Fronts and bumps in spatially extended Kuramoto networks
type: Item
instance of: scholarly article
title: Fronts and bumps in spatially extended Kuramoto networks
zbMATH Open document ID: 1262.34038
published in: Physica D
publication date: 2012-01-04T00:00:00Z
review text: The author considers two models of coupled phase oscillators, using the continuum limit description and the Ott-Antonsen approach. The first model is a chain of \(N\) phase oscillators equally spaced on the one-dimensional spatial domain \([0,L]\) \[ \frac{d \theta_j}{d t} = \omega_j + k \operatorname{Im}( z_j(t) e^{-i \theta_j} ),\quad j=1,\dotsc,N, \tag{1} \] and coupled via a complex, non-local mean-field \(z_m\) given by \[ \begin{aligned}\frac{d z_m(t)}{d t} = - z_m(t) + \frac{L}{N} \sum\limits_{j=1}^N K_{mj} e^{i \theta_j(t)}, \\ K_{mj} = K\left( \frac{L |m - j|}{N - 1} \right),\quad K(x) = (1/2) e^{- |x|}.\end{aligned} \tag{2} \] The natural frequencies \(\omega_j\) of the oscillators are chosen from a Lorentzian distribution \(g(\omega) = \pi^{-1} [ (\omega - \omega_0)^2 + 1 ]^{-1}\) centred at \(\omega = \omega_0\), and \(k>0\) is an overall coupling strength. In the continuum limit \(N\to\infty\), the dynamics of system (1), (2) can be effectively described in terms of the probability density function \(F(x,\omega,\theta,t)\) (the probability that an oscillator at position \(x\) has intrinsic frequency \(\omega\) and state \(\theta\), at time \(t\)) that obeys some evolution integro-differential equation. It is shown that, for certain parameter values, the latter equation exhibits bistability between the completely asynchronous state (with constant probability distribution) and spatially uniform partially synchronous state (with an explicitly known, time-periodic probability distribution). Based on this observation, the author constructs moving front solutions, which connect regions of high local synchrony with regions of complete asynchrony, and bump solutions consisting of spatially-localized regions of partially-synchronous oscillators surrounded by complete asynchrony. Numerical continuation is used to determine parameter regions in which such solutions exist and are stable. An extension of the results to two spatial dimensions is discussed. The second…
Mathematics Subject Classification ID: 34C15
Mathematics Subject Classification ID: 35Q83
Mathematics Subject Classification ID: 60H99
zbMATH DE Number: 5994489
zbMATH Keywords: phase oscillators
zbMATH Keywords: non-local coupling
zbMATH Keywords: partial synchronization
zbMATH Keywords: moving fronts
zbMATH Keywords: spatially localized states
author: Carlo Laing
reviewed by: Oleh Omel'chenko
MaRDI profile type: Publication
full work available at URL: j.physd.2011.09.009
OpenAlex ID: W2043830451
cites work: Synchronization: a universal concept in nonlinear sciences
cites work: Chemical oscillations, waves, and turbulence
cites work: From Kuramoto to Crawford: Exploring the onset of synchronization in population of coupled oscillators
cites work: Identical phase oscillators with global sinusoidal coupling evolve by Möbius group action
cites work: Low dimensional behavior of large systems of globally coupled oscillators
cites work: Monotonicity of Phaselocked Solutions in Chains and Arrays of Nearest-Neighbor Coupled Oscillators
cites work: Stationary bumps in networks of spiking neurons
cites work: The dynamics of chimera states in heterogeneous Kuramoto networks
cites work: CHIMERA STATES IN A RING OF NONLOCALLY COUPLED OSCILLATORS
cites work: Chimera states in heterogeneous networks
cites work: Spectral properties of Chimera states
cites work: Long time evolution of phase oscillator systems
cites work: Dynamics and pattern formation in large systems of spatially-coupled oscillators with finite response times
cites work: Freezing Solutions of Equivariant Evolution Equations
cites work: Homoclinic orbits in reversible systems and their applications in mechanics, fluids and optics
cites work: Snakes, Ladders, and Isolas of Localized Patterns
cites work: Multiple Bumps in a Neuronal Model of Working Memory
cites work: PDE Methods for Nonlocal Models
cites work: Stability diagram for the forced Kuramoto model
cites work: Comment on “Long time evolution of phase oscillator systems” [Chaos 19, 023117 (2009)]
cites work: Spiral Waves in Nonlocal Equations
DOI: 10.1016/J.PHYSD.2011.09.009
Recommended article: Fronts in extended systems of bistable maps coupled via convolutions
Recommended article: Bifurcation Study of the Kuramoto Transition in Random Oscillator Networks
Recommended article: Chaos in Kuramoto oscillator networks
Recommended article: BIFURCATIONS IN A FREQUENCY-WEIGHTED KURAMOTO OSCILLATORS NETWORK
Recommended article: The Kuramoto model in complex networks
Recommended article: Propagating fronts in a bistable coupled map lattice
Recommended article: Bifurcations of front dynamics in a reaction-diffusion system with spatial inhomogeneities
Recommended article: Front Bifurcations in an Excitatory Neural Network
Recommended article: Partial Phase Cohesiveness in Networks of Networks of Kuramoto Oscillators
Recommended article: Fixed points and stability in the two-network frustrated Kuramoto model
description: scientific article; zbMATH DE number 5994489