# Asymptotic formation and orbital stability of phase-locked states in
Kuramoto–Lohe type synchronization models on Lie groups

Sang Woo Ryoo
Mathematics Department, Princeton University,
Princeton, New Jersey 08544-1000, United States<EMAIL_ADDRESS>
###### Abstract.

It is known that the Kuramoto model has a critical coupling strength above
which phase-locked states exist, and, by the work of Choi, Ha, Jung, and Kim
(2012), that these phase-locked states are orbitally stable. This property of
admitting orbitally stable phase-locked states is preserved under the
nonabelian generalizations of the Kuramoto model pioneered by Lohe (2009). We
provide a framework for understanding these phenomena, by formulating the
aforementioned models as special cases of a general ODE model describing
populations of particles on Lie groups with pairwise attractive Kuramoto-type
interactions. This general model can be used in producing many specific models
with orbitally stable nonabelian phase-locked states.

###### Key words and phrases:

Kuramoto model, Lohe model, Lie group, Gronwall inequality, functional
calculus

###### 1991 Mathematics Subject Classification:

34D06, 34H15, 82C22

Acknowledgment: This work was written while the author was visiting Seoul
National University under the auspices of the Hyperbolic and Kinetic Equations
(HYKE) group. We thank Hangjun Cho and Myeongju Kang of the HYKE group for
helpful discussions. This work was partially supported by the Korea Foundation
for Advanced Studies.

###### Contents

  1. 1 Introduction
    1. 1.1 The Kuramoto model and its nonabelian generalizations
    2. 1.2 The generalized Kuramoto–Lohe model and our main theorems
    3. 1.3 Directions for further research
  2. 2 Notation and preliminaries on Lie groups and Lie algebras
  3. 3 Proof of Theorems 1.7 and 1.8
    1. 3.1 Proof of Theorem 1.7 assuming Theorem 1.8
    2. 3.2 Reducing Theorem 1.8 to Theorem 3.1
    3. 3.3 Reducing from a model on the Lie group to a model on the Lie algebra
    4. 3.4 Two Gronwall inequalities

##  1\. Introduction

###  1.1. The Kuramoto model and its nonabelian generalizations

To motivate the paper, we begin with a description of previous works; those
who are curious of our main results may skip to subsection 1.2.

Starting with Huygens’s observation in the mid-seventeenth century of the
asynchronization of two pendulum clocks hanging on a common bar, synchronous
phenomena have often been observed in our nature and have been reported in
scientific literature; see Pikovsky, Rosenblum, and Kurths [26] for a
comprehensive survey. However, systematic approaches to synchronization based
on mathematical models were only relatively recently made by A. Winfree [30]
and Y. Kuramoto [21, 22] in the 1960s and 1970s, respectively. After Winfree
and Kuramoto’s works, many phenomenological and mechanical synchronization
models were proposed in the biology, engineering, and statistical physics
communities. Amongst other models, in this paper we begin our discussion with
Kuramoto’s model, which often serves as a prototype model in the study of
synchronization.

The _Kuramoto model_ , introduced in [21], is a simple first-order dynamical
system for the synchronization of an ensemble of weakly coupled phase
oscillators. In [21] it was shown that the corresponding mean-field kinetic
equation exhibits a phase-transition from the disordered state to the ordered
state, as the coupling strength increases from zero to some large value; this
led to an extensive investigation of the Kuramoto model, such as [1, 2, 3, 4,
5, 6, 10, 11, 13, 17, 18, 28, 29], to name a few. In this paper, we focus on
the Kuramoto model without passing the system size to infinity; the phase-
transition is still present in this finitary setting.

More precisely, the Kuramoto model is the Cauchy problem for the real
variables $\\{\theta_{i}(t)\\}_{i=1}^{N}$ given by the nonlinear system of
ordinary differential equations

(1.1)
$\begin{cases}\dot{\theta}_{i}(t)=\omega_{i}+\frac{\kappa}{N}\sum_{j=1}^{N}\sin(\theta_{j}(t)-\theta_{i}(t))\\\
\theta_{i}(0)=\theta_{i}^{0}\end{cases},\quad i=1,\cdots,N,$

where $\omega_{1},\cdots,\omega_{N}\in\mathbb{R}$ are the intrinsic
frequencies, $\kappa>0$ is the coupling strength, and
$\theta_{1}^{0},\cdots,\theta_{N}^{0}\in\mathbb{R}$ are the initial data.
Existence and global uniqueness of solutions follow from the Cauchy-Lipschitz
theory.

In nonlinear dynamical systems theory, one of the central problems other than
well-posedness is to understand the large-time dynamics such as existence of
equilibria or limit-cycles and their stability. Thus, the first step in this
direction is to identify the possible limiting configurations of the given
dynamical system, and the next step is to study the stability of the limiting
configurations.

In order to do this, it is worth observing the following conservation law for
(1.1):

(1.2)
$\frac{d}{dt}\Big{(}\sum_{i=1}^{N}\theta_{i}(t)-t\sum_{i=1}^{N}\nu_{i}\Big{)}=0,\quad
t>0.$

Thus, if $\sum_{i=1}^{N}\nu_{i}\neq 0$, then the Kuramoto model (1.1) cannot
have an equilibrium. However, the Kuramoto model has a related Galilean
invariance, namely (1.1) is invariant under Galilean transformations of the
form

(1.3) $\theta_{i}^{\prime}=\theta_{i}-\nu
t-\vartheta,~{}\nu_{i}^{\prime}=\nu_{i}-\nu,\quad i=1,\cdots,N,$

for $\nu,\vartheta\in\mathbb{R}$. In particular, if we take

$\nu=\frac{1}{N}\sum_{j=1}^{N}\nu_{j},\quad\vartheta=0,$

then the conservation law (1.2) now becomes conservation of total phase:

(1.4) $\sum_{i=1}^{N}\theta_{i}^{\prime}=\sum_{i=1}^{N}\theta_{i}^{0}.$

Thus it makes sense to discuss equilibria of the transformed variables
$\theta_{i}^{\prime}$. In terms of the pre-transform variables $\theta_{i}$,
an equilibrium of $\theta_{i}^{\prime}$ means an equilibrium relative to a
frame rotating with velocity given by the average phase velocity
$\frac{1}{N}\sum_{i=1}^{N}\nu_{i}$. By the conservation law (1.2), this is
equivalent to a relative equilibrium, namely a state where the relative phases
$\theta_{i}-\theta_{j}$ are constant. We call these relative equilibria phase-
locked states, and we call convergence to such equilibria asymptotic phase-
locking.

######  Definition 1.1 ([3, 5, 17, 28, 29]).

Let $\\{\theta_{i}(t)\\}_{i=1}^{N}$ be a solution to the Kuramoto model (1.1).

  1. (1)

$\\{\theta_{i}(t)\\}_{i=1}^{N}$ is a phase-locked state of (1.1) if all the
relative phase differences are constant:

$\theta_{i}(t)-\theta_{j}(t)=\theta_{i}(0)-\theta_{j}(0),\quad t\geq 0,\qquad
1i,j=1,\cdots,N.$

  2. (2)

$\\{\theta_{i}(t)\\}_{i=1}^{N}$ exhibits asymptotic phase-locking if the
relative phase differences converge as $t\to\infty$:

$\exists\lim_{t\to\infty}(\theta_{i}(t)-\theta_{j}(t)),\quad i,j=1,\cdots,N.$

######  Remark 1.1.

It is easy to see that asymptotic phase-locking implies that the relative
frequencies must tend to zero asymptotically:

$\lim_{t\to\infty}|{\dot{\theta}}_{i}(t)-{\dot{\theta}}_{j}(t)|=0,\quad
i,j=1,\cdots,N.$

This is sometimes used as the definition for asymptotic phase-locking.

The existence and stability of phase-locked states are important topics in the
nonlinear dynamics of the Kuramoto model, as discussed in [1, 2, 4, 6, 11].

Heuristically, as the coupling strength $\kappa$ gets larger and larger, it
should become “easier” for the system (1.1) to admit phase-locked states and
exhibit asymptotic phase-locking. Indeed, the interaction term
$\sin(\theta_{i}-\theta_{j})$ is pairwise attractive, and so if the coupling
strength $\kappa$ is large enough compared to the differences
$\nu_{i}-\nu_{j}$ in the intrinsic frequencies, the pairwise interaction terms
$\sin(\theta_{k}-\theta_{l})$ should dominate in the expression
$\dot{\theta}_{i}-\dot{\theta}_{j}$, leading to an “ease” of the existence of
phase-locked states and an “ease” of approaching them.

This heuristic argument may be quantified into the following precise theorems.
First, when the $\nu_{i}$’s are identical, we have the following theorem.

######  Theorem 1.1 ([13, Theorem 3.1]).

Suppose that the coupling strength and natural frequencies satisfy

$\kappa>0,\quad\nu_{i}=\nu_{j},~{}i,j=1,\cdots,N,$

and that the initial configuration satisfies

$D^{0}\coloneqq\max_{i,j=1,\cdots,N}|\theta_{i}^{0}-\theta_{j}^{0}|<\pi.$

Then

$\max_{i,j=1,\cdots,N}|\theta_{i}(t)-\theta_{j}(t)|\leq
D^{0}\exp\left(-\frac{\kappa t\sin D^{0}}{D^{0}}\right),\quad t\geq 0.$

Second, when the $\nu_{i}$’s are nonidentical, we have the following theorem.

######  Theorem 1.2 ([4, 10, 11]).

Suppose that the coupling strength $\kappa$ and an auxiliary parameter
$\kappa_{e}$ satisfy

$\kappa>\kappa_{e}>\max_{i,j=1,\cdots,N}|\nu_{i}-\nu_{j}|>0$

and let $\\{\theta_{i}(t)\\}_{i=1}^{N}$ be a solution to (1.1) such that

$\max_{i,j=1,\cdots,N}|\theta_{i}^{0}-\theta_{j}^{0}|<\pi-{{\arcsin\left(\frac{\max_{i,j=1,\cdots,N}|\nu_{i}-\nu_{j}|}{\kappa_{e}}\right)}}.$

Then, the following assertions hold:

  1. (1)

The phase diameter is bounded in the sense that there exists a finite time
$T\geq 0$ such that

$\max_{i,j=1,\cdots,N}|\theta_{i}(t)-\theta_{j}(t)|\leq\arcsin\left(\frac{\max_{i,j=1,\cdots,N}|\nu_{i}-\nu_{j}|}{\kappa}\right),\quad
for~{}t\geq T.$

  2. (2)

Asymptotic phase-locking occurs at an exponential rate: there exist positive
constants $C_{0}$ and $\Lambda=O(\kappa)$ such that

$D({\dot{\Theta}}(t))\leq C_{0}\exp(-\Lambda t),\quad\mbox{for }t\geq 0.$

  3. (3)

The emergent phase-locked state is unique up to $U(1)$-symmetry, and is
ordered according to the ordering of their natural frequencies. Namely, if
$\\{\tilde{\theta}_{i}(t)\\}_{i=1}^{N}$ were another solution to (1.1) with

$\max_{i,j=1,\cdots,N}|\tilde{\theta}_{i}^{0}-\tilde{\theta}_{j}^{0}|<\pi-{{\arcsin\left(\frac{\max_{i,j=1,\cdots,N}|\nu_{i}-\nu_{j}|}{\kappa_{e}}\right)}},$

then there exists $\phi\in\mathbb{R}$ such that

$\lim_{t\to\infty}\left(\theta_{i}(t)-\tilde{\theta}_{i}(t)\right)=\phi,\quad
i=1,\cdots,N.$

Also, there are universal constants $U$ and $L$ such that for any indices
$i,j$ with $\nu_{i}\geq\nu_{j}$,

$\sin^{-1}\left(\frac{\nu_{i}-\nu_{j}}{\kappa
U}\right)\leq\lim_{t\to\infty}\left(\theta_{i}(t)-\theta_{j}(t)\right)\leq\sin^{-1}\left(\frac{\nu_{i}-\nu_{j}}{\kappa
L}\right).$

######  Remark 1.2.

The threshold $\max_{i,j=1,\cdots,N}|\nu_{i}-\nu_{j}|$ in Theorem 1.2 is
equivalent up to universal constant factors to the actual threshold for the
existence of phase-locked states. More precisely, by [28], for each fixed
$N\geq 2$ and $\\{\nu_{i}\\}_{i=1}^{N}$ there exists
$\kappa_{c}(\\{\nu_{i}\\}_{i=1}^{N})\geq 0$ such that (1.1) admits a phase-
locked state if and only if $\kappa\geq\kappa_{c}(\\{\nu_{i}\\}_{i=1}^{N})$;
this $\kappa_{c}(\\{\nu_{i}\\}_{i=1}^{N})$ satisfies [4, 6, 18] the following:

$\frac{N}{2(N-1)}\max_{i,j=1,\cdots,N}|\nu_{i}-\nu_{j}|\leq\kappa_{c}(\\{\nu_{i}\\}_{i=1}^{N})\leq\max_{i,j=1,\cdots,N}|\nu_{i}-\nu_{j}|.$

This concludes our discussion of the Kuramoto model; to recapitulate, in the
case of identical oscillators, the Kuramoto model completely synchronizes in
the sense of Theorem 1.1, and in the case of nonidentical oscillators, the
Kuramoto model admits an orbitally stable phase-locked state in the sense of
Theorem 1.2. The purpose of this paper is to obtain a better understanding of
Theorems 1.1 and 1.2 by proving them in a more general setting.

