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17
On the Stability of the Kuramoto Model of Coupled Nonlinear Oscillators
 In Proceedings of the American Control Conference
, 2004
"... We provide a complete analysis of the Kuramoto model of coupled nonlinear oscillators with uncertain natural frequencies and arbitrary interconnection topology. Our work extends and supersedes existing, partial results for the case of an alltoall connected network. Using tools from spectral gra ..."
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Cited by 58 (8 self)
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We provide a complete analysis of the Kuramoto model of coupled nonlinear oscillators with uncertain natural frequencies and arbitrary interconnection topology. Our work extends and supersedes existing, partial results for the case of an alltoall connected network. Using tools from spectral graph theory and control theory, we prove that for couplings above a critical value all the oscillators synchronize, resulting in convergence of all phase di#erences 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 7 (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.
Neural oscillators and integrators in the dynamics of decision tasks
, 2004
"... In this dissertation I develop both general results on the dynamics of neural oscillators and integrators and specific applications of these results to brain areas involved in simple cognitive tasks. The scientific motivation is broad: neural networks inside our brains are able to adapt to changing ..."
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Cited by 1 (0 self)
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In this dissertation I develop both general results on the dynamics of neural oscillators and integrators and specific applications of these results to brain areas involved in simple cognitive tasks. The scientific motivation is broad: neural networks inside our brains are able to adapt to changing information processing demands by exercising cognitive control, for example focussing on salient components of noisy sensory inputs when making specific decisions based on these inputs, but relaxing this focus when other needs become prominent. But what free variables or parameters can account for the observed adaptability? And does this adaptation occur optimally, with respect to simple economic metrics and physiological limitations? Here I address these questions via reduced models of neurons and populations near bifurcations, which characterize the dynamics of a brainstem nucleus involved in adaptive cognitive control, and via variational problems arising from neural signal processing, which clarify the role of this nucleus, and other dynamical mechanisms in decision tasks. First, I study and apply nonlinear oscillator dynamics. I develop and extend phase
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
Dynamics Of Kinks And Vortices In JosephsonJunction Arrays
, 1998
"... . We present an experimental as well as theoretical study of kink motion in onedimensional arrays of Josephson junctions connected in parallel by superconducting wires. The boundaries are closed to form a ring, and the waveform and stability of an isolated circulating kink is discussed. Two onedim ..."
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. We present an experimental as well as theoretical study of kink motion in onedimensional arrays of Josephson junctions connected in parallel by superconducting wires. The boundaries are closed to form a ring, and the waveform and stability of an isolated circulating kink is discussed. Two onedimensional rings can be coupled which provides an interesting and clean platform to study interactions between kinks. These studies form foundations for investigating the more difficult twodimensional arrays in which vortices move along rows but with some interrow coupling. We introduce recent progress in the analysis of vortex dynamics in 2D arrays. Key words. Josephson junction, kink, vortex, patterns, experiments. 1. Introduction. Over several decades, many applied mathematicians have studied coupled systems of Josephson junctions. They have certainly been intrigued by the apparent simplicity of the governing equations in contrast to the variety of the dynamics they exhibit. Josephson ar...
.4.3 Belyakov's scaling
"... to a homoclinic orbit to a saddle. The system we study is the following x = y \Gamma z y = 2:657466x +2:328733y + x 2 + xy + x 2 +0:83893461z (8.14) z = e 2:657466x \Gamma 0:83893461y 3:361277 \Gamma 0:83893461z This system was derived by augmenting a truncated unfolding of the normal ..."
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to a homoclinic orbit to a saddle. The system we study is the following x = y \Gamma z y = 2:657466x +2:328733y + x 2 + xy + x 2 +0:83893461z (8.14) z = e 2:657466x \Gamma 0:83893461y 3:361277 \Gamma 0:83893461z This system was derived by augmenting a truncated unfolding of the normal form of the Takens Bogdanov bifurcation x = y y = 2cx + (1 + c)y + x 2 + xy (8.15) which is numerically observed to have a homoclinic connection to the origin at c 1:328733,
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.