Consensus and balancing on the three-sphere

Crnkić, Aladin; Jaćimović, Vladimir

doi:10.1007/s10898-018-0723-1

Cited by 9 publications

(7 citation statements)

References 20 publications

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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%

“…As an example, we have plotted u i in figure 4(b) for N = 80, in which the last component has been dropped, showing a closed sequence of points in S 2 , in contrast to the d = 4 case in figure 4(a). Under the Hopf fibration (37) we have u i → cos 6πi N , sin 6πi N , 0 and so all nodes are mapped to points on the equator.…”

Section: The Homogeneous Five-body Systemmentioning

confidence: 99%

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Higher-order synchronization on the sphere

Lohe

2021

J. Phys. Complex.

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We construct a system of $N$ interacting particles on the unit sphere $S^{d-1}$ in $d$-dimensional space, which has $d$-body interactions only. The equations have a gradient formulation derived from a rotationally-invariant potential of a determinantal form summed over all nodes, with antisymmetric coefficients. For $d=3$, for example, all trajectories lie on the $2$-sphere and the potential is constructed from the triple scalar product summed over all oriented $2$-simplices. We investigate the cases $d=3,4,5$ in detail, and find that the system synchronizes from generic initial values, for both positive and negative coupling coefficients, to a static final configuration in which the particles lie equally spaced on $S^{d-1}$. Completely synchronized configurations also exist, but are unstable under the $d$-body interactions. We compare the relative effect of $2$-body and $d$-body forces by adding the well-studied $2$-body interactions to the potential, and find that higher-order interactions enhance the synchronization of the system, specifically, synchronization to a final configuration consisting of equally spaced particles occurs for all $d$-body and $2$-body coupling constants of any sign, unless the attractive $2$-body forces are sufficiently strong relative to the $d$-body forces. In this case the system completely synchronizes as the $2$-body coupling constant increases through a positive critical value, with either a continuous transition for $d=3$, or discontinuously for $d=5$. Synchronization also occurs if the nodes have distributed natural frequencies of oscillation, provided that the frequencies are not too large in amplitude, even in the presence of repulsive 2-body interactions which by themselves would result in asynchronous behaviour.

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Section: Pairwise Interactions On the Unit Spherementioning

confidence: 99%

Section: The Homogeneous Systemmentioning

confidence: 99%

Section: The Homogeneous Five-body Systemmentioning

confidence: 99%

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Higher-order synchronization on the sphere

Lohe

2021

J. Phys. Complex.

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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%

Systems of matrix Riccati equations, linear fractional transformations, partial integrability and synchronization

Lohe

2019

Journal of Mathematical Physics

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“…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%