Lohe [23, 24] suggested a nonabelian generalization of the Kuramoto model to
the unitary groups $U(d)$ as follows: the _Lohe model_ describes the dynamics
of $\\{U_{i}(t)\\}_{i=1}^{N}\subset U(d)$ as

(1.5)
$\begin{cases}\dot{U}_{i}U_{i}^{\dagger}=iH_{i}+\frac{\kappa}{2N}\sum_{j=1}^{N}\left(U_{j}U_{i}^{\dagger}-U_{i}U_{j}^{\dagger}\right),\\\
U_{i}(0)=U_{i}^{0},\end{cases}\quad i=1,\cdots,N,$

where $\kappa>0$ is again the coupling strength, the Hermitian matrix $H_{i}$
is the Hamiltonian of $U_{i}$, $\dagger$ denotes the Hermitian conjugate, and
$\\{U_{i}^{0}\\}_{i=1}^{N}\subset U(d)$ is the initial data. The usual Cauchy-
Lipschitz theory again guarantees uniqueness and global existence of
solutions. It can be checked that the Kuramoto model is the $U(1)$ case of the
Lohe model (1.5).

Model (1.5) is a mathematical model for quantum synchronization [24]: consider
a quantum network [20, 31, 32] consisting of $N$ quantum nodes and quantum
channels connecting all possible pairs of quantum nodes. Each quantum node can
be viewed as a component of a physical system interacting via quantum
channels. For instance, atoms at nodes can have effect spin-spin interactions
generated by a single photon pulse traveling along the channels (see [20] for
a detailed explanation). Lohe [23, 24] proposed the model (1.5) for
synchronization over such quantum networks and studied the existence of phase-
locked states of the proposed model based on numerical simulations.

As explained earlier, once well-posedness is out of the way, the central task
in studying model (1.5) is to understand the large-time dynamics, namely the
existence of equilibria or limit-cycles and their stability. In the nonabelian
setting, one can see that [23, 24] the correct definition for phase-locked
states are solutions $\\{U_{i}(t)\\}_{i=1}^{N}$ such that the nonabelian
ratios $U_{i}U_{j}^{\dagger}$ are constant, or equivalently right-invariant
flows of the form $U_{i}=U_{i}^{\infty}e^{-\mathrm{i}\Lambda t}$, where
$U_{i}^{\infty}\in U(d)$, $i=1,\cdots,N$ are unitary matrices, and $\Lambda$
is a constant $d\times d$ Hermitian matrix. Solving the corresponding
nonlinear algebraic system, one can see that $U_{i}^{\infty}$, $i=1,\cdots,N$,
and $\Lambda$ must satisfy the following algebraic constraints:

$U_{i}^{\infty}\Lambda
U_{i}^{\infty\dagger}=H_{i}-\frac{\mathrm{i}\kappa}{2N}\sum_{k=1}^{N}\left(U_{i}^{\infty}U_{k}^{\infty\dagger}-U_{k}^{\infty}U_{i}^{\infty\dagger}\right),\quad
i=1,\cdots,N.$

The orbital stability of these phase-locked states roughly means that if
$\\{V_{i}\\}_{i=1}^{N}$ is another solution with similar initial conditions,
then the limiting values of $U_{i}U_{j}^{\dagger}$ and $V_{i}V_{j}^{\dagger}$
are the same and that $V_{i}^{\dagger}U_{i}$ converges to a value common for
all $i$. Note that the stability issue can be studied in an a priori setting
without knowing the existence of phase-locked states and implies the
uniqueness of phase-locked states.

Lohe [23] numerically observed a nonlinear version of Theorem 1.2 for the
$U(2)$ model, and the author and Ha [16] gave a rigorous proof of a nonlinear
version of Theorems 1.1 and 1.2 for all unitary groups $U(d)$. These theorems
are stated as follows.

######  Theorem 1.3 ([16, Theorem 1]).

Suppose that the coupling strength $\kappa$ and Hermitian matrices $H_{i}$,
$i=1,\cdots,N$, satisfy

$\kappa>0,\quad H_{i}=H_{j},~{}i,j=1,\cdots,N,$

and suppose the initial data $\\{U_{i}^{0}\\}_{i=1}^{N}$ satisfies

$D^{0}\coloneqq\max_{i,j=1,\cdots,N}\|U_{i}^{0}-U_{j}^{0}\|_{\mathrm{Frob}}<\sqrt{2},$

where $\|\cdot\|_{\mathrm{Frob}}$ denotes the Frobenius norm. Then

$\max_{i,j=1,\cdots,N}\|U_{i}(t)-U_{j}(t)\|_{\mathrm{Frob}}\leq\sqrt{\frac{2(D^{0})^{2}}{(D^{0})^{2}+(2-(D^{0})^{2})e^{2\kappa
t}}}.$

######  Theorem 1.4 ([16, Theorem 3]).

Suppose that the coupling strength $\kappa$ and Hermitian matrices $H_{i}$,
$i=1,\cdots,N$, satisfy

$\kappa>\frac{54}{17}\max_{i,j=1,\cdots,N}\|H_{i}-H_{j}\|_{\mathrm{Frob}}$

and the initial data $\\{U_{i}^{0}\\}_{i=1}^{N}$ satisfies

$\max_{i,j=1,\cdots,N}\|U_{i}^{0}-U_{j}^{0}\|_{\mathrm{Frob}}<\sqrt{2}.$

Then the following assertions are true.

  1. (1)

The flow $\\{U_{i}(t)\\}_{i=1}^{N}$ achieves asymptotic phase-locking, i.e.,

$\lim_{t\to\infty}U_{i}(t)U_{j}(t)^{\dagger}$

converges exponentially fast.

  2. (2)

There exists a phase-locked state $\\{V_{i}\\}$ and a unitary matrix $L\in
U(d)$ such that

$\max_{i,j=1,\cdots,N}\|V_{i}-V_{j}\|_{\mathrm{Frob}}<\sqrt{2},\quad\lim_{t\to\infty}\|U_{i}-V_{i}L\|_{\mathrm{Frob}}=0,$

the convergence happening exponentially fast.

  3. (3)

Phase-locked states $\\{V_{i}\\}$ with
$\max_{i,j=1,\cdots,N}\|V_{i}-V_{j}\|_{\mathrm{Frob}}<\sqrt{2}$ are unique up
to right-multiplication, that is, if $\\{W_{i}\\}$ is another phase-locked
state with $\max_{i,j=1,\cdots,N}\|W_{i}-W_{j}\|_{\mathrm{Frob}}<\sqrt{2}$,
then there exists a unitary matrix $L\in U(d)$ such that $W_{i}=V_{i}L$.

######  Remark 1.3.

One can see from Theorem 1.4 that if $\\{U_{i}(t)\\}_{i=1}^{N}$ and
$\\{\tilde{U}_{i}(t)\\}_{i=1}^{N}$ are two solutions to (1.5) with

$\kappa>\frac{54}{17}\max_{i,j=1,\cdots,N}\|H_{i}-H_{j}\|_{\mathrm{Frob}},\quad\|U_{i}^{0}-U_{j}^{0}\|_{\mathrm{Frob}},\|\tilde{U}_{i}^{0}-\tilde{U}_{j}^{0}\|_{\mathrm{Frob}}<\sqrt{2},~{}i,j=1,\cdots,N,$

that the following orbital stability statements hold:

  1. (1)

$\lim_{t\to\infty}U_{i}(t)U_{j}(t)^{\dagger}=\lim_{t\to\infty}\tilde{U}_{i}(t)\tilde{U}_{j}(t)^{\dagger}$,
$i,j=1,\cdots,N$,

  2. (2)

there exists a unitary matrix $L_{\infty}\in U(d)$ such that
$\lim_{t\to\infty}U_{i}(t)^{\dagger}\tilde{U}_{i}(t)=L_{\infty}$.

Theorems 1.3 and 1.4 can be viewed as $U(d)$-versions of Theorems 1.1 and 1.2.

It was suggested by Lohe [23] that the model (1.5) can be extended to other
matrix groups. Here is a direct quotation from [23, p.5]:

> …There also exist non-Abelian Kuramoto models for the symplectic groups,
> with defining equations similar to [(1.5)], and hence for all classical
> compact Lie groups. By choosing an indefinite metric one may extend these
> models to the noncompact case, for which trajectories are unbounded,
> however, we do not discuss synchronization or other properties of these
> particular models here…

In [15], the author, Ha, and Ko interpreted this as a model on
$GL_{n}(\mathbb{C})$ as follows:

(1.6)
$\begin{cases}\dot{X}_{i}X_{i}^{-1}=H_{i}+\frac{\kappa}{2N}\sum_{j=1}^{N}\left(X_{j}X_{i}^{-1}-X_{i}X_{j}^{-1}\right),\\\
X_{i}(0)=X_{i}^{0},\end{cases}\quad i=1,\cdots,N,$

where $H_{i}\in M_{n,n}(\mathbb{C})$ are $n\times n$ complex matrices and
$X_{i}^{0}\in GL_{n}(\mathbb{C})$ are the initial data, and characterized
various matrix Lie groups, such as $U(d)$, $O(m,n)$, $SO(m,n)$, $SP(d)$,
$SL_{2}(\mathbb{R})$, on which this model defines a closed trajectory. (More
precisely, if a matrix Lie group $G\leq GL_{n}(\mathbb{C})$ has Lie algebra
$\mathfrak{g}$ satisfying the closure condition

$X-X^{-1}\in\mathfrak{g},\quad\forall X\in G,$

then with $H_{i}\in\mathfrak{g}$ and $X_{i}^{0}\in G$ the local solution
$\\{X_{i}(t)\\}_{i=1}^{N}$ to (1.6) satisfies $X_{i}(t)\in G$ for its time of
existence.) Note that the Cauchy-Lipschitz theory now only guarantees the
uniqueness and _local_ existence of solutions to (1.6), for
$GL_{n}(\mathbb{C})$ is noncompact. The following example illustrates finite-
time blowup.

######  Example 1.1 ([15, Remark 2.3]).

We have the divergent initial condition

$n=N=2,~{}H_{1}=H_{2}=0,~{}X_{1}^{0}=\operatorname{diag}(1,1),~{}X_{2}^{0}=\operatorname{diag}(-0.5,-2).$

Indeed, the solution takes the form

$X_{1}=\operatorname{diag}(x_{1},x_{1}^{-1}),\quad
X_{2}=\operatorname{diag}(x_{2},x_{2}^{-1}),$

and $x_{1}/x_{2}$ solves

$\frac{d}{dt}(x_{1}/x_{2})=\kappa\left(1-(x_{1}/x_{2})^{2}\right),\quad
x_{1}(0)/x_{2}(0)=-2$

and hence diverges to $-\infty$ in finite time.

Thus, care is needed in arguing global existence of solutions for special
initial data.

One can define phase-locked states of (1.6) to be solutions
$\\{X_{i}(t)\\}_{i=1}^{N}$ such that $X_{i}(t)X_{j}(t)^{-1}$ is constant, and
can see that they are right-invariant flows of the form
$X_{i}(t)=X_{i}^{\infty}e^{\Lambda t}$, where $X_{i}^{\infty}\in
GL_{n}(\mathbb{C})$, $i=1,\cdots,N$, and $\Lambda\in M_{n,n}(\mathbb{C})$
satisfy

$X_{i}^{\infty}\Lambda(X_{i}^{\infty})^{-1}=H_{i}+\frac{\kappa}{2N}\sum_{j=1}^{N}\left[X_{j}^{\infty}(X_{i}^{\infty})^{-1}-X_{i}^{\infty}(X_{j}^{\infty})^{-1}\right],\quad
i=1,\cdots,N.$

The work [15] was able to establish a nonlinear version of Theorems 1.1 and
1.2 for the model (1.6), as follows. First, for identical oscillators, we have
the following theorem.

######  Theorem 1.5 ([15, Proposition 4.1, Theorems 4.1 and 4.2]).

Let the coupling strength and intrinsic Hamiltonians satisfy

$\kappa>0,\quad H_{i}=H\in M_{n,n}(\mathbb{C}),~{}i=1,\cdots,N,$

and suppose the initial data satisfies

$D^{0}\coloneqq\|X_{i}^{0}(X_{j}^{0})^{-1}-I_{n}\|_{\mathrm{Frob}}<1.$

Then

  1. (1)

(Global existence of solutions) There is a global solution
$\\{X_{i}(t)\\}_{i=1}^{N}$ to (1.6).

  2. (2)

(Solution operator splitting) $\tilde{X}_{i}(t)=e^{-Ht}X_{i}(t)$ is a solution
to (1.6) with $H_{i}=0$, $\tilde{X}_{i}(0)=X_{i}^{0}$.

  3. (3)

(Exponential synchronization) We have

$\|e^{-Ht}X_{i}(t)X_{j}(t)^{-1}e^{Ht}-I_{n}\|_{\mathrm{Frob}}\leq\frac{D^{0}}{(1-D^{0})e^{\kappa
t}+D^{0}}.$

In particular, if $\kappa>2\|H\|_{\mathrm{Frob}}$, then
$X_{i}(t)X_{j}(t)^{-1}\to I_{n}$ exponentially.

