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19
On the Stability of the Kuramoto Model of Coupled Nonlinear Oscillators
, 2005
"... We provide an analysis of the classic Kuramoto model of coupled nonlinear oscillators that goes beyond the existing results for alltoall networks of identical oscillators. Our work is applicable to oscillator networks of arbitrary interconnection topology with uncertain natural frequencies. Using ..."
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Cited by 62 (8 self)
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We provide an analysis of the classic Kuramoto model of coupled nonlinear oscillators that goes beyond the existing results for alltoall networks of identical oscillators. Our work is applicable to oscillator networks of arbitrary interconnection topology with uncertain natural frequencies. Using tools from spectral graph theory and control theory, we prove that for couplings above a critical value, the synchronized state is locally asymptotically stable, resulting in convergence of all phase differences to a constant value, both in the case of identical natural frequencies as well as uncertain ones. We further explain the behavior of the system as the number of oscillators grows to infinity.
Collective motion and oscillator synchronization
 Proc. Block Island Workshop on Cooperative Control
, 2003
"... Summary. This paper studies connections between phase models of coupled oscillators and kinematic models of groups of selfpropelled particles. These connections are exploited in the analysis and design of feedback control laws for the individuals that stabilize collective motions for the group. 1 ..."
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Cited by 23 (7 self)
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Summary. This paper studies connections between phase models of coupled oscillators and kinematic models of groups of selfpropelled particles. These connections are exploited in the analysis and design of feedback control laws for the individuals that stabilize collective motions for the group. 1
Symmetry and phaselocking in a ring of pulsecoupled oscillators with distributed delays
 Physica D
, 1999
"... Phaselocking in a ring of pulsecoupled integrateandfire oscillators with distributed delays is analysed using group theory. The period of oscillation of a solution and those related by symmetry is determined selfconsistently. Numerical continuation of maximally symmetric solutions in characteri ..."
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Cited by 11 (4 self)
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Phaselocking in a ring of pulsecoupled integrateandfire oscillators with distributed delays is analysed using group theory. The period of oscillation of a solution and those related by symmetry is determined selfconsistently. Numerical continuation of maximally symmetric solutions in characteristic system length and timescales yields bifurcation diagrams with spontaneous symmetry breaking. The stability of phaselocked solutions is determined via a linearisation of the oscillator firing map. In the weakcoupling regime, averaging leads to an effective phasecoupled model with distributed phaseshifts and the analysis of the system is considerably simplified. In particular, the collective period of a solution is now slaved to the relative phases. For odd numbered rings, spontaneous symmetry breaking can lead to a change of stability of a travelling wave state via a simple Hopf bifurcation. The resulting nonphaselocked solutions are constructed via numerical continuation at these bifurcation points. The corresponding behaviour in the integrateandfire system is explored with simulations showing bifurcations to
An ODE Model of the Motion of Pelagic Fish
, 2007
"... A system of ordinary differential equations (ODEs) is derived from a discrete system of Vicsek, Czirók et al. [35], describing the motion of a school of fish. Classes of linear and stationary solutions of the ODEs are found and their stability explored using equivariant bifurcation theory. The exist ..."
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Cited by 8 (1 self)
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A system of ordinary differential equations (ODEs) is derived from a discrete system of Vicsek, Czirók et al. [35], describing the motion of a school of fish. Classes of linear and stationary solutions of the ODEs are found and their stability explored using equivariant bifurcation theory. The existence of periodic and toroidal solutions is also proven under deterministic perturbations and structurally stable heteroclinic connections are found. Applications of the model to the migration of the capelin, a pelagic fish that undertakes an extensive migration in the North Atlantic, are discussed and simulation of the ODEs presented. 1
TongueLike Bifurcation Structures of the MeanField Dynamics in a Network of Chaotic Elements”, Physica 124D
, 1998
"... Collective behavior is studied in globally coupled maps. Several coherent motions exist, even in fully desynchronized state. To characterize the collective behavior, we introduce scaling transformation of parameter, and detect the tonguelike structure of collective motions in parameter space. Such ..."
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Cited by 5 (0 self)
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Collective behavior is studied in globally coupled maps. Several coherent motions exist, even in fully desynchronized state. To characterize the collective behavior, we introduce scaling transformation of parameter, and detect the tonguelike structure of collective motions in parameter space. Such collective motion is supported by the separation of time scale, given by the selfconsistent relationship between the collective motion and chaotic dynamics of each element. It is shown that the change of collective motion is related with the window structure of a single onedimensional map. Formation and collapse of regular collective motion are understood as the internal bifurcation structure. Coexistence of multiple attractors with different collective behaviors is also found in fully desynchronized state.
Chimera states in heterogeneous networks
, 2008
"... Chimera states in networks of coupled oscillators occur when some fraction of the oscillators synchronise with one another, while the remaining oscillators are incoherent. Several groups have studied chimerae in networks of identical oscillators, but here we study these states in heterogeneous model ..."
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Cited by 1 (0 self)
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Chimera states in networks of coupled oscillators occur when some fraction of the oscillators synchronise with one another, while the remaining oscillators are incoherent. Several groups have studied chimerae in networks of identical oscillators, but here we study these states in heterogeneous models for which the natural frequencies of the oscillators are chosen from a distribution. For a model consisting of two subnetworks we obtain exact results by reduction to a finite set of differential equations, and for a network of oscillators in a ring we generalise known results. We find that heterogeneity can destroy chimerae, destroy all states except chimerae, or destabilise chimerae in Hopf bifurcations, depending on the form of the heterogeneity. Synchronisation of interacting oscillators is a problem of fundamental importance, with applications from Josephson junction circuits to neuroscience [19, 24, 22, 27]. Since oscillators are unlikely to be identical, the effects of heterogeneity on their collective behaviour is of interest. One
unknown title
, 1996
"... Stability of periodic solutions in series arrays of Josephson junctions with internal capacitance ..."
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Stability of periodic solutions in series arrays of Josephson junctions with internal capacitance
Partially integrable dynamics of ensembles of nonidentical oscillators
, 2010
"... We consider ensembles of sinecoupled phaseoscillators consisting of subpopulations of identical units, with a general heterogeneous coupling between subpopulations. Using the WatanabeStrogatz ansatz we reduce the dynamics of the ensemble to a relatively smallnumberofdynamicalvariables plusmicrosco ..."
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We consider ensembles of sinecoupled phaseoscillators consisting of subpopulations of identical units, with a general heterogeneous coupling between subpopulations. Using the WatanabeStrogatz ansatz we reduce the dynamics of the ensemble to a relatively smallnumberofdynamicalvariables plusmicroscopicconstants ofmotion. This reduction is independent of the sizes of subpopulations and remains valid in the thermodynamic limits, where these sizes or/and the number of subpopulations are infinite. We demonstrate that the approach to the dynamics of such systems, recently proposed by Ott and Antonsen, correspondsto a particular choice of microscopic constants of motion. The theory is applied to the standard Kuramoto model and to the description of two interacting subpopulations, exhibiting a chimera state. Furthermore, we analyze the dynamics of the extension of the Kuramoto model for the case of nonlinear coupling and demonstrate the multistability of synchronous states.
unknown title
"... This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal noncommercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or sel ..."
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This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal noncommercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier’s archiving and manuscript policies are encouraged to visit: