2018
DOI: 10.1007/s10898-018-0723-1
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Consensus and balancing on the three-sphere

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Cited by 9 publications
(7 citation statements)
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“…The system (2) has been intensively investigated for general d, usually with global coupling and for the homogeneous case Ω i = 0, whether as a model of synchronization [24][25][26][27][28][29][30], or opinion formation and consensus studies on the unit sphere [31][32][33][34][35][36][37], or for the modelling of swarming behaviour [16,38,39]. Many synchronization properties have been established for any d [25,26,[40][41][42], in particular for the case of identical frequencies Ω i = Ω, it is known that for κ 2 > 0 the order parameter r = X av evolves exponentially quickly to the value r ∞ = 1, in which case all nodes are co-located to form a completely synchronized state, or for κ 2 < 0 to a state with r ∞ = 0.…”
Section: Pairwise Interactions On the Unit Spherementioning
confidence: 99%
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“…The system (2) has been intensively investigated for general d, usually with global coupling and for the homogeneous case Ω i = 0, whether as a model of synchronization [24][25][26][27][28][29][30], or opinion formation and consensus studies on the unit sphere [31][32][33][34][35][36][37], or for the modelling of swarming behaviour [16,38,39]. Many synchronization properties have been established for any d [25,26,[40][41][42], in particular for the case of identical frequencies Ω i = Ω, it is known that for κ 2 > 0 the order parameter r = X av evolves exponentially quickly to the value r ∞ = 1, in which case all nodes are co-located to form a completely synchronized state, or for κ 2 < 0 to a state with r ∞ = 0.…”
Section: Pairwise Interactions On the Unit Spherementioning
confidence: 99%
“…Equal spacing of the nodes, for example, which follows from the rotationally invariant expression (33), is not evident. However, if we rotate the final configuration to obtain (34), then under the Hopf fibration (37) we find that x i → cos 4πi N , sin 4πi N , 0 , and so for odd N the nodes are mapped to N equally spaced distinct points on the equator of S 2 , and to N/2 distinct points for even N.…”
Section: The Homogeneous Systemmentioning
confidence: 99%
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“…Models of synchronization on the unit sphere S d−1 have been extensively developed, including proofs of synchronization properties together with various applications 20,23-27,65-67 , including consensus properties in opinion dynamics [68][69][70][71] . It has been previously observed that the system with identical frequency matrices can be partially integrated in any dimension d using the vector transform 28 , and also for S 3 by using quaternionic variables 29 .…”
Section: Vector Models On the Unit Spherementioning
confidence: 99%
“…As the fundamental and important issue of flocking and synchronization, consensus problem has attracted a larger number of scientists' attention, e.g. [1,[4][5][6][7][8][9][10][11][12].…”
Section: Introductionmentioning
confidence: 99%