Hopf bifurcations and oscillatory instabilities of spike solutions for the one-dimensional Gierer-Meinhardt model
type: Item
instance of: scholarly article
title: Hopf bifurcations and oscillatory instabilities of spike solutions for the one-dimensional Gierer-Meinhardt model
zbMATH Open document ID: 1030.35011
published in: Journal of Nonlinear Science
publication date: 2003-08-27T00:00:00Z
review text: The authors investigate the stability of symmetric \(k\)-spike equilibrium solutions to the Gierer-Meinhardt reaction-diffusion system in one space dimension in the limit of small activator diffusity and for various ranges of the reaction-time constant \(\tau\) and diffusitivity \(D\) of the inhibitor field dynamics. A nonlocal eigenvalue problem is derived that determines the stability on an \(O(1)\) time scale of these \(k\)-spike equilibrium patterns. For \(k=1\) and for various exponent sets of the nonlinear terms, they show that a one-spike solution is stable only when \(0 \leq \tau < \tau_0(D)\) and at \(\tau_0(D)\) a pair of complex conjugate eigenvalues crosses the imaginary axis. For \(k > 1\) positive real eigenvalues occur which lead to an instability of competition type; also synchronous oscillatory instability in the amplitudes of the spikes can occur for positive values of \(\tau\).
Mathematics Subject Classification ID: 35B32
Mathematics Subject Classification ID: 35B35
Mathematics Subject Classification ID: 35K57
zbMATH DE Number: 1971876
zbMATH Keywords: nonlocal eigenvalue problem
zbMATH Keywords: synchronous oscillatory instability
zbMATH Keywords: one space dimension
zbMATH Keywords: small activator diffusity
author: Michael J. Ward
author: Juncheng Wei
reviewed by: Alois Steindl
describes a project that uses: COLSYS
MaRDI profile type: Publication
full work available at URL: s00332-002-0531-z
OpenAlex ID: W1556426368
DOI: 10.1007/S00332-002-0531-Z
description: scientific article; zbMATH DE number 1971876