Second, for nonidentical oscillators, we have the following theorem.

######  Theorem 1.6 ([15, Theorems 5.1, 5.2]).

Suppose that the coupling strength $\kappa$ and initial data
$\\{X_{i}^{0}\\}_{i=1}^{N}$ satisfy

$\kappa>(6+4\sqrt{2})\max_{i=1,\cdots,N}\|H_{i}\|_{\mathrm{Frob}},\quad\|X_{i}^{0}(X_{j}^{0})^{-1}-I_{d}\|_{\mathrm{Frob}}<\sqrt{2}-1,~{}i,j=1,\cdots,N.$

Then the following assertions hold.

  1. (1)

The solution $\\{X_{i}(t)\\}_{i=1,\cdots,N,t\geq 0}$ to (1.6) exists for all
$t\geq 0$.

  2. (2)

The solution $\\{X_{i}(t)\\}_{i=1,\cdots,N,t\geq 0}$ achieves asymptotic
phase-locking in the sense that the limit
$\lim_{t\to\infty}X_{i}(t)X_{j}(t)^{-1}$ exists for all $i,j=1,\cdots,N$. This
convergence is exponentially fast.

  3. (3)

There exists a phase-locked state $\\{Y_{i}(t)\\}_{i=1,\cdots,N,t\geq 0}$ with

$\|Y_{i}Y_{j}^{-1}-I_{d}\|_{\mathrm{Frob}}<\sqrt{2}-1,$

and it is unique up to right-multiplication: if
$\\{Z_{i}(t)\\}_{i=1,\cdots,N,t\geq 0}$ is another phase-locked state with

$\|Z_{i}Z_{j}^{-1}-I_{d}\|_{\mathrm{Frob}}<\sqrt{2}-1,$

then there exists some $S\in GL_{n}(\mathbb{C})$ such that

$Z_{i}=Y_{i}S,\quad i=1,\cdots,N.$

Moreover, under the stronger assumption

$\kappa>\frac{16}{7}(6+4\sqrt{2})\max_{i=1,\cdots,N}\|H_{i}\|_{\mathrm{Frob}},$

we have the following assertion.

  1. (4)

For the above phase-locked state $\\{Y_{i}(t)\\}_{i=1,\cdots,N,t\geq 0}$ there
exists some $R\in GL_{n}(\mathbb{C})$ such that

$\lim_{t\to\infty}Y_{i}(t)^{-1}X_{i}(t)=R.$

This convergence is exponentially fast.

All in all, Theorems 1.5 and 1.6 give analogues of Theorems 1.1 and 1.2 for
system (1.6). Compared to the previous Theorems 1.1, 1.2, 1.3, 1.4, we start
to see that the threshold for $\kappa$ for synchronous behavior start to
depend on the maximum _absolute_ magnitude
$\max_{i=1,\cdots,N}\|H_{i}\|_{\mathrm{Frob}}$, instead of the maximum
_relative_ magnitude $\max_{i,j=1,\cdots,N}\|H_{i}-H_{j}\|_{\mathrm{Frob}}$.
This is because the right-invariant quantity
$\|XY^{-1}-I_{d}\|_{\mathrm{Frob}}$, which serves as a proxy for a metric, is
not left-invariant.

Given this line of reasoning so far, it is natural to imagine whether there is
a version of the Kuramoto and Lohe models on general Lie groups and whether it
still enjoys the stability properties outlined so far. We will now present an
abstraction of the ideas presented so far in the setting of Lie groups.

###  1.2. The generalized Kuramoto–Lohe model and our main theorems

Let $G$ be a Lie group with Lie algebra $(\mathfrak{g},[,])$. We wish to model
the behavior of particles $\\{X_{i}(t)\\}_{i=1}^{N}$ on $G$. As before, each
particle $X_{i}$ has an intrinsic Hamiltonian $H_{i}\in\mathfrak{g}$, and,
under the absence of other particles, has equation of motion

$\dot{X}_{i}=(dR_{X_{i}})_{e}(H_{i}),$

where $e\in G$ is the identity element of $G$, and for $g\in G$,
$R_{g}:G\rightarrow G$ denotes right multiplication $R_{g}(x):=xg$ (and
similarly $L_{g}:G\rightarrow G$ denotes left multiplication $L_{g}(x):=gx$).
The interaction between particles $X_{i}$ and $X_{j}$ will happen via their
relative phases $X_{j}X_{i}^{-1}$, and in order to properly define the
equation of motion, we will need a function $\phi:G\to\mathfrak{g}$ to choose
the direction $\phi(X_{j}X_{i}^{-1})\in\mathfrak{g}$ that $X_{i}$ ‘heads
toward’ $X_{j}$.

Thus, we define the _generalized Kuramoto–Lohe model on_ $G$ as the Cauchy
problem

(1.7)
$\begin{cases}\displaystyle\dot{X}_{i}=(dR_{X_{i}})_{e}\left(H_{i}+\frac{\kappa}{N}\sum_{j=1}^{N}\phi(X_{j}X_{i}^{-1})\right),\\\
X_{i}(0)=X_{i}^{0},\end{cases}\quad i=1,\cdots,N,$

for the solution $\\{X_{1},\cdots,X_{N}\\}$, where
$X_{1}^{0},\cdots,X_{N}^{0}\in G$ are the initial data,
$H_{1},\cdots,H_{N}\in\mathfrak{g}$ are the intrinsic Hamiltonians, and
$\kappa>0$ is the coupling strength. Also, $\phi:G\to\mathfrak{g}$ is a map
describing the pairwise interaction of the oscillators and satisfies

(H) $\phi\mbox{ is
}C^{1},\quad\phi(e)=0,\quad(d\phi)_{e}:\mathfrak{g}\to\mathfrak{g}\mbox{ has
only (complex) eigenvalues with positive real parts}.$

It is clear that this model generalizes the Kuramoto model (1.1) and the Lohe
matrix models (1.5) and (1.6). Indeed, for the Kuramoto model we take
$G=\mathbb{S}^{1}$ and interaction function $\phi(\theta)=\sin\theta$ (we are
considering the dynamics on the universal cover $\mathbb{R}$ of
$\mathbb{S}^{1}$). The unitary Lohe model (1.5) is the case $G=U(d)$,
$\phi(U)=\frac{1}{2}(U-U^{\dagger})$, and the matrix Lohe model (1.6) is the
case $G=GL_{n}(\mathbb{C})$, $\phi(X)=\frac{1}{2}(X-X^{-1})$ of the
generalized Kuramoto–Lohe model.

DeVille [9] considered the following model on $GL_{n}(\mathbb{C})$:

(1.8)
$\dot{X}_{i}X_{i}^{-1}=H_{i}+\frac{1}{2}\sum_{j=1}^{N}\gamma_{ij}\left(f(X_{j}X_{i}^{-1})-f(X_{i}X_{j}^{-1})\right).$

For functions $f$ satisfying (H), DeVille’s model also is of the form (1.7).

The standard Cauchy–Lipschitz theory guarantees the existence of a local
$C^{1}$-solution to (1.7). However, global existence of a solution may fail if
the initial data are far apart, as demonstrated in Example 1.1.

Note that, for $g\in G$ and $J$ in the center of $\mathfrak{g}$, the system
(1.7) admits the Galilean transformation

$X_{i}^{0}\mapsto X_{i}^{0}g,\quad X_{i}(t)\mapsto X_{i}(t)ge^{Jt},\quad
H_{i}\mapsto H_{i}+J.$

Therefore, it is natural to consider right-invariant metrics on $G$ for the
purposes of analyzing (1.7), and to consider relative equilibria of (1.7) as
the solutions for which the ratios $X_{i}X_{j}^{-1}$ are constant. More
precisely, endow $\mathfrak{g}$ with a Euclidean norm $|\cdot|$, extend this
to a right-invariant Riemannian metric on $G$, and let $d:G\times
G\to[0,\infty)$ be the corresponding right-invariant geodesic distance. Also,
we will call relative equilibria _phase-locked states_ , as defined below.

######  Definition 1.2.

A solution $\\{X_{i}\\}_{i=1}^{N}$ to (1.7) is a _phase-locked state_ if
$X_{i}(t)X_{j}(t)^{-1}$ is constant in $t$ for each $i,j=1,\cdots,N$.

We may again characterize the phase-locked states as follows. This is
essentially due to Lohe [23].

######  Proposition 1.1 (Characterization of phase-locked states, [23, p.6]).

A solution $\\{X_{i}\\}_{i=1}^{N}$ to (1.7) is a phase-locked state if and
only if $X_{i}(t)=X_{i}^{\infty}\exp(\Lambda t)$, where $\exp$ is the Lie
group exponential and $X_{i}^{\infty}\in G$, $i=1,\cdots,N$, and
$\Lambda\in\mathfrak{g}$ satisfy

(1.9)
$\operatorname{Ad}_{X_{i}^{\infty}}\Lambda=H_{i}+\frac{\kappa}{N}\sum_{j=1}^{N}\phi(X_{j}^{\infty}(X_{i}^{\infty})^{-1}),\quad
i=1,\cdots,N,$

where $\operatorname{Ad}_{g}=(dL_{g})_{g^{-1}}\circ(dR_{g^{-1}})_{e}$ is the
adjoint.

The main theorems of this paper are as follows. The first main theorem says
that if the intrinsic Hamiltonians are identical, then a unique global
solution exists and exhibits complete synchronization.

######  Theorem 1.7.

There exist constants $C,c,c_{1},c_{2}>0$ depending on $G$, $\mathfrak{g}$,
and $\phi$ such that the following is true.

  1. (1)

Let the coupling strength $\kappa$ and intrinsic Hamiltonians $H_{i}$ satisfy

$\kappa>c|H|,\quad H_{i}=H,~{}i=1,\cdots,N,$

for some fixed $H\in\mathfrak{g}$, and let the initial condition
$\\{X_{i}^{0}\\}_{i=1}^{N}$ satisfy

$d(X_{i}^{0},X_{j}^{0})<c_{1},\quad i,j=1,\cdots,N.$

Then the following statements hold.

    1. (a)

(Global existence of solutions) There is a global solution
$\\{X_{i}(t)\\}_{i=1}^{N}$ to (1.7).

    2. (b)

(Exponential synchronization) We have

$d(X_{i}(t),X_{j}(t))\leq Ce^{-c_{2}\kappa t},\quad i,j=1,\cdots,N.$

  2. (2)

Suppose that the interaction function $\phi$ satisfies

$\operatorname{Ad}_{g}\phi(h)=\phi(ghg^{-1}),\quad g,h\in G.$

Let the coupling strength $\kappa$ and intrinsic Hamiltonians $H_{i}$ satisfy

$\kappa>0,\quad H_{i}=H,~{}i=1,\cdots,N,$

for some fixed $H\in\mathfrak{g}$, and the initial condition
$\\{X_{i}^{0}\\}_{i=1}^{N}$ satisfy

$d(X_{i}^{0},X_{j}^{0})<c_{1}.$

Then the following statements hold.

    1. (a)

(Global existence of solutions) There is a global solution
$\\{X_{i}(t)\\}_{i=1}^{N}$ to (1.7).

    2. (b)

(Solution operator splitting) $\tilde{X}_{i}(t)=\exp(-Ht)X_{i}(t)$ is a
solution to (1.7) with $H_{i}=0$, $\tilde{X}_{i}(0)=X_{i}^{0}$.

    3. (c)

(Exponential synchronization) We have

$d(\exp(-Ht)X_{i}(t),\exp(-Ht)X_{j}(t))\leq Ce^{-c_{2}\kappa t},\quad
i,j=1,\cdots,N.$

The second main theorem says that if the coupling strength $\kappa$ is
sufficiently large compared to the magnitude of the Hamiltonians $|H_{i}|$,
then phase-locked states, with each particle being sufficiently close together
(measured by $d$), exist and are unique up to right-multiplication. Also, if
the initial data are sufficiently close together, then a global solution
exists and exponentially converges towards a right-multiplication of the
aforementioned phase-locked state.

######  Theorem 1.8.

There exist constants $C,c,c_{1},c_{2}>0$ depending on $G$, $\mathfrak{g}$,
and $\phi$ such that if

(1.10)
$\kappa>c\|H\|_{\infty},\quad\|H\|_{\infty}\coloneqq\max_{i=1,\cdots,N}|H_{i}|,$

then the following holds.

  1. (1)

(Existence and uniqueness of a phase-locked state) There exists a phase-locked
state $\\{X_{i}^{\infty}\exp(\Lambda t)\\}_{i=1}^{N}$, with $X_{i}^{\infty}\in
G$, $i=1,\cdots,N$, $\Lambda\in\mathfrak{g}$ satisfying (1.9), such that

$d(X_{i}^{\infty},X_{j}^{\infty})\leq\frac{C\|H\|_{\infty}}{\kappa}.$

This is unique up to right-multiplication in a larger set, i.e., if
$\\{X_{i}(t)\\}_{i=1}^{N}$ is a phase-locked state with

$d(X_{i}^{0},X_{j}^{0})<c_{1},\quad\forall i,j=1,\cdots,N,$

then $X_{i}(t)=X_{i}^{\infty}\exp(\Lambda t)g$, $i=1,\cdots,N$, for some fixed
$g\in G$. (This larger set will be the basin of attraction for this phase-
locked state.)

