A lifting method for analyzing distributed synchronization on the unit sphere

Thunberg, Johan; Markdahl, Johan; Bernard, Florian; Gonçalves, Jorge

doi:10.1016/j.automatica.2018.07.007

Cited by 34 publications

(20 citation statements)

References 25 publications

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“…The aforementioned models are generalizations of the Kuramoto model [2,5,16,17,21,22,23,24,30,31,36,37,44,45,48,53,54] in the context of higherdimensional sphere, unitary group and quaternions [10,11,12,13,15,18,19,20,25,27,35,41,50,58]…”

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confidence: 99%

Emergent behaviors of the generalized Lohe matrix model

Ha¹,

Park²

2021

DCDS-B

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We present a first-order aggregation model on the space of complex matrices which can be derived from the Lohe tensor model on the space of tensors with the same rank and size. We call such matrix-valued aggregation model as "the generalized Lohe matrix model". For the proposed matrix model with two cubic coupling terms, we study several structural properties such as the conservation laws, solution splitting property. In particular, for the case of only one coupling, we reformulate the reduced Lohe matrix model into the Lohe matrix model with a diagonal frustration, and provide several sufficient frameworks leading to the complete and practical aggregations. For the estimates of collective dynamics, we use a nonlinear functional approach using an ensemble diameter which measures the degree of aggregation.

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confidence: 99%

Emergent behaviors of the generalized Lohe matrix model

Ha¹,

Park²

2021

DCDS-B

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“…Theorem 3. Consider the high-dimensional Kuramoto model (28). Assume that the interconnection graph  is a strongly connected digraph, and there exists a vector p ∈ R n satisfying Ωp = 0 and ‖p‖ = 1.…”

Section: Theorem 2 If the Digraph  Of The High-dimensional Kuramoto Modelmentioning

confidence: 99%

“…Example 2. Consider the high-dimensional Kuramoto model (28) with proportional nonidentical oscillators, where n = 3, m = 4, the interconnection digraph is a directed cycle, and the dynamics is described by…”

Section: Simulationmentioning

confidence: 99%

“…where r i ∈ R n is the state of the i th oscillator, Ω i is an n × n skew-symmetric matrix, k > 0 is the interconnection gain, and A = (a i j ) ∈ R m×m is the weighted adjacency matrix of the interconnecting graph. There are many theoretical results on synchronization behaviors of the high-dimensional Kuramoto model and its variations with identical oscillators [23][24][25][26][27][28][29]. In [30], Ha et [34].…”

Section: Introductionmentioning

confidence: 99%

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Synchronization of high‐dimensional Kuramoto models with nonidentical oscillators and interconnection digraphs

Zhang

Zhu

2021

IET Control Theory & Appl

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This paper investigates some synchronization behaviors of high-dimensional Kuramoto models with nonidentical oscillators and interconnection digrphs. By using the matrix Riccati differential equation of the state error variables, it is proved that a high-dimensional Kuramoto model can achieve local practical synchronization when the interconnection digraph is strongly connected or has a spanning tree. Compared with the existing practical synchronization literature, the results are based on general digraphs instead of complete graphs. Moreover, the complete synchronization is proved for proportional nonidentical oscillators limited on a half-sphere and interconnected by a strongly connected digraph. Finally, some numerical simulations are given to validate the obtained theoretical results.

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“…In this paper, we continue studies begun in [8,24] on the emergent dynamics of the LHS model. The LHS model corresponds to the complex counterpart of the Lohe sphere(LS) model which has been extensively studied in previous literature [11,26,32,33,37,43]. The LHS model is the first-order aggregation model describing continuous-time dynamics of particle's position on the Hermitian unit sphere HS…”

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confidence: 99%

Emergent behaviors of Lohe Hermitian sphere particles under time-delayed interactions

Ha¹,

Hwang²,

Park³

2021

NHM

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We study emergent behaviors of the Lohe Hermitian sphere(LHS) model with a time-delay for a homogeneous and heterogeneous ensemble. The LHS model is a complex counterpart of the Lohe sphere(LS) aggregation model on the unit sphere in Euclidean space, and it describes the aggregation of particles on the unit Hermitian sphere in C d with d ≥ 2. Recently it has been introduced by two authors of this work as a special case of the Lohe tensor model. When the coupling gain pair satisfies a specific linear relation, namely the Stuart-Landau(SL) coupling gain pair, it can be embedded into the LS model on R 2d . In this work, we show that if the coupling gain pair is close to the SL coupling pair case, the dynamics of the LHS model exhibits an emergent aggregate phenomenon via the interplay between time-delayed interactions and nonlinear coupling between states. For this, we present several frameworks for complete aggregation and practical aggregation in terms of initial data and system parameters using the Lyapunov functional approach.

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A lifting method for analyzing distributed synchronization on the unit sphere

Cited by 34 publications

References 25 publications

Emergent behaviors of the generalized Lohe matrix model

Emergent behaviors of the generalized Lohe matrix model

Synchronization of high‐dimensional Kuramoto models with nonidentical oscillators and interconnection digraphs

Emergent behaviors of Lohe Hermitian sphere particles under time-delayed interactions

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