# Complete aggregation of the Lohe tensor model with the same free flow

@article{Ha2020CompleteAO,
title={Complete aggregation of the Lohe tensor model with the same free flow},
author={Seung‐Yeal Ha and Hansol Park},
journal={arXiv: Mathematical Physics},
year={2020}
}
• Published 11 April 2020
• Computer Science
• arXiv: Mathematical Physics
The Lohe tensor model is a first-order tensor-valued continuous-time model for the aggregation of tensors with the same rank and size. It reduces to well-known aggregation models such as the Kuramoto model, the Lohe sphere model and the Lohe matrix model as special cases for low-rank tensors. We present a sufficient and necessary framework for the solution splitting property(SSP) and analyze two possible asymptotic states(completely aggregate state and bi-polar state) which can emerge from a…
10 Citations
Emergent Behaviors of Lohe Tensor Flocks
• Mathematics
• 2020
We present a new aggregation model on the space of rank-m tensors with the same size, and study emergent dynamics of the proposed model. Our proposed new aggregation model is general enough to
Existence and Emergent Dynamics of Quadratically Separable States to the Lohe Tensor Model
• Mathematics
SIAM Journal on Applied Dynamical Systems
• 2022
A tensor is a multi-dimensional array of complex numbers, and the Lohe tensor model is an aggregation model on the space of tensors with the same rank and size. It incorporates previously
On the Schrödinger–Lohe Hierarchy for Aggregation and Its Emergent Dynamics
• Mathematics, Computer Science
Journal of Statistical Physics
• 2020
This paper provides an explicit connection between the Schrodinger-Lohe model and the complex Lohe sphere model, and then by exploiting this explicit relation, it constructs infinite-dimensional liftings of the Lohe matrix and theLohe tensor models.
Collective behaviors of the Lohe Hermitian sphere model with inertia
• Mathematics
• 2020
We present a second-order extension of the first-order Lohe hermitian sphere(LHS) model and study its emergent asymptotic dynamics. Our proposed model incorporates an inertial effect as a
Emergent behaviors in group ring flocks
• Mathematics, Economics
• 2020
We present a first-order aggregation model on a group ring, and study its asymptotic dynamics. In a positive coupling strength regime, we show that the flow generated by the proposed model tends to
A Dynamical Systems Approach for the Shape Matching of Polytopes Along Rigid-Body Motions
• Mathematics
SIAM Journal on Applied Mathematics
• 2021
We present a dynamical systems approach for geometric matchings in an ensemble of polytopes along rigid-body motions. Each polytope can be characterized by a vertex set and edge or faces determined
An algebraic approach for the weak coupling of multiple Lohe tensor models
• Mathematics
• 2021
We present a systematic algebraic approach for the weak coupling of Cauchy problems to multiple Lohe tensor models. For this, we identify an admissible Cauchy problem to the Lohe tensor (LT) model
Collective Dynamics of Lohe type aggregation models
• Mathematics
• 2021
In this paper, we review state-of-the-art results on the collective behaviors for Lohe type first-order aggregation models. Collective behaviors of classical and quantum many-body systems have
On the Gradient Flow Formulation of the Lohe Matrix Model with High-Order Polynomial Couplings
• Mathematics
• 2020
We present a generalized Lohe matrix model for a homogeneous ensemble with higher order couplings via the gradient flow approach. For the homogeneous free flow with the same hamiltonian, it is well

## References

SHOWING 1-10 OF 46 REFERENCES
Emergent behaviors of the generalized Lohe matrix model
• Mathematics
• 2020
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
Emergent Behaviors of Lohe Tensor Flocks
• Mathematics
• 2020
We present a new aggregation model on the space of rank-m tensors with the same size, and study emergent dynamics of the proposed model. Our proposed new aggregation model is general enough to
From the Lohe Tensor Model to the Lohe Hermitian Sphere Model and Emergent Dynamics
• Computer Science
SIAM J. Appl. Dyn. Syst.
• 2020
This work studies how the LHS model appears as a special case of the Lohe tensor model and provides a cross-ratio like conserved quantity, a sufficient framework for the complete aggregation and a uniform $\ell^p$-stability estimate with respect to initial data.
Emergent Behavior of a Second-Order Lohe Matrix Model on the Unitary Group
• Mathematics
Journal of Statistical Physics
• 2019
We study a second-order extension to the first-order Lohe matrix model on the unitary group which can be reduced to the second-order Kuramoto model with inertia as a special case. For the proposed
On the Relaxation Dynamics of Lohe Oscillators on Some Riemannian Manifolds
• Mathematics
Journal of Statistical Physics
• 2018
We study the collective relaxation dynamics appearing in weakly coupled Lohe oscillators in a large coupling regime. The Lohe models on the unit sphere and unitary group were proposed as a nonabelian
Hand-waving and Interpretive Dance: An Introductory Course on Tensor Networks
• Physics
• 2016
The curse of dimensionality associated with the Hilbert space of spin systems provides a significant obstruction to the study of condensed matter systems. Tensor networks have proven an important
Asymptotic behavior and stability for the Schrödinger-Lohe model
• Mathematics
Journal of Mathematical Physics
• 2018
The Schrodinger-Lohe (S-L) model is an infinite-dimensional non-Abelian generalization of the Kuramoto model which serves as a prototype model for quantum synchronization. In this paper, we study
Non-Abelian Kuramoto models and synchronization
We describe non-Abelian generalizations of the Kuramoto model for any classical compact Lie group and identify their main properties. These models may be defined on any complex network where the
Systems of matrix Riccati equations, linear fractional transformations, partial integrability and synchronization
• M. A. Lohe
• Mathematics
Journal of Mathematical Physics
• 2019
We partially integrate a system of rectangular matrix Riccati equations which describe the synchronization behavior of a nonlinear complex system of N globally connected oscillators. The equations