Now let $\\{X_{i}(t)\\}_{i=1}^{N}$ be a solution to (1.7) with initial data
$\\{X_{i}^{0}\\}_{i=1}^{N}$ satisfying

$d(X_{i}^{0},X_{j}^{0})<c_{1},\quad\forall i,j=1,\cdots,N.$

  1. (2)

(Global existence and local stability) There exists a unique global solution
$X(t)=\\{X_{i}(t)\\}_{i=1}^{N}$ to (1.7), with

$d(X_{i}(t),X_{j}(t))<c_{1},~{}t\geq 0,\quad\forall i,j=1,\cdots,N,$

and

$\limsup_{t\to\infty}d(X_{i}(t),X_{j}(t))\leq\frac{C\|H\|_{\infty}}{\kappa},~{}t\geq
0,\quad\forall i,j=1,\cdots,N.$

  2. (3)

(Asymptotic phase-locking) We have

$\lim_{t\to\infty}X_{i}(t)X_{j}(t)^{-1}=X_{i}^{\infty}(X_{j}^{\infty})^{-1},$

the convergence rate being $O_{G}(e^{-\kappa c_{2}t})$.111We will use the
following (standard) asymptotic notation. For $P,Q>0$, $P=O(Q)$ and
$Q=\Omega(P)$ mean that $P\leq KQ$ for a universal constant $K\in(0,\infty)$.
If we need to allow for dependence on parameters, we indicate this by
subscripts. For example, in the presence of auxiliary parameters $\psi,\xi$,
the notations $P=O_{\psi,\xi}(Q)$ and $Q=\Omega_{\psi,\xi}(P)$ mean that
$P\leq K(\psi,\xi)Q$ where $K(\psi,\xi)\in(0,\infty)$ may depend only on
$\psi$ and $\xi$.

  3. (4)

(Orbital stability) The solution $X(t)=\\{X_{i}(t)\\}_{i=1}^{N}$ exponentially
converges to a right-multiplication of the phase-locked state of (1), i.e.,
there is a phase-locked state $\\{Z_{i}^{\infty}\exp(Mt)\\}_{i=1}^{N}$ such
that

$d(X_{i}(t),Z_{i}^{\infty}\exp(Mt))=O_{G,\phi}(e^{-\kappa c_{2}t}),\quad t\geq
0.$

  4. (5)

(Exponential synchronization of the normalized speeds) We have

$(dL_{X_{i}(t)}^{-1})_{e}\dot{X}_{i}(t)-M=O_{G,\phi}(\|H\|_{\infty}e^{-\kappa
c_{2}t}),\quad i=1,\cdots,N.$

Theorem 1.7 is a generalization of Theorems 1.1, 1.3, and 1.5, and Theorem 1.8
is a generalization of Theorems 1.2, 1.4, and 1.6. Theorem 1.7 and part (2) of
Theorem 1.8 are relatively unsurprising: as the particles are pairwise
attractive, if they are initially close together, then they should stay close
together for all times, guaranteeing global existence, and should attract each
other as much as they can. The author finds part (4) of Theorem 1.8, the
orbital stability statement, to be somewhat surprising: it only requires (H),
namely that $\phi$ is $C^{1}$ and that it is locally attractive in the sense
that $(d\phi)_{e}$ has only eigenvalues with positive real parts. Note that
higher order considerations, such as $C^{2}$ regularity of $\phi$ or curvature
properties of $G$, are irrelevant.

Thus, as long as we define any model in the form of (1.7), by choosing a Lie
group $G$ and an interaction function $\phi:G\to\mathfrak{g}$ to satisfy (H),
this model will automatically have the stability properties of Theorem 1.8.
This grants us significant liberty in defining locally well-behaved
synchronization models: we only need to choose a Lie group $G$ and an
interaction function $\phi:G\to\mathfrak{g}$ satisfying (H).

For example, it was suggested by an anonymous referee for the previous paper
[16] that we consider the model on
$G=SU(d),SL_{d}(\mathbb{C}),SL_{d}(\mathbb{R})$ given by

(1.11)
$\dot{X}_{i}X_{i}^{-1}=H_{i}+\frac{\kappa}{2N}\sum_{j=1}^{N}\left(X_{j}X_{i}^{-1}-X_{i}X_{j}^{-1}-\frac{1}{d}\mathrm{Tr}[X_{j}X_{i}^{-1}]I_{d}+\frac{1}{d}\mathrm{Tr}[X_{i}X_{j}^{-1}]I_{d}\right).$

One can check that (H) holds for this interaction function, so that Theorem
1.8 holds for (1.11).

Here is an intuitive heuristic as to why Theorem 1.8 should be true. If the
particles are concentrated locally, say around the identity $e\in G$, then by
setting

$X_{i}=\exp(Y_{i})\quad i=1,\cdots,N,$

we can linearize the governing equation (1.7) to

(1.12)
$\dot{Y}_{i}=H_{i}+\frac{\kappa}{N}\sum_{k=1}^{N}(d\phi)_{e}(Y_{k}-Y_{i}),\quad
i=1,\cdots,N.$

This is an exactly solvable linear differential equation. In particular, the
dynamics for the differences $Y_{i}-Y_{j}$ is simple:

$\dot{Y}_{i}-\dot{Y}_{j}=H_{i}-H_{j}-\kappa(d\phi)_{e}(Y_{i}-Y_{j}),\quad
i,j=1,\cdots,N.$

As $(d\phi)_{e}$ has only eigenvalues with positive real parts, if we make
$\kappa$ large enough compared to $|H_{i}-H_{j}|$, then $Y_{i}-Y_{j}$ should
converge exponentially to a fixed value.

Thus, the proof of Theorem 1.8 for the nonlinear model (1.7) will be a
“nonlinearization” of the above heuristic argument, i.e., we will check that
the nonlinear effects are controllable compared to the dominant linear term.
One caveat of this nonlinearization is that, as mentioned after Theorem 1.6,
we will need to require that $\kappa$ is large compared to the $|H_{i}|$’s as
in (1.10), instead of requiring $\kappa$ to be large compared to the
$|H_{i}-H_{j}|$’s. There will also be the complication that the $X_{i}$’s may
not stay in the neighborhood of a single point for all time, so we will work
with the logarithms of the _ratios_ $X_{i}X_{j}^{-1}$, which will stay close
to the identity $e\in G$.

###  1.3. Directions for further research

One reason for considering generalizations of the Kuramoto model (1.1) should
come from a search for new phenomena not observed in the Kuramoto model. Thus,
since Theorem 1.8 is true not only for the Kuramoto model but also for its Lie
group generalizations in the form of (1.7), the merit of studying such models
in the form of (1.7) will have to come from the behavior of the model when the
particles are not extremely concentrated. For example, DeVille [9] presents
several special solutions to (1.8) and shows that it has multiple attractors
under rich network topologies, while Ritchie, Lohe, and Williams [27] present
a detailed study of the global solutions of the special case $G=SO(1,1)$.

In the proof of Theorem 1.8, one proves the statements (2), (3), (1), (4), and
(5) in this order, with possibly smaller $c_{1}$ and larger $c$ (and
consequently larger $\kappa$) along the way. Thus there might be a sequence of
phase transitions

$X_{i}X_{j}^{-1}\mbox{ bounded }\Rightarrow X_{i}X_{j}^{-1}\mbox{ converges
}\Rightarrow X_{i}\to X_{j}^{\infty}\exp(\Lambda t),$

as we increase the coupling strength $\kappa$. It would be interesting to see
whether the first phase-transition actually occurs. That is, for small
$\kappa$, the solutions might stay bounded from each other with their ratios
not converging, possibly exhibiting chaotic behavior. Observing such
‘synchronous chaos’ in a prototypical model such as (1.5) would be
interesting.

######  Question 1.1 (Synchronous chaos).

Is it possible that in a semi-local regime with intermediate coupling strength
$\kappa$:

$c_{1}^{\prime}<\max_{i,j=1,\cdots,N}d(X_{i}^{0},X_{j}^{0})<c_{1},\quad
c\|H\|_{\infty}<\kappa<c^{\prime}\|H\|_{\infty},$

with different constants $c_{1}^{\prime}$ and $c^{\prime}$, $X_{i}X_{j}^{-1}$
stay bounded yet fail to converge? Does this occur for the model (1.5)?

See Remark 3.3 for another formulation of this question.

We remark that some models which are qualitatively different from the Kuramoto
model can still be written in the form (1.7) if we abandon hypothesis (H). For
example, the Cucker-Smale model [7, 8]

(1.13) $\begin{cases}\dot{x}_{i}=v_{i},\\\
\dot{v}_{i}=\frac{\kappa}{N}\sum_{j=1}^{N}\psi(|x_{i}-x_{j}|)(v_{j}-v_{i}),\end{cases}t>0,~{}i=1,\cdots,N$

where $\psi:\mathbb{R}^{+}\to\mathbb{R}$ is a function, is, up to a uniform
translation in the $x$-variable, in the form of (1.7) with $(x,v)\in
G=T\mathbb{R}^{d}=\mathbb{R}^{d}\times\mathbb{R}^{d}$, with the interaction
function $\phi$ now failing to have only eigenvalues with positive real parts,
and fails to be smooth if $\psi$ possesses a singularity at $0$. Another
example is the Swarmalator model proposed by O’Keeffe, Hong, and Strogatz
[25]:

(1.14)
$\begin{cases}\dot{x}_{i}=v_{i}+\frac{1}{N}\sum_{j=1}^{N}[I_{att}(x_{j}-x_{i})F(\theta_{j}-\theta_{i})-I_{rep}(x_{j}-x_{i})],\\\
\dot{\theta}_{i}=\omega_{i}+\frac{\kappa}{N}\sum_{j=1}^{N}H_{att}(\theta_{j}-\theta_{i})G(x_{j}-x_{i}),\end{cases}t>0,~{}i=1,\cdots,N.$

This is again in the form of (1.7) with $(x,\theta)\in
G=\mathbb{R}^{d}\times\mathbb{S}^{1}$, but with $\phi$ failing to satisfy (H).

Thus it is natural to pose the following question.

######  Question 1.2.

What is the behavior of the model (1.7) for different $\phi$?

  1. (1)

Let $\phi:G\to\mathfrak{g}$ be smooth but not satisfy (H). To what extent can
the local behavior of (1.7) be explained by the behavior of the linearized
model (1.12)?

  2. (2)

Let $G=\mathbb{R}^{d}$ and let
$\phi:\mathbb{R}^{d}\setminus\\{0\\}\to\mathbb{R}^{d}$ possess a singularity
at the origin. Are there certain types of singularities of $\phi$ for which
solutions to (1.7) near the origin exhibit nice behavior?

Another drawback of Theorem 1.8 is a possible deterioration of the statements
for large dimensions. In many cases, we have a family of Lie groups, such as
the family of unitary groups $\\{U(d)\\}_{d=1}^{\infty}$, whose dimension
grows to infinity. In this case, the size of the stability basin of Theorem
1.8 compared to the whole group diminishes exponentially as the dimension
parameter grows to infinity. Thus, in practice, it would be desirable to
obtain dimension-independent bounds for Theorem 1.8. This is possible for the
Lohe model (1.5), by using the operator norm instead of the Frobenius norm, as
can be checked by modifying the arguments of [16] in a straightforward manner.

######  Question 1.3.

For incarnations of the model (1.7) other than (1.5), is it possible to obtain
dimension-independent versions of Theorem 1.8?

It would also be interesting to study infinite limits such as the mean-field
limit, as performed for the Lohe model (1.5) by Golse and Ha [12], or to study
stochastic variations, as performed for the Lohe matrix model (1.6) by Kim and
Kim [19]. One could also study the effects of frustrations as in the author’s
joint work with Ha, Kim, and Park [14].

The rest of this paper is organized as follows. We recall basic Lie group
theory and establish some notation in Section 2, and then we prove Theorem 1.8
in Section 3.

##  2\. Notation and preliminaries on Lie groups and Lie algebras

We establish some notation about our Lie group $(G,\mathfrak{g})$ and recall
some standard facts from the literature.

As $(d\phi)_{e}:\mathfrak{g}\to\mathfrak{g}$ has only eigenvalues with
positive real parts, we may assume that the right-invariant Riemannian metric
on $G$, which is given as an inner product $\langle,\rangle$ on
$\mathfrak{g}$, with corresponding norm
$|\cdot|=\langle\cdot,\cdot\rangle^{1/2}$, is so that for some $\lambda>0$,

(2.1) $\langle v,(d\phi)_{e}(v)\rangle\geq\lambda|v|^{2},\quad\forall
v\in\mathfrak{g}.$

(This change of Riemannian metric only distorts the geodesic metric of $G$ by
a constant factor, and thus does not affect the validity of Theorem 1.8.) We
define, for $r>0$, the ball of radius $r$

$B_{r}=\\{v\in\mathfrak{g}:|v|<r\\}.$

The Lie group exponential map $\exp:\mathfrak{g}\to G$ is a local
diffeomorphism at $0\in\mathfrak{g}$, with
$(d\exp)_{0}=\operatorname{id}_{\mathfrak{g}}$. We may choose a radius $r>0$
depending on $G$ so that $\exp:B_{r}\to\exp(B_{r})$ is a diffeomorphism, and
so that consequently the logarithm $\log:\exp(B(0,r))\to B_{r}$ is well
defined and smooth; we will use this as a coordinate chart for $G$ in this
paper.

