Oscillator death in coupled functional differential equations near Hopf bifurcation
type: Item
instance of: scholarly article
title: Oscillator death in coupled functional differential equations near Hopf bifurcation
zbMATH Open document ID: 1099.34066
author: Fatihcan M. Atay
published in: Journal of Differential Equations
publication date: 2006-03-20T00:00:00Z
review text: Stability of the equilibrium solution is analyzed for coupled systems of retarded functional-differential equations \[ \dot x_i(t) = L_i x_i^{t} + \varepsilon f_i(x_i^{t},\varepsilon)+\varepsilon \kappa g_i(x_1^{t},\dots,x_N^{t},\varepsilon), \quad i=1,\dots,N, \] near a supercritical Hopf bifurcation. Here, \(x^{t}\) denotes the function defined on \([-\tau,0]\) by \(x^{t}(\theta)=x(t+\theta)\) for \(x: \mathbb{R}\to \mathbb{R}^{n}\) and \(\tau\geq 0\). For each \(i\), the operator \(L_i: C([-\tau,0],\mathbb{R}^{n})\to \mathbb{R}^{n}\) is linear and continuous. Moreover, the equation \(\dot x_i(t)=L_i x_i^{t}\) has a pair of characteristic values \(\pm i \omega_i\). \(f_i\) and \(g_i\) vanish at \(x^{t}=0\). The author gives conditions in terms of the coupling rate \(\kappa\) for the stability of the zero equilibrium solution. In particular, it is shown that the largest eigenvalue of the graph Laplacian completely characterizes the effect of the connection topology on the stability of the diffusively and symmetrically coupled identical systems.
reviewed by: S. Yanchuk
Mathematics Subject Classification ID: 34K20
Mathematics Subject Classification ID: 34K18
zbMATH DE Number: 5013577
zbMATH Keywords: amplitude death
zbMATH Keywords: time delay
zbMATH Keywords: Hopf bifurcation
zbMATH Keywords: stability
zbMATH Keywords: Laplacian
Wikidata QID: Q60148228
MaRDI profile type: Publication
full work available at URL: j.jde.2005.01.007
OpenAlex ID: W2027719610
cites work: Amplitude response of coupled oscillators
cites work: Total and partial amplitude death in networks of diffusively coupled oscillators
cites work: Open problems of Paul Erd�s in graph theory
cites work: Oscillator death in populations of ''all to all'' coupled nonlinear oscillators
cites work: Oscillator Death in Systems of Coupled Neural Oscillators
cites work: Q2716030
cites work: Q5519283
cites work: Averaging methods for differential equations with retarded arguments and a small parameter
cites work: Q3860165
cites work: Matrix Analysis
cites work: Amplitude death in an array of limit-cycle oscillators.
cites work: Q5838364
DOI: 10.1016/J.JDE.2005.01.007
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description: scientific article; zbMATH DE number 5013577