Emergence and bifurcations of Lyapunov manifolds in nonlinear wave equations
type: Item
instance of: scholarly article
title: Emergence and bifurcations of Lyapunov manifolds in nonlinear wave equations
zbMATH Open document ID: 1186.37058
author: Taoufik Bakri
author: H. G. E. Meijer
author: Ferdinand Verhulst
published in: Journal of Nonlinear Science
publication date: 2009-12-02T00:00:00Z
review text: Persistence behavior and bifurcations of Lyapunov manifolds are studied for classes of a parametrically excited wave equations. The semilinear IVP \[ \frac{d w}{ dt } + A w =\varepsilon f(w,t,\varepsilon), \, w(0)=w_0 \] is considered, where \(A\) generates a uniformly bounded \(C_0\)-group \(G(t)\) on the Banach space \(X.\) The second object of this paper is the following wave equation: \[ u_{tt} - c^2 u_{xx} + \varepsilon \beta u_t +(\omega_0^2 + \varepsilon \gamma \cos t) u = \varepsilon \alpha u^3,\quad t\geq 0,\;0<x<\pi \] \[ u_x(0,t) = u_x(\pi,t)=0,\quad \beta>0 \] The fast dynamics behavior is segregated for a finite, resonant part of the spectrum and slow dynamics is found elsewhere. The manifold where the fast dynamics takes place is almost-invariant. It is to be noted that a number of the phenomena have been found in this paper, i.e. periodic and quasi-periodic solutions, are stable and open for experimental investigation.
Mathematics Subject Classification ID: 37G15
Mathematics Subject Classification ID: 35L70
Mathematics Subject Classification ID: 35B32
Mathematics Subject Classification ID: 74J30
zbMATH DE Number: 5641443
zbMATH Keywords: bifurcation
zbMATH Keywords: parametrically excited wave equations
zbMATH Keywords: nonlinear waves
zbMATH Keywords: resonance
reviewed by: Denis Sidorov
describes a project that uses: MATCONT
describes a project that uses: CONTENT
MaRDI profile type: Publication
full work available at URL: s00332-009-9045-2
OpenAlex ID: W1999697308
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DOI: 10.1007/S00332-009-9045-2
description: scientific article; zbMATH DE number 5641443