The Baker–Campbell–Hausdorff formula describes the group operation on this
coordinate chart. Given $v,w\in\mathfrak{g}$, with $|v|,|w|$ sufficiently
small, if we define

(2.2) $\displaystyle\begin{aligned}
u\coloneqq&\sum_{m=1}^{\infty}\frac{(-1)^{m-1}}{m}\sum_{\begin{smallmatrix}r_{1}+s_{1}>0\\\
\vdots\\\
r_{m}+s_{m}>0\end{smallmatrix}}\frac{[v^{r_{1}}w^{s_{1}}v^{r_{2}}w^{s_{2}}\dotsm
v^{r_{m}}w^{s_{m}}]}{(\sum_{j=1}^{m}(r_{j}+s_{j}))\cdot\prod_{i=1}^{m}r_{i}!s_{i}!}\\\
=&v+w+a(v,w),\end{aligned}$

where we have used the notation

$[g^{r_{1}}h^{s_{1}}\dotsm
g^{r_{m}}h^{s_{m}}]=[\underbrace{g,[g,\dotsm[g}_{r_{1}},[\underbrace{h,[h,\dotsm[h}_{s_{1}},\,\dotsm\,[\underbrace{g,[g,\dotsm[g}_{r_{m}},[\underbrace{h,[h,\dotsm
h}_{s_{m}}]]\dotsm]],$

and $a(v,w)$ denotes the sum of the terms of second or higher degree, then the
power series defining $u$ is absolutely convergent, and satisfies

$\exp(u)=\exp(v)\exp(w).$

We may assume that the radius $r>0$ is sufficiently small so that for $u,v\in
B_{r}$, the Baker–Campbell–Hausdorff formula for $u$ and $v$ converges
absolutely, with the map $a:B_{r}\times B_{r}\to\mathfrak{g}$ being smooth
with bounded derivatives:

(2.3)
$a(u,v)=O_{G}(|u||v|),~{}a(u,v)-a(u^{\prime},v^{\prime})=O_{G}((|u|+|u^{\prime}|)|v-v^{\prime}|+(|v|+|v^{\prime}|)|u-u^{\prime}|),\quad\forall
u,v,u^{\prime},v^{\prime}\in B_{r}.$

We denote the adjoint map
$\operatorname{Ad}:G\to\operatorname{Aut}(\mathfrak{g})$, which is defined to
be
$\operatorname{Ad}_{g}=(dL_{g})_{g^{-1}}\circ(dR_{g^{-1}})_{e}=(dR_{g^{-1}})_{g}\circ(dL_{g})_{e}$
for $g\in G$. The derivative of the adjoint map,
$\operatorname{ad}:\mathfrak{g}\to\operatorname{Der}(\mathfrak{g})$, is given
by $\operatorname{ad}_{v}=[v,\cdot]$, for all $v\in\mathfrak{g}$. It is well-
known that

(2.4) $\operatorname{Ad}_{\exp(v)}=e^{\operatorname{ad}_{v}},\forall
v\in\mathfrak{g},$

where the right-hand side denotes the element of
$\operatorname{Aut}(\mathfrak{g})$ given by

$e^{\operatorname{ad}_{v}}\coloneqq\sum_{m=0}^{\infty}\frac{1}{m!}(\operatorname{ad}_{v})^{m}.$

We will need to take derivatives under the exponential map. If
$Y:I\to\mathfrak{g}$ is a differentiable curve in $\mathfrak{g}$ defined on
some interval $I\subset\mathbb{R}$, then it is well-known that

(2.5)
$\frac{d}{dt}\exp(Y)=(dR_{\exp(Y)})_{e}\frac{e^{\operatorname{ad}_{Y}}-1}{\operatorname{ad}_{Y}}\left(\frac{dY}{dt}\right),$

where, on the right-hand side, the operator
$\frac{e^{\operatorname{ad}_{Y}}-1}{\operatorname{ad}_{Y}}$ is defined using
the power series expansion for $\frac{e^{z}-1}{z}$. By functional calculus, if
$r>0$ is sufficiently small and $v\in B_{r}$, then the operator
$\frac{e^{\operatorname{ad}_{v}}-1}{\operatorname{ad}_{v}}$ is invertible with
the well-defined inverse
$\frac{\operatorname{ad}_{v}}{e^{\operatorname{ad}_{v}}-1}$. Thus, if $Y(t)\in
B_{r}$ for all $t\in I$, then denoting $X=\exp(Y)$, we have

(2.6)
$\frac{d}{dt}\log(X)=\frac{\operatorname{ad}_{Y}}{e^{\operatorname{ad}_{Y}}-1}(dR_{X^{-1}})_{X}\left(\frac{dX}{dt}\right).$

Again by functional calculus, if $r>0$ is sufficiently small, we have

(2.7)
$\frac{\operatorname{ad}_{v}}{e^{\operatorname{ad}_{v}}-1},\frac{\operatorname{ad}_{v}}{1-e^{-\operatorname{ad}_{v}}}=\operatorname{id}_{\mathfrak{g}}+O(|v|),\quad
v\in B_{r},$

and

(2.8)
$\frac{\operatorname{ad}_{v}}{e^{\operatorname{ad}_{v}}-1}-\frac{\operatorname{ad}_{w}}{e^{\operatorname{ad}_{w}}-1},\frac{\operatorname{ad}_{v}}{1-e^{-\operatorname{ad}_{v}}}-\frac{\operatorname{ad}_{w}}{1-e^{-\operatorname{ad}_{w}}}=O(|v-w|),\quad
v,w\in B_{r}.$

We will later use (2.7) and (2.8) to control (2.6).

##  3\. Proof of Theorems 1.7 and 1.8

In this section we prove Theorems 1.7 and 1.8.

Theorem 1.7 is proven directly assuming Theorem 1.8.

Theorem 1.8 is proven by two reductions. First, we will simplify Theorem 1.8
into the more dynamically tractable statement of Theorem 3.1. Then, to prove
Theorem 3.1, instead of working with the system (1.7) on the Lie group $G$, we
will work with the system (3.6) on the Lie algebra $\mathfrak{g}$ of the
logarithms of the ratios of the particles. Proving Theorem 3.1 on the Lie
group $G$ reduces to proving the corresponding Theorem 3.2 on the Lie algebra
$\mathfrak{g}$. We will then prove Theorem 3.2 by introducing two Lyapunov
functionals and establishing Gronwall inequalities for them.

###  3.1. Proof of Theorem 1.7 assuming Theorem 1.8

  1. (1)

This is a direct consequence of Theorem 1.8 (2) and (3).

  2. (2)

Statement (b) is a straightforward computation. Statements (a) and (c) are
immediate from Theorem 1.8 (2) and (3) for $\\{\tilde{X}_{i}(t)\\}_{i=1}^{N}$.

###  3.2. Reducing Theorem 1.8 to Theorem 3.1

Now we begin the proof of Theorem 1.8. We first remark on the order of
constants. We will first select $C$ and $c_{2}$ depending on $G$,
$\mathfrak{g}$, and $\phi$, then select $c_{1}$ sufficiently small depending
on $C$ and $c_{2}$, then select $c$ sufficiently large depending on $C$,
$c_{2}$, and $c_{1}$.

We first reduce Theorem 1.8 to the following theorem.

######  Theorem 3.1.

There exist a universal constant $c_{3}>0$, a sufficiently large constant
$c>0$ and a sufficiently small constant $c_{1}>0$ depending on $G$,
$\mathfrak{g}$, and $\phi$ such that if

$\kappa>c\|H\|_{\infty},$

then the following holds.

  1. (1)

(Local stability) Let $\\{X_{i}(t)\\}_{i=1}^{N}$ be a solution to (1.7) with
initial data $\\{X_{i}^{0}\\}_{i=1}^{N}$ satisfying

$d(X_{i}^{0},X_{j}^{0})<c_{1},\quad\forall i,j=1,\cdots,N.$

Then there exists a unique global solution $X(t)=\\{X_{i}(t)\\}_{i=1}^{N}$ to
(1.7), with

$d(X_{i}(t),X_{j}(t))<c_{1},~{}t\geq 0,\quad\forall i,j=1,\cdots,N,$

and

$\limsup_{t\to\infty}d(X_{i}(t),X_{j}(t))\leq\frac{c_{3}\|H\|_{\infty}}{\kappa\lambda},~{}t\geq
0,\quad\forall i,j=1,\cdots,N.$

  2. (2)

(Convergence of two flows) Let $\\{\tilde{X_{i}}\\}_{i=1}^{N}$ be a solution
to (1.7) with initial data $\\{\tilde{X}_{i}^{0}\\}_{i=1}^{N}$ satisfying

(3.1) $d(\tilde{X}_{i}^{0},\tilde{X}_{j}^{0})<c_{1},\quad\forall
i,j=1,\cdots,N.$

Then

$d(X_{i}(t)X_{j}(t)^{-1},\tilde{X}_{i}(t)\tilde{X}_{j}(t)^{-1})=O_{G}(d(X_{i}^{0}(X_{j}^{0})^{-1},\tilde{X}_{i}^{0}(\tilde{X}_{j}^{0})^{-1})e^{-\kappa\lambda
t/3}).$

###### Proof of Theorem 1.8 assuming Theorem 3.1.

We prove Theorem 1.8 in the following order.

  * (2)

We note that Theorem 1.8 (2) is just Theorem 3.1 (1).

  * (1)+(3)

We prove statements (1) and (3) together.

Let $\\{\tilde{X}_{i}\\}_{i=1}^{N}$ be a solution to (1.7) with initial data
$\\{\tilde{X}_{i}^{0}\\}_{i=1}^{N}$ satisfying

$d(\tilde{X}_{i}^{0},\tilde{X}_{j}^{0})<c_{1},\quad\forall i,j=1,\cdots,N.$

By Theorem 3.1 (2), we know that

(3.2) $d(X_{i}(t)X_{j}(t)^{-1},\tilde{X}_{i}(t)\tilde{X}_{j}(t)^{-1})\leq
Ce^{-\kappa\lambda t/3},$

for some constant $C$ depending on $G$. By considering the time-delayed
solution $\tilde{X}(t)=X(t+1)$ to (1.7), Theorem 3.1 (1) tells us that (3.1)
is true, so (3.2) tells us that $\\{X_{i}(n)X_{j}(n)^{-1}\\}_{n\in\mathbb{N}}$
is a Cauchy sequence and hence is exponentially convergent. By considering the
time-delayed solutions $\tilde{X}(t)=X(t+T)$ for $0\leq T\leq 1$, again
Theorem 3.1 (1) tells us that (3.1) is true, so we have by (3.2) that

$\displaystyle\sup_{0\leq T\leq
1}d(X_{i}(n)X_{j}(n)^{-1},X_{i}(n+T)X_{j}(n+T)^{-1})$
$\displaystyle\leq\sup_{0\leq T\leq
1}d(X_{i}(0)X_{j}(0)^{-1},X_{i}(T)X_{j}(T)^{-1})\cdot Ce^{-\kappa\lambda n/3}$
$\displaystyle\leq O_{G}(Ce^{-\kappa\lambda n/3}),\quad n\in\mathbb{N},$

which tells us that $\\{X_{i}(t)X_{j}(t)^{-1}\\}_{t\geq 0}$ is convergent to
some value $X_{ij}^{\infty}$, $i,j=1,\cdots,N$.

Now it remains to verify the existence of a phase-locked state
$X_{i}^{\infty}\exp(\Lambda t)$ satisfying (1), and that
$X_{ij}=X_{i}^{\infty}(X_{j}^{\infty})^{-1}$.

