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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 107 (13 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.
Distributed geodesic control laws for flocking of nonholonomic agents
 IEEE Transaction on Automatic Control
, 2005
"... Abstract—We study the problem of flocking and velocity alignment in a group of kinematic nonholonomic agents in 2 and 3 dimensions. By analyzing the velocity vectors of agents on a circle (for planar motion) or sphere (for 3D motion), we develop a geodesic control law that minimizes a misalignment ..."
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Cited by 35 (6 self)
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Abstract—We study the problem of flocking and velocity alignment in a group of kinematic nonholonomic agents in 2 and 3 dimensions. By analyzing the velocity vectors of agents on a circle (for planar motion) or sphere (for 3D motion), we develop a geodesic control law that minimizes a misalignment potential and results in velocity alignment and flocking. The proposed control laws are distributed and will provably result in flocking when the underlying proximity graph which represents the neighborhood relation among agents is connected. We further show that flocking is possible even when the topology of the proximity graph changes over time, so long as a weaker notion of joint connectivity is preserved. Index Terms—Cooperative control, distributed coordination, flocking, multiagent systems. I.