Self-excited vibrations for damped and delayed 1-dimensional wave equations
type: Item
instance of: scholarly article
title: Self-excited vibrations for damped and delayed 1-dimensional wave equations
zbMATH Open document ID: 1421.35021
author: Nemanja Kosovalić
author: B. Pigott
published in: Journal of Dynamics and Differential Equations
publication date: 2019-04-25T00:00:00Z
review text: This nice paper concerns 1-dimensional damped and delayed semilinear wave equations of sine-Gordon type with homogeneous Dirichlet boundary conditions. Taking the time delay as bifurcation parameter it is shown that there exists a sequence of Hopf bifurcations from the trivial stationary solution, and a formula for the corresponding bifurcation direction is given. The authors apply the Lyapunov-Schmidt method and the implicit function theorem by working in Sobolev spaces on the two-torus and with corresponding Fourier expansions and by using some special symmetry properties of the nonlinearities. This approach is well-known for ODEs and parabolic PDEs, but for hyperbolic PDEs there appear essential and deep technical difficulties connected with the derivative loss property (small divisors) of hyperbolic PDEs. For hyperbolic systems without delay, but with general nonlinearities, these difficulties have been overcome already by \textit{I. Kmit} and \textit{L. Recke} [J. Differ. Equations 257, No. 1, 264--309 (2014; Zbl 1310.35030), and ibid. 259, No. 11, 6287--6337 (2015; Zbl 1343.35023)].
reviewed by: Lutz Recke
Mathematics Subject Classification ID: 35B32
Mathematics Subject Classification ID: 35L71
Mathematics Subject Classification ID: 35L20
Mathematics Subject Classification ID: 35B10
zbMATH DE Number: 7047662
zbMATH Keywords: Hopf bifurcation
zbMATH Keywords: sine-Gordon equation
zbMATH Keywords: Lyapunov-Schmidt method
zbMATH Keywords: implicit function theorem
zbMATH Keywords: Fourier expansions
zbMATH Keywords: derivative loss property
zbMATH Keywords: small divisors
MaRDI profile type: Publication
full work available at URL: s10884-018-9654-2
OpenAlex ID: W2793453534
cites work: Q4786565
cites work: KAM for reversible derivative wave equations
cites work: Q4945128
cites work: Complex dynamics and multistability in a damped harmonic oscillator with delayed negative feedback
cites work: Limit cycles, tori, and complex dynamics in a second-order differential equation with delayed negative feedback
cites work: Newton's method and periodic solutions of nonlinear wave equations
cites work: Periodic solutions of nonlinear hyperbolic evolution systems
cites work: Self-oscillation
cites work: Bifurcation of periodic solutions for a semilinear wave equation
cites work: Bifurcation theory. An introduction with applications of PDEs
cites work: Stable periods of a semilinear wave equation and bifurcation of periodic solutions
cites work: Effect of resonance on the existence of periodic solutions for strongly damped wave equation
cites work: Center manifolds for semilinear equations with non-dense domain and applications to Hopf bifurcation in age structured models
cites work: Well posedness and exponential stability in a wave equation with a strong damping and a strong delay
cites work: Stability and Instability Results of the Wave Equation with a Delay Term in the Boundary or Internal Feedbacks
cites work: Variational and topological methods in nonlinear problems
cites work: Periodic Solutions of Nonlinear Hyperbolie Partial Differential Equations
cites work: Periodic solutions of nonlinear hyperbolic partial differential equations. II
DOI: 10.1007/S10884-018-9654-2
description: scientific article; zbMATH DE number 7047662