Note that, as $\frac{d}{dt}X^{-1}=-(dL_{X^{-1}})_{e}(dR_{X^{-1}})_{X}\dot{X}$,

$\frac{d}{dt}X_{i}^{-1}=-(dL_{X_{i}^{-1}})_{e}\left(H_{i}+\frac{\kappa}{N}\sum_{k=1}^{N}\phi(X_{k}X_{i}^{-1})\right).$

So

(3.3) $\displaystyle\begin{aligned}
\frac{d}{dt}\left(X_{i}X_{j}^{-1}\right)=&(dR_{X_{j}^{-1}})_{X_{i}}\left(\frac{d}{dt}X_{i}\right)+(dL_{X_{i}})_{X_{j}^{-1}}\left(\frac{d}{dt}X_{j}^{-1}\right)\\\
=&(dR_{X_{i}X_{j}^{-1}})_{e}\left(H_{i}+\frac{\kappa}{N}\sum_{k=1}^{N}\phi(X_{k}X_{i}^{-1})\right)-(dL_{X_{i}X_{j}^{-1}})_{e}\left(H_{j}+\frac{\kappa}{N}\sum_{k=1}^{N}\phi(X_{k}X_{j}^{-1})\right)\end{aligned}$

Since the ratios $X_{i}X_{j}^{-1}$ converge, both sides of (3.3) must converge
to zero. Hence

(3.4)
$(dR_{X_{ij}^{\infty}})_{e}\left(H_{i}+\frac{\kappa}{N}\sum_{k=1}^{N}\phi(X_{ki}^{\infty})\right)=(dL_{X_{ij}^{\infty}})_{e}\left(H_{j}+\frac{\kappa}{N}\sum_{k=1}^{N}\phi(X_{kj}^{\infty})\right),\quad
i,j=1,\cdots,N.$

Now, by continuity, we must have $X_{ii}^{\infty}=e$ and
$X_{ij}^{\infty}X_{jk}^{\infty}=X_{ik}^{\infty}$, $i,j,k=1,\cdots,N$.
Therefore if we set

$X_{i}^{\infty}\coloneqq X_{i1}^{\infty},\quad i=1,\cdots,N,$

then $X_{ij}=X_{i}^{\infty}(X_{j}^{\infty})^{-1}$, and (3.4) becomes

$(\operatorname{Ad}_{X_{i}^{\infty}})^{-1}\left(H_{i}+\frac{\kappa}{N}\sum_{k=1}^{N}\phi(X_{k}^{\infty}(X_{i}^{\infty})^{-1})\right)=(\operatorname{Ad}_{X_{j}^{\infty}})^{-1}\left(H_{j}+\frac{\kappa}{N}\sum_{k=1}^{N}\phi(X_{k}^{\infty}(X_{j}^{\infty})^{-1})\right),\quad
i,j=1,\cdots,N.$

Thus we can define the common value $\Lambda\in\mathfrak{g}$ as

$\Lambda=(\operatorname{Ad}_{X_{i}^{\infty}})^{-1}\left(H_{i}+\frac{\kappa}{N}\sum_{k=1}^{N}\phi(X_{k}^{\infty}(X_{i}^{\infty})^{-1})\right),\quad
i=1,\cdots,N.$

Then $\Lambda$ and $X_{i}^{\infty}$ satisfy (1.9), so by Proposition 1.1,
$\\{X_{i}^{\infty}\exp(\Lambda t)\\}_{i=1}^{N}$ is indeed a phase-locked
state.

We have just proved (3) and the existence statement of (1). It remains to
verify the uniqueness statement of (1).

Let $\\{\tilde{X}_{i}^{\infty}\exp(\tilde{\Lambda}t)\\}_{i=1}^{N}$ be another
phase-locked state, with

$d(\tilde{X}_{i}^{\infty},\tilde{X}_{j}^{\infty})<c_{1}.$

By Theorem 3.1 (2), we have

$X_{i}^{\infty}(X_{j}^{\infty})^{-1}=\tilde{X}_{i}^{\infty}(\tilde{X}_{j}^{\infty})^{-1},\quad
i,j=1,\cdots,N.$

So
$\tilde{X}_{i}^{\infty}(X_{i}^{\infty})^{-1}=\tilde{X}_{j}^{\infty}(X_{j}^{\infty})^{-1}$,
$i,j=1,\cdots,N$, and if we set $g$ as this common value, we have

$\tilde{X}_{i}^{\infty}=X_{i}^{\infty}g,\quad i=1,\cdots,N.$

Now from (1.9), we have
$\operatorname{Ad}_{X_{i}^{\infty}}\Lambda=\operatorname{Ad}_{X_{i}^{\infty}g}\tilde{\Lambda}$,
so

$\tilde{\Lambda}=\operatorname{Ad}_{g^{-1}}\Lambda.$

Therefore

$\tilde{X}_{i}^{\infty}\exp(\tilde{\Lambda}t)=X_{i}^{\infty}g\exp(\operatorname{Ad}_{g^{-1}}\Lambda
t)=X_{i}^{\infty}\exp(\Lambda t)g,\quad i=1,\cdots,N,t\geq 0,$

as desired.

  * (4)

By (3), we may define the quantities

$\Lambda_{i}=H_{i}+\frac{\kappa}{N}\sum_{j=1}^{N}\phi(\lim_{t\to\infty}X_{j}(t)X_{i}(t)^{-1}),\quad
i=1,\cdots,N.$

(By the discussion of the proof of (3),
$\Lambda_{i}=\operatorname{Ad}_{X_{i}^{\infty}}\Lambda$.) By (2), we have
$|\Lambda_{i}|\leq O_{G,\phi}(\|H\|_{\infty})$ for some constant $C$.

Now define

$Z_{i}(t)=\exp(-\Lambda_{i}t)X_{i}(t),\quad i=1,\cdots,N.$

Then by (2.4),

$(dR_{Z_{i}^{-1}})_{Z_{i}}\dot{Z}_{i}=e^{-\mathrm{ad}_{\Lambda_{i}t}}\left(H_{i}+\frac{\kappa}{N}\sum_{j=1}^{N}\phi(X_{j}X_{i}^{-1})-\Lambda_{i}\right)=O_{G,\phi}(e^{O_{G}(\|H\|_{\infty})t-\kappa\lambda
t/3}),$

so if $\kappa>\Omega_{G}(\|H\|_{\infty}/\lambda)$, $Z_{i}$ exponentially
converges, say to $Z_{i}^{\infty}$.

Now

$\displaystyle d(X_{i},\exp(\Lambda_{i}t)Z_{i}^{\infty})$
$\displaystyle=d(\exp(\Lambda_{i}t)Z_{i}\exp(-\Lambda_{i}t),\exp(\Lambda_{i}t)Z_{i}^{\infty}\exp(-\Lambda_{i}t))$
$\displaystyle\leq
e^{\|\operatorname{ad}_{\Lambda_{i}}\|t}d(Z_{i},Z_{i}^{\infty})$
$\displaystyle=e^{O_{G}(\|H\|_{\infty})t}O_{G,\phi}(e^{O_{G}(\|H\|_{\infty})t-\kappa\lambda
t/3})$ $\displaystyle\rightarrow 0,$

where the first equality follows from right-invariance of $d$, and the first
inequality follows from (2.4) ($\|\operatorname{ad}_{\Lambda_{i}}\|$ is the
operator norm). As
$\exp(\Lambda_{i}t)Z_{i}^{\infty}=Z_{i}^{\infty}\exp(\operatorname{Ad}_{(Z_{i}^{\infty})^{-1}}(\Lambda_{i})t)$,
we have

$\displaystyle
d(Z_{i}^{\infty}\exp(\operatorname{Ad}_{(Z_{i}^{\infty})^{-1}}(\Lambda_{i})t)(Z_{j}^{\infty}\exp(\operatorname{Ad}_{(Z_{j}^{\infty})^{-1}}(\Lambda_{j})t))^{-1},\lim_{t\to\infty}X_{i}(t)X_{j}(t)^{-1})$
$\displaystyle\leq
d(\exp(\Lambda_{i}t)Z_{i}^{\infty}(\exp(\Lambda_{j}t)Z_{j}^{\infty})^{-1},\exp(\Lambda_{i}t)Z_{i}^{\infty}X_{j}(t)^{-1})$
$\displaystyle\quad+d(\exp(\Lambda_{i}t)Z_{i}^{\infty}X_{j}(t)^{-1},X_{i}(t)X_{j}(t)^{-1})$
$\displaystyle\quad+d(X_{i}(t)X_{j}(t)^{-1},\lim_{t\to\infty}X_{i}(t)X_{j}(t)^{-1})$
$\displaystyle\leq
e^{\|\operatorname{ad}_{\Lambda_{i}}\|t}\|\operatorname{Ad}_{Z_{i}^{\infty}}\|d((\exp(\Lambda_{j}t)Z_{j}^{\infty})^{-1},X_{j}(t)^{-1})$
$\displaystyle\quad+d(\exp(\Lambda_{i}t)Z_{i}^{\infty},X_{i}(t))+d(X_{i}(t)X_{j}(t)^{-1},\lim_{t\to\infty}X_{i}(t)X_{j}(t)^{-1})$
$\displaystyle\leq
e^{\|\operatorname{ad}_{\Lambda_{i}}\|t}\|\operatorname{Ad}_{Z_{i}^{\infty}}\|e^{\|\operatorname{ad}_{\Lambda_{j}}\|t}\|\operatorname{Ad}_{Z_{j}^{\infty}}\|d(\exp(\Lambda_{j}t)Z_{j}^{\infty},X_{j}(t))$
$\displaystyle\quad+d(\exp(\Lambda_{i}t)Z_{i}^{\infty},X_{i}(t))+d(X_{i}(t)X_{j}(t)^{-1},\lim_{t\to\infty}X_{i}(t)X_{j}(t)^{-1}).$

The last two terms clearly converge to zero, while the first term is
$O_{G}(e^{O_{G}(\|H\|_{\infty})t})O_{G,\phi}(e^{O_{G}(\|H\|_{\infty})t-\kappa\lambda
t/3})$ and thus converges to zero with large $\kappa$. Thus we have the
convergence

$Z_{i}^{\infty}\exp(\operatorname{Ad}_{(Z_{i}^{\infty})^{-1}}(\Lambda_{i})t)(Z_{j}^{\infty}\exp(\operatorname{Ad}_{(Z_{j}^{\infty})^{-1}}(\Lambda_{j})t))^{-1}\to\lim_{t\to\infty}X_{i}(t)X_{j}(t)^{-1},\quad
t\to\infty,$

for all $i,j=1\cdots,N$, which is only possible if
$Z_{i}^{\infty}(Z_{j}^{\infty})^{-1}=\lim_{t\to\infty}X_{i}(t)X_{j}(t)^{-1}$
and
$\operatorname{Ad}_{(Z_{i}^{\infty})^{-1}}(\Lambda_{i})=\operatorname{Ad}_{(Z_{j}^{\infty})^{-1}}(\Lambda_{j})$,
$i,j=1,\cdots,N$. If we denote this latter common value by $M\in\mathfrak{g}$,
then
$\operatorname{Ad}_{Z_{i}^{\infty}}M=\Lambda_{i}=H_{i}+\frac{\kappa}{N}\sum_{j}\phi(Z_{j}^{\infty}(Z_{i}^{\infty})^{-1})$
satisfies (1.9). Therefore $\\{Z_{i}^{\infty}\exp(Mt)\\}_{i=1}^{N}$ is a
phase-locked state such that $d(X_{i}(t),Z_{i}^{\infty}\exp(Mt))$ converges
exponentially to zero.

  * (5)

We compute

$\displaystyle(dL_{X_{i}(t)}^{-1})_{e}\dot{X}_{i}(t)-M$
$\displaystyle=\operatorname{Ad}_{X_{i}(t)^{-1}}\left(H_{i}+\frac{\kappa}{N}\sum_{j=1}^{N}\phi(X_{j}X_{i}^{-1})\right)-M$
$\displaystyle=(\operatorname{Ad}_{X_{i}(t)^{-1}Z_{i}^{\infty}\exp(Mt)}-I)\operatorname{Ad}_{(Z_{i}^{\infty}\exp(Mt))^{-1}}\left(H_{i}+\frac{\kappa}{N}\sum_{j=1}^{N}\phi(X_{j}X_{i}^{-1})\right)$
$\displaystyle\quad+\operatorname{Ad}_{(Z_{i}^{\infty}\exp(Mt))^{-1}}\left(H_{i}+\frac{\kappa}{N}\sum_{j=1}^{N}\phi(X_{j}X_{i}^{-1})-\Lambda_{i}\right)$
$\displaystyle\quad(\because\operatorname{Ad}_{(Z_{i}^{\infty}\exp(Mt))^{-1}}\Lambda_{i}=\operatorname{Ad}_{\exp(-Mt)}M=M).$

The first term is of the form
$O_{G,\phi}(\|H\|e^{(O_{G}(\|H\|_{\infty})-\kappa\lambda/3)t})$ and thus
converges to zero. Similarly, the second term is of the form
$O(e^{(O_{G}(\|H\|_{\infty})-\kappa\lambda/3)t})$ and thus converges to zero.

∎

###  3.3. Reducing from a model on the Lie group to a model on the Lie algebra

Our goal now is to prove Theorem 3.1. Theorem 3.1 roughly says that if the
particles $X_{i}$ are initially close together, they will stay together for
all time and will exhibit good stability behavior. However, we have little
control on where the whole aggregate will head toward; it is possible that the
entire population heads away from its initial position exponentially fast.

Thus, if we restrict our attention to the ratios $X_{i}X_{j}^{-1}$ of the
particles, we should be able to analyze them in a small neighborhood of the
identity.

From (3.3), we may formulate the Cauchy problem for the relative phases
$X_{i}X_{j}^{-1}$.

(3.5)
$\begin{cases}\displaystyle\frac{d}{dt}(X_{i}X_{j}^{-1})=(dR_{X_{i}X_{j}^{-1}})_{e}\left(H_{i}-\operatorname{Ad}_{X_{i}X_{j}^{-1}}H_{j}+\frac{\kappa}{N}\sum_{k=1}^{N}[\phi(X_{k}X_{i}^{-1})-\operatorname{Ad}_{X_{i}X_{j}^{-1}}\phi(X_{k}X_{j}^{-1})]\right),\\\
X_{i}X_{j}^{-1}(0)=X_{i}^{0}(X_{j}^{0})^{-1},\quad i,j=1,\cdots,N,\end{cases}$

Again, the standard Cauchy–Lipschitz theory guarantees the existence of a
local $C^{1}$-solution to (3.5). Observe that a global solution to (1.7)
exists if and only if a global solution to (3.5) exists. Indeed, the only if
direction is trivial, and conversely if the $X_{i}X_{j}^{-1}$’s are globally
defined, we may consider the right-hand side of (1.7) to be a time-dependent
bounded $C^{1}$-forcing term, and so by a standard argument (using the fact
that $G$ as a manifold is complete) the solutions $X_{i}(t)$ must exist for
all time.

