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

Lohe, M. A.

doi:10.1063/1.5085248

Cited by 42 publications

(25 citation statements)

References 72 publications

(224 reference statements)

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“…In this paper, we propose a generalized first-order aggregation model on the space of complex matrices with the same size(not necessarily a square matrix) and study its emergent dynamics. For the Stiefel manifold consisting of d 1 × d 2 matrix S with the property S t S = I d2 , emergent dynamics of an aggregation model on the Stiefel manifold was extensively studied in [26,42,43], and we also refer to [38] for an related issue. In fact, for U(d), our proposed model can be reduced to the Lohe matrix model [30,34,38,39,40].…”

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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%

“…Preliminaries. In this section, we briefly introduce the Lohe tensor model [33], and discuss relations between the generalized complex Lohe sphere model [32] and the Lohe matrix model [38,39,40].…”

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confidence: 99%

Emergent behaviors of the generalized Lohe matrix model

Ha¹,

Park²

2021

DCDS-B

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“…To provide various constants of motion, we define some special type of natural frequency matrices set SN d . Constants of motion for matrix Riccati equations are introduced in [26,31]. We also refer to constants of motion of the Lohe sphere model and Lohe hermitian sphere model in [14] and [18], respectively.…”

Section: 22mentioning

confidence: 99%

“…Now we want to construct the natural frequency term of the high-dimensional Winfree model. To generalize the natural frequency terms of the Winfree model to the natural frequency terms of the Winfree sphere model, we refer to a relation between the Kuramoto model [1,2,11,22,23] and the Lohe sphere model [14,26,27,28].…”

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Generalization of the Winfree model to the high-dimensional sphere and its emergent dynamics

Park¹

2022

DCDS

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<p style='text-indent:20px;'>We present a high-dimensional Winfree model in this paper. The Winfree model is a mathematical model for synchronization on the unit circle. We generalize this model compare to the high-dimensional sphere and we call it the Winfree sphere model. We restricted the support of the influence function in the neighborhood of the attraction point to a small diameter to mimic the influence function as the Dirac delta distribution. We can obtain several new conditions of the complete phase-locking states for the identical Winfree sphere model from restricting the support of the influence function. We also prove the complete oscillator death(COD) state from the exponential <inline-formula><tex-math id="M1">\begin{document}$ \ell^1 $\end{document}</tex-math></inline-formula>-stability and the existence of the equilibrium solution.</p>

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“…Lohe sphere model On the other hand, the emergent dynamics of Lohe type matrix models has also been investigated in literature, e.g., sufficient conditions for complete aggregations [6,14], mean-field approach for quaternion's collective dynamics [13], generalized Lohe sphere model on Riemannian manifolds [20], a gradient flow approach for the Lohe matrix model [27], conserved quantities and non-abelian generalization of the Kuramoto model [30,31,32] etc.…”

Section: Conservation Lawsmentioning

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

Cited by 42 publications

References 72 publications

Emergent behaviors of the generalized Lohe matrix model

Emergent behaviors of the generalized Lohe matrix model

Generalization of the Winfree model to the high-dimensional sphere and its emergent dynamics

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

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