf bifurcation theorem for semilinear FDEs in general Banach spaces, but also in a natural way derive criteria for the existence, stability, and bifurcation direction of branches of bifurcating periodic solutions. We employ these general results to obtain the existence of infinite many small-amplitude wave solutions for a delayed Ginzburg-Landau equation on a two-dimensional disk with the homogeneous Dirichlet boundary condition.' Sections of the paper on decomposition of the phase space, equivariant normal form, and Hopf bifurcation with symmetry provide detail and useful discussion.
Mathematics Subject Classification ID: 34K30
Mathematics Subject Classification ID: 34K17
Mathematics Subject Classification ID: 34K18
Mathematics Subject Classification ID: 34K13
Mathematics Subject Classification ID: 34K20
Mathematics Subject Classification ID: 34K19
zbMATH DE Number: 7483926
zbMATH Keywords: invariant manifold
zbMATH Keywords: normal form
zbMATH Keywords: stability
zbMATH Keywords: Hopf bifurcation
reviewed by: Neville J. Ford
MaRDI profile type: Publication
full work available at URL: j.jde.2022.02.016
OpenAlex ID: W4212812376
cites work: Unique normal forms for planar vector fields
cites work: Stability and Hopf bifurcation for a population delay model with diffusion effects
cites work: Interaction of spatial diffusion and delays in models of genetic control by repression
cites work: Q5521733
cites work: A bifurcation problem for a functional differential equation of finitely retarded type
cites work: Stability and Hopf bifurcation in a diffusive logistic population model with nonlocal delay effect
cites work: Toroidal normal forms for bifurcations in retarded functional differential equations. I: Multiple Hopf and transcritical/multiple Hopf interaction
cites work: Q3948194
cites work: A simple global characterization for normal forms of singular vector fields
cites work: Stability and bifurcation for a delayed predator-prey model and the effect of diffusion
cites work: Normal forms for semilinear functional differential equations in Banach spaces and applications. II.
cites work: Normal forms and Hopf bifurcation for partial differential equations with delays
cites work: Normal forms for retarded functional differential equations with parameters and applications to Hopf bifurcation
cites work: Normal forms for retarded functional differential equations and applications to Bogdanov-Takens singularity
cites work: Global asymptotics in some quasimonotone reaction-diffusion systems with delays
cites work: Singularities and groups in bifurcation theory. Volume II
cites work: Nonlinear oscillations, dynamical systems, and bifurcations of vector fields
cites work: Bifurcation in a reaction-diffusion model with nonlocal delay effect and nonlinear boundary condition
cites work: Patterns in a nonlocal time-delayed reaction-diffusion equation
cites work: Spatio-temporal patterns in a diffusive model with non-local delay effect
cites work: Generalized Hopf Bifurcation for Neutral Functional Differential Equations
cites work: Stability and bifurcation in a reaction-diffusion model with nonlocal delay effect
cites work: Equivariant normal forms for neutral functional differential equations
cites work: Oscillatory and Stationary Patterns in a Diffusive Model with Delay Effect
cites work: Bifurcation theory of functional differential equations
cites work: Theory of functional differential equations. 2nd ed
cites work: Existence results for partial neutral functional differential equations with unbounded delay
cites work: Existence of periodic solutions of partial neutral functional differential equations with unbounded delay
cites work: Q3892399
cites work: Q4266071
cites work: Hopf Bifurcation for Semilinear FDEs in General Banach Spaces
cites work: Stability and Hopf bifurcation in a Hutchinson model
cites work: Centre manifolds for partial differential equations with delays
cites work: Normal forms for semilinear equations with non-dense domain with applications to age structured models
cites work: Q3194770
cites work: Hopf bifurcation for non-densely defined Cauchy problems
cites work: Theory and applications of abstract semilinear Cauchy problems
cites work: Center manifolds for semilinear equations with non-dense domain and applications to Hopf bifurcation in age structured models
cites work: Abstract Functional Differential Equations and Reaction-Diffusion Systems
cites work: Q3492051
cites work: Bifurcation and Asymptotic Behavior of Solutions of a Delay-Differential Equation with Diffusion
cites work: Stable and unstable manifolds for partial functional differential equations
cites work: Normal Forms and Unfoldings for Local Dynamical Systems
cites work: Invariant manifolds of partial functional differential equations.
cites work: Q5813396
cites work: Hopf bifurcations in a reaction-diffusion population model with delay effect
cites work: Singularities of vector fields
cites work: Existence and Stability for Partial Functional Differential Equations
cites work: Normal forms for singularities of vector fields
cites work: Q4856171
cites work: Normal forms for neutral functional differential equations
cites work: Theory and applications of partial functional differential equations
cites work: The Hopf bifurcation and its stability for semilinear diffusion equations with time delay arising in ecology
cites work: Hopf bifurcation in delayed van der Pol oscillators
DOI: 10.1016/J.JDE.2022.02.016
description: scientific article; zbMATH DE number 7483926