Assuming $X_{i}X_{j}^{-1}\in\exp(B_{r})$ for all $i,j=1,\cdots,N$, define
$Y_{ij}=\log(X_{i}X_{j}^{-1})$. We will analyze the $Y_{ij}$’s on
$\mathfrak{g}$ instead of the $X_{i}X_{j}^{-1}$’s on $G$; we are using $B_{r}$
as a coordinate chart for $\exp(B_{r})$. Now, by (2.4) and (2.6), the
$Y_{ij}$’s form the solution to the Cauchy problem

(3.6)
$\begin{cases}\displaystyle\dot{Y}_{ij}=\frac{\operatorname{ad}_{Y_{ij}}}{e^{\operatorname{ad}_{Y_{ij}}}-1}H_{i}-\frac{\operatorname{ad}_{Y_{ij}}}{1-e^{-\operatorname{ad}_{Y_{ij}}}}H_{j}+\frac{\kappa}{N}\sum_{k=1}^{N}\left[\frac{\operatorname{ad}_{Y_{ij}}}{e^{\operatorname{ad}_{Y_{ij}}}-1}\phi\circ\exp(Y_{ki})-\frac{\operatorname{ad}_{Y_{ij}}}{1-e^{-\operatorname{ad}_{Y_{ij}}}}\phi\circ\exp(Y_{kj})\right],\\\
Y_{ij}(0)=\log(X_{i}^{0}(X_{j}^{0})^{-1}),\quad i,j=1,\cdots,N.\end{cases}$

Once again, the standard Cauchy–Lipschitz theory guarantees the existence of a
local $C^{1}$-solution to (3.6). This time, we can only observe that the
existence of a global solution to (3.6) guarantees the existence of a global
solution to (3.5), but not the other way around.

Note that, by the Baker-Campbell-Hausdorff formula, the $Y_{ij}$’s satisfy the
compatibility equations

(3.7) $Y_{ji}=-Y_{ij},\quad Y_{ik}=Y_{ij}+Y_{jk}+a(Y_{ij},Y_{jk}),\quad
i,j,k=1,\cdots,N$

for all times of their existence.

We now translate Theorem 3.1, a statement about the $X_{i}$’s, into a
statement about the $Y_{ij}$’s.

######  Theorem 3.2.

There exist a universal constant $C>0$, a sufficiently large constant $c>0$
and a sufficiently small constant $c_{1}>0$ depending on $G$, $\mathfrak{g}$,
and $\phi$ such that if

(3.8) $\kappa>c\|H\|_{\infty}$

then the following holds.

  1. (1)

(Local stability) If

$|Y_{ij}(0)|<c_{1},\quad\forall i,j=1,\cdots,N,$

then there exists a unique global solution $Y(t)=\\{Y_{ij}(t)\\}_{i,j=1}^{N}$
to (3.6) satisfying

$|Y_{ij}(t)|<c_{1},~{}t\geq 0,\quad\forall i,j=1,\cdots,N,$

and

$\limsup_{t\to\infty}|Y_{ij}(t)|\leq\frac{C\|H\|_{\infty}}{\kappa\lambda},\quad\forall
i,j=1,\cdots,N.$

  2. (2)

(Convergence of two flows) Let
$\tilde{Y}(t)=\\{\tilde{Y}_{ij}(t)\\}_{i,j=1}^{N}$ be a solution to (3.6) with
initial data $\\{\tilde{Y}_{ij}^{0}\\}_{i,j=1}^{N}$ satisfying the
compatibility equations (3.7) and

$|\tilde{Y}_{ij}^{0}|<c_{1},\quad\forall i,j=1,\cdots,N.$

Then

$|Y_{ij}(t)-\tilde{Y}_{ij}(t)|\leq|Y_{ij}^{0}-\tilde{Y}_{ij}^{0}|\cdot
e^{-\kappa\lambda t/3},\quad t\geq 0.$

It is clear that Theorem 3.2 implies Theorem 3.1, so our goal is now to prove
Theorem 3.2.

###  3.4. Two Gronwall inequalities

The proof of Theorem 3.2 depends on two Gronwall inequalities for two Lyapunov
functionals for $Y$, each respectively corresponding to statements (1) and (2)
of Theorem 3.2.

First, we consider the Lyapunov functional

(3.9) $\|Y\|_{\infty}\coloneqq\max_{i,j=1,\cdots,N}|Y_{ij}|.$

######  Lemma 3.1 (First Gronwall inequality).

Let $Y(t)=\\{Y_{ij}(t)\\}_{i,j=1}^{N}$, $t\in[0,T)$, be a solution to (3.6),
with $Y_{ij}(t)\in B_{r}$ for all $i,j=1,\cdots,N$ and $t\in[0,T)$. Then

(3.10)
$\dot{Y}_{ij}=-\kappa\left[d\phi_{e}(Y_{ij})+o_{G,\phi}(\|Y\|_{\infty})\right]+O(\|H\|_{\infty}),\quad
i,j=1,\cdots,N,$

and thus

(3.11)
$\left.\frac{d}{dt}\right|^{+}\|Y\|_{\infty}\leq-\kappa\left[\lambda\|Y\|_{\infty}+o_{G,\phi}(\|Y\|_{\infty})\right]+O(\|H\|_{\infty}),$

where $\left.\frac{d}{dt}\right|^{+}$ denotes the upper right Dini derivative,
and the little $o$ notation is with respect to $\|Y\|_{\infty}\to 0$ and has
constants depending on $G$, $\mathfrak{g}$, and $\phi$.

######  Remark 3.1.

Intuitively, since the original model (1.7) is locally attractive, it is
expected and unsurprising that a measure such as $\|Y\|_{\infty}$ for the
diameter of a concentrated population should stay small; the ‘first-order
force’ $-\kappa d\phi_{e}(Y_{ij})$ in (3.10) represents this local attraction.

###### Proof of Lemma 3.1.

By functional calculus (2.7),

$\frac{\operatorname{ad}_{Y_{ij}}}{e^{\operatorname{ad}_{Y_{ij}}}-1},\frac{\operatorname{ad}_{Y_{ij}}}{1-e^{-\operatorname{ad}_{Y_{ij}}}}=\operatorname{id}_{\mathfrak{g}}+O(|Y_{ij}|),$

and

$\phi\circ\exp(Y_{ij})=d\phi_{e}(Y_{ij})+o_{G,\phi}(|Y_{ij}|),$

so by (3.6)

$\dot{Y}_{ij}=O(\|H\|_{\infty})+\frac{\kappa}{N}\sum_{k=1}^{N}\left[d\phi_{e}(Y_{ki})-d\phi_{e}(Y_{kj})+o_{G,\phi}(\|Y\|_{\infty})\right].$

Applying the Baker-Campbell-Hausdorff formula (3.7),

$Y_{ki}-Y_{kj}\stackrel{{\scriptstyle\eqref{compatibility}}}{{=}}-Y_{ij}-a(Y_{jk},Y_{ki})\stackrel{{\scriptstyle\eqref{a-bound}}}{{=}}-Y_{ij}+O_{G}(\|Y\|_{\infty}^{2}),$

so

$\dot{Y}_{ij}=O(\|H\|_{\infty})-\kappa\left[\phi_{e}(Y_{ij})+o_{G,\phi}(\|Y\|_{\infty})\right]$

which is (3.10). Now (3.11) follows from (3.10) by (2.1). ∎

This Gronwall inequality allows us to prove the first part of Theorem 3.2.

###### Proof of Theorem 3.2 (1).

Let $c_{1}$ be small enough so that $|Y_{ij}|<c_{1}$, $i,j=1,\cdots,N$,
guarantees that $Y_{ij}\in B_{r}$ and that the $o(\|Y\|_{\infty})$ in (3.11)
is smaller than $\frac{\lambda}{2}\|Y\|_{\infty}$. Then (3.11) becomes

$\left.\frac{d}{dt}\right|^{+}\|Y(t)\|_{\infty}\leq-\frac{\kappa\lambda}{2}\|Y(t)\|_{\infty}+C\|H\|_{\infty}$

for some absolute constant $C$, for all times $t\geq 0$ such that
$\|Y(t)\|<c_{1}$. If $\kappa$ is large enough so that

$\frac{2C\|H\|_{\infty}}{\kappa\lambda}<c_{1},$

then by a standard exit-time argument, $Y(t)<c_{1}$ for all $t\geq 0$ and
$\limsup_{t\to\infty}\|Y(t)\|_{\infty}\leq\frac{2C\|H\|_{\infty}}{\kappa\lambda}$.
∎

Our second Lyapunov functional measures the maximal mismatch between the two
flows.

(3.12)
$\|Y-\tilde{Y}\|_{\infty}\coloneqq\max_{i,j=1,\cdots,N}|Y_{ij}-\tilde{Y}_{ij}|.$

######  Lemma 3.2 (Second Gronwall inequality).

Let $Y(t)=\\{Y_{ij}(t)\\}_{i,j=1}^{N}$ and
$\tilde{Y}(t)=\\{\tilde{Y}_{ij}(t)\\}_{i,j=1}^{N}$, $t\in[0,T)$ be solutions
to (3.6) with initial data $\\{Y_{ij}^{0}\\}_{i,j=1}^{N}$ and
$\\{\tilde{Y}_{ij}^{0}\\}_{i,j=1}^{N}$, respectively, with
$Y_{ij}(t),\tilde{Y}_{ij}(t)\in B_{r}$ for all $i,j=1,\cdots,N$ and
$t\in[0,T)$. Then

(3.13)
$\dot{Y}_{ij}-\dot{\tilde{Y}}_{ij}=-\kappa\left[d\phi_{e}(Y_{ij}-\tilde{Y}_{ij})+o_{G,\phi}(\|Y-\tilde{Y}\|_{\infty})\right]+O(\|Y-\tilde{Y}\|_{\infty}\cdot\|H\|_{\infty}),$

and thus

(3.14)
$\left.\frac{d}{dt}\right|^{+}\|Y-\tilde{Y}\|_{\infty}\leq-\kappa\left[\lambda\|Y-\tilde{Y}\|_{\infty}+o_{G,\phi}(\|Y-\tilde{Y}\|_{\infty})\right]+O(\|Y-\tilde{Y}\|_{\infty}\cdot\|H\|_{\infty}),$

where the little o notation is with respect to $\|Y\|,\|\tilde{Y}\|\to 0$.

######  Remark 3.2.

Observe that (3.13) is not the difference of (3.10) for $Y$ and $\tilde{Y}$.
One can accurately guess the leading term $-\kappa
d\phi_{e}(Y_{ij}-\tilde{Y}_{ij})$ in this manner, but cannot get the correct
error terms. It is crucial that the error terms are controlled in terms of
$\|Y-\tilde{Y}\|$, as opposed to simply $\|Y\|$ and $\|\tilde{Y}\|$, in order
to use (3.14) to fruition; this fact is the core mathematical content of the
proof of Theorem 3.1 (2).

###### Proof of Lemma 3.14.

We expand

$\displaystyle\dot{Y}_{ij}-\dot{\tilde{Y}}_{ij}$
$\displaystyle=\frac{\operatorname{ad}_{Y_{ij}}}{e^{\operatorname{ad}_{Y_{ij}}}-1}H_{i}-\frac{\operatorname{ad}_{Y_{ij}}}{1-e^{-\operatorname{ad}_{Y_{ij}}}}H_{j}+\frac{\kappa}{N}\sum_{k=1}^{N}\left[\frac{\operatorname{ad}_{Y_{ij}}}{e^{\operatorname{ad}_{Y_{ij}}}-1}\phi\circ\exp(Y_{ki})-\frac{\operatorname{ad}_{Y_{ij}}}{1-e^{-\operatorname{ad}_{Y_{ij}}}}\phi\circ\exp(Y_{kj})\right]$
$\displaystyle\quad-\frac{\operatorname{ad}_{\tilde{Y}_{ij}}}{e^{\operatorname{ad}_{\tilde{Y}_{ij}}}-1}H_{i}+\frac{\operatorname{ad}_{\tilde{Y}_{ij}}}{1-e^{-\operatorname{ad}_{\tilde{Y}_{ij}}}}H_{j}-\frac{\kappa}{N}\sum_{k=1}^{N}\left[\frac{\operatorname{ad}_{\tilde{Y}_{ij}}}{e^{\operatorname{ad}_{\tilde{Y}_{ij}}}-1}\phi\circ\exp(\tilde{Y}_{ki})-\frac{\operatorname{ad}_{\tilde{Y}_{ij}}}{1-e^{-\operatorname{ad}_{\tilde{Y}_{ij}}}}\phi\circ\exp(\tilde{Y}_{kj})\right]$
$\displaystyle=\left(\frac{\operatorname{ad}_{Y_{ij}}}{e^{\operatorname{ad}_{Y_{ij}}}-1}-\frac{\operatorname{ad}_{\tilde{Y}_{ij}}}{e^{\operatorname{ad}_{\tilde{Y}_{ij}}}-1}\right)\left(H_{i}+\frac{\kappa}{N}\sum_{k=1}^{N}\phi\circ\exp(\tilde{Y}_{ki})\right)$
$\displaystyle\quad-\left(\frac{\operatorname{ad}_{Y_{ij}}}{1-e^{-\operatorname{ad}_{Y_{ij}}}}-\frac{\operatorname{ad}_{\tilde{Y}_{ij}}}{1-e^{-\operatorname{ad}_{\tilde{Y}_{ij}}}}\right)\left(H_{j}+\frac{\kappa}{N}\sum_{k=1}^{N}\phi\circ\exp(\tilde{Y}_{kj})\right)$
$\displaystyle\quad+\frac{\kappa}{N}\sum_{k=1}^{N}\left[\frac{\operatorname{ad}_{Y_{ij}}}{e^{\operatorname{ad}_{Y_{ij}}}-1}\Big{(}\phi\circ\exp(Y_{ki})-\phi\circ\exp(\tilde{Y}_{ki})\Big{)}-\frac{\operatorname{ad}_{Y_{ij}}}{1-e^{-\operatorname{ad}_{Y_{ij}}}}\Big{(}\phi\circ\exp(Y_{kj})-\phi\circ\exp(\tilde{Y}_{kj})\Big{)}\right].$

By functional calculus (2.8),

$\frac{\operatorname{ad}_{Y_{ij}}}{e^{\operatorname{ad}_{Y_{ij}}}-1}-\frac{\operatorname{ad}_{\tilde{Y}_{ij}}}{e^{\operatorname{ad}_{\tilde{Y}_{ij}}}-1},\frac{\operatorname{ad}_{Y_{ij}}}{1-e^{-\operatorname{ad}_{Y_{ij}}}}-\frac{\operatorname{ad}_{\tilde{Y}_{ij}}}{1-e^{-\operatorname{ad}_{\tilde{Y}_{ij}}}}=O(|Y_{ij}-\tilde{Y}_{ij}|),$

and

$\displaystyle\phi\circ\exp(Y_{ij})-\phi\circ\exp(\tilde{Y}_{ij})=$
$\displaystyle
d\phi_{\exp(Y_{ij})}(Y_{ij}-\tilde{Y}_{ij})+o_{G,\phi}(|Y_{ij}-\tilde{Y}_{ij}|)$
$\displaystyle=$ $\displaystyle
d\phi_{e}(Y_{ij}-\tilde{Y}_{ij})+o_{G,\phi}(|Y_{ij}-\tilde{Y}_{ij}|).$

So we may simplify

$\displaystyle\dot{Y}_{ij}-\dot{\tilde{Y}}_{ij}$
$\displaystyle=O(\|Y-\tilde{Y}\|_{\infty}(\|H\|_{\infty}+\kappa\|Y\|_{\infty}+\kappa\|\tilde{Y}\|_{\infty}))$
$\displaystyle\quad+\frac{\kappa}{N}\sum_{k=1}^{N}\left[\Big{(}\phi\circ\exp(Y_{ki})-\phi\circ\exp(\tilde{Y}_{ki})\Big{)}-\Big{(}\phi\circ\exp(Y_{kj})-\phi\circ\exp(\tilde{Y}_{kj})\Big{)}\right]$
$\displaystyle=O(\|Y-\tilde{Y}\|_{\infty}\|H\|_{\infty})+\kappa\cdot
o_{G}(\|Y-\tilde{Y}\|_{\infty})$
$\displaystyle\quad+\frac{\kappa}{N}\sum_{k=1}^{N}\left[d\phi_{e}(Y_{ki}-\tilde{Y}_{ki}-Y_{kj}+\tilde{Y}_{kj})+o_{G,\phi}(\|Y-\tilde{Y}\|_{\infty})\right].$

Again, by the Baker-Campbell-Hausdorff formula (3.7),

$\displaystyle Y_{ki}-Y_{kj}-\tilde{Y}_{ki}+\tilde{Y}_{kj}$
$\displaystyle\stackrel{{\scriptstyle\eqref{compatibility}}}{{=}}-Y_{ij}+\tilde{Y}_{ij}-a(Y_{jk},Y_{ki})+a(\tilde{Y}_{jk},\tilde{Y}_{ki})$
$\displaystyle\stackrel{{\scriptstyle\eqref{a-bound}}}{{=}}-Y_{ij}+\tilde{Y}_{ij}+O_{G}((\|Y\|_{\infty}+\|\tilde{Y}\|_{\infty})\|Y-\tilde{Y}\|_{\infty}),$

and this last big O term is again $o_{G}(\|Y-\tilde{Y}\|_{\infty})$. Thus

$\dot{Y}_{ij}-\dot{\tilde{Y}}_{ij}=-\kappa\left[d\phi_{e}(Y_{ij}-\tilde{Y}_{ij})+o_{G,\phi}(\|Y-\tilde{Y}\|_{\infty})\right]+O(\|Y-\tilde{Y}\|_{\infty}\cdot\|H\|_{\infty}).$

∎

###### Proof of Theorem 3.2(2).

If $c_{1}$ is small enough so that $\|Y\|,\|\tilde{Y}\|<c_{1}$ implies that
$Y,\tilde{Y}\in B_{r}$ and that the $o(\|Y-\tilde{Y}\|_{\infty})$ term in
(3.14) is at most $\frac{\lambda\|Y-\tilde{Y}\|_{\infty}}{3}$, (3.14) becomes

$\left.\frac{d}{dt}\right|^{+}\|Y-\tilde{Y}\|_{\infty}\leq-\frac{2\kappa\lambda}{3}\|Y-\tilde{Y}\|_{\infty}+C\|Y-\tilde{Y}\|_{\infty}\cdot\|H\|_{\infty}$

for some absolute constant $C>0$. By choosing
$\kappa>\frac{3C}{\lambda}\|H\|_{\infty}$ we have

$\left.\frac{d}{dt}\right|^{+}\|Y-\tilde{Y}\|_{\infty}\leq-\frac{\kappa\lambda}{3}\|Y-\tilde{Y}\|_{\infty}$

for times $t$ at which $\|Y(t)\|,\|\tilde{Y}(t)\|<c_{1}$. This hypothesis is
satisfied by Theorem 3.2 (1) if $\|Y(0)\|,\|\tilde{Y}(0)\|<c_{1}$, in which
case we have

(3.15)
$\|Y(t)-\tilde{Y}(t)\|_{\infty}\leq\|Y(0)-\tilde{Y}(0)\|_{\infty}\exp\left(-\frac{\kappa\lambda}{3}t\right).$

∎

######  Remark 3.3.

Question 1.1 may be reformulated as follows. Is it possible that in a semi-
local regime with intermediate coupling strength $\kappa$:

$c_{1}^{\prime}<\max_{i,j=1,\cdots,N}d(X_{i}^{0},X_{j}^{0})<c_{1},\quad
c\|H\|_{\infty}<\kappa<c^{\prime}\|H\|_{\infty},$

with different constants $c_{1}^{\prime}$ and $c^{\prime}$, (3.10) holds yet
(3.13) fails?

## References

  * [1] Acebrón, J. A., Bonilla, L. L., Vicente, C. J. P., Ritort, F., and Spigler, R. The Kuramoto model: A simple paradigm for synchronization phenomena.  Reviews of modern physics 77, 1 (2005), 137.
  * [2] Aeyels, D., and Rogge, J. Stability of phase locking and existence of entrainment in networks of globally coupled oscillators.  IFAC Proceedings Volumes 37, 13 (2004), 763–768.
  * [3] Aeyels, D., and Rogge, J. A. Existence of partial entrainment and stability of phase locking behavior of coupled oscillators.  Progress of Theoretical Physics 112, 6 (2004), 921–942.
  * [4] Choi, Y.-P., Ha, S.-Y., Jung, S., and Kim, Y. Asymptotic formation and orbital stability of phase-locked states for the Kuramoto model.  Physica D: Nonlinear Phenomena 241, 7 (2012), 735–754.
  * [5] Chopra, N., and Spong, M. W. On synchronization of kuramoto oscillators.  In Proceedings of the 44th IEEE Conference on Decision and Control (2005), IEEE, pp. 3916–3922.
  * [6] Chopra, N., and Spong, M. W. On exponential synchronization of Kuramoto oscillators.  IEEE transactions on Automatic Control 54, 2 (2009), 353–357.
  * [7] Cucker, F., and Smale, S. Emergent behavior in flocks.  IEEE Transactions on Automatic Control 52, 5 (2007), 852–862.
  * [8] Cucker, F., and Smale, S. On the mathematics of emergence.  Japanese Journal of Mathematics 2, 1 (2007), 197–227.
  * [9] DeVille, L. Synchronization and stability for quantum Kuramoto.  Journal of Statistical Physics 174, 1 (2019), 160–187.
  * [10] Dörfler, F., and Bullo, F. On the critical coupling for Kuramoto oscillators.  SIAM Journal on Applied Dynamical Systems 10, 3 (2011), 1070–1099.
  * [11] Dörfler, F., and Bullo, F. Synchronization in complex networks of phase oscillators: A survey.  Automatica 50, 6 (2014), 1539–1564.
  * [12] Golse, F., and Ha, S.-Y. A mean-field limit of the Lohe matrix model and emergent dynamics.  Archive for Rational Mechanics and Analysis 234, 3 (2019), 1445–1491.
  * [13] Ha, S.-Y., Ha, T., and Kim, J.-H. On the complete synchronization of the Kuramoto phase model.  Physica D: Nonlinear Phenomena 239, 17 (2010), 1692–1700.
  * [14] Ha, S.-Y., Kim, D., Park, H., and Ryoo, S. W. Constants of motion for the finite-dimensional Lohe type models with frustration and applications to emergent dynamics.  Physica D: Nonlinear Phenomena 416 (2021), 132781.
  * [15] Ha, S.-Y., Ko, D., and Ryoo, S. W. Emergent dynamics of a generalized Lohe model on some class of Lie groups.  Journal of Statistical Physics 168, 1 (2017), 171–207.
  * [16] Ha, S.-Y., and Ryoo, S. W. On the emergence and orbital stability of phase-locked states for the Lohe model.  Journal of Statistical Physics 163, 2 (2016), 411–439.
  * [17] Ha, S.-Y., and Ryoo, S. W. Asymptotic phase-locking dynamics and critical coupling strength for the Kuramoto model.  Communications in Mathematical Physics 377, 2 (2020), 811–857.
  * [18] Jadbabaie, A., Motee, N., and Barahona, M. On the stability of the Kuramoto model of coupled nonlinear oscillators.  In Proceedings of the 2004 American Control Conference (2004), vol. 5, IEEE, pp. 4296–4301.
  * [19] Kim, D., and Kim, J. Stochastic Lohe matrix model on the Lie group and mean-field limit.  Journal of Statistical Physics 178, 6 (2020), 1467–1514.
  * [20] Kimble, H. J. The quantum internet.  Nature 453, 7198 (2008), 1023–1030.
  * [21] Kuramoto, Y. International symposium on mathematical problems in theoretical physics.  Lecture notes in Physics 30 (1975), 420.
  * [22] Kuramoto, Y. Chemical oscillations, waves, and turbulence.  Courier Corporation, 2003.
  * [23] Lohe, M. Non-Abelian Kuramoto models and synchronization.  Journal of Physics A: Mathematical and Theoretical 42, 39 (2009), 395101.
  * [24] Lohe, M. Quantum synchronization over quantum networks.  Journal of Physics A: Mathematical and Theoretical 43, 46 (2010), 465301.
  * [25] O’Keeffe, K. P., Hong, H., and Strogatz, S. H. Oscillators that sync and swarm.  Nature Communications 8, 1 (2017), 1–13.
  * [26] Pikovsky, A., Rosenblum, M., and Kurths, J. Synchronization: a universal concept in nonlinear science, 2002.
  * [27] Ritchie, L. M., Lohe, M., and Williams, A. G. Synchronization of relativistic particles in the hyperbolic Kuramoto model.  Chaos: An Interdisciplinary Journal of Nonlinear Science 28, 5 (2018), 053116.
  * [28] Verwoerd, M., and Mason, O. Global phase-locking in finite populations of phase-coupled oscillators.  SIAM Journal on Applied Dynamical Systems 7, 1 (2008), 134–160.
  * [29] Verwoerd, M., and Mason, O. On computing the critical coupling coefficient for the Kuramoto model on a complete bipartite graph.  SIAM Journal on Applied Dynamical Systems 8, 1 (2009), 417–453.
  * [30] Winfree, A. T. Biological rhythms and the behavior of populations of coupled oscillators.  Journal of theoretical biology 16, 1 (1967), 15–42.
  * [31] Xu, M., Tieri, D., Fine, E., Thompson, J., and Holland, M. Quantum synchronization of two ensembles of atoms.  In APS Division of Atomic, Molecular and Optical Physics Meeting Abstracts (2014), pp. H6–008.
  * [32] Zhu, B., Schachenmayer, J., Xu, M., Herrera, F., Restrepo, J. G., Holland, M. J., and Rey, A. M. Synchronization of interacting quantum dipoles.  New Journal of Physics 17, 8 (2015), 083063.