# Asymptotic convergence of heterogeneous first-order aggregation models: from
the sphere to the unitary group

Dohyun Kim
School of Mathematics, Statistics and Data Science,
Sungshin Women’s University, Seoul 02844, Republic of Korea
<EMAIL_ADDRESS>and  Hansol Park
Department of Mathematics,
Simon Fraser University, 8888 University Dr, Burnaby, BC V5A 1S6, Canada
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We provide the detailed asymptotic behavior for first-order aggregation models
of heterogeneous oscillators. Due to the dissimilarity of natural frequencies,
one could expect that all relative distances converge to definite positive
value and furthermore that each oscillator converges to a possibly different
stationary point. In order to establish the desired results, we introduce a
novel method, called dimension reduction method that can be applied to a
specific situation when the degree of freedom of the natural frequency is one.
In this way, we would say that although a small perturbation is allowed,
convergence toward an equilibrium of the gradient flow is still guaranteed.
Several first-order aggregation models are provided as concrete examples by
using the dimension reduction method to study the structure of the
equilibrium, and numerical simulations are conducted to support theoretical
results.

###### Key words and phrases:

Aggregation, Kuramoto model, locked states, swarm sphere model,
synchronization, heterogeneous system

###### 2020 Mathematics Subject Classification:

34C15, 34D06, 34C40

Acknowledgment. The work of D. Kim was supported by the National Research
Foundation of Korea (NRF) grant funded by the Korea government (MSIT)
(No.2021R1F1A1055929) and the work of H. Park is supported by Pacific
Institute for the Mathematical Science(PIMS), Canada postdoctoral fellowship.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Kuramoto model
    2. 2.2 Real swarm sphere model
    3. 2.3 Complex swarm sphere model
    4. 2.4 Lohe matrix model
  3. 3 Convergence toward equilibrium on the real unit sphere
    1. 3.1 Complexification
    2. 3.2 Dimension reduction method
    3. 3.3 Numeric simulations
  4. 4 Convergence toward equilibrium on the complex unit sphere
    1. 4.1 Schrödinger–Lohe model
    2. 4.2 Lohe Hermitian sphere model
  5. 5 Convergence toward equilibrium on the unitary group
  6. 6 Conclusion

##  1\. Introduction

Collective behaviors of synchronous many-body systems have been widely studied
after the work of two pioneers Kuramoto [9, 10] and Winfree [15, 16].
Depending on the dissimilarity of particles (or agents), a synchronous system
is classified into two types: homogeneous and heterogeneous, and such
dissimilarity is modeled as natural frequencies of particles. If natural
frequencies are all the same, then particles are called identical and the
corresponding system is said to be homogeneous. Otherwise, particles are
called nonidentical and the system becomes heterogeneous.

In literature, there have been lots of developments for homogeneous systems
compared to heterogeneous ones. As will be shown later, we use the Kuramoto
model [9], swarm sphere models on real and complex spheres [5, 11, 13] and the
Lohe matrix model [12] as concrete examples. All these models with homogeneous
structures can be represented as gradient flows with total relative distances
as (analytic) potentials. Then, since underlying manifolds are compact, it
follows from dynamical system theories that all solutions converge to
stationary points. However, when a heterogeneous system is considered, natural
frequencies are incorporating into the system and this leads to the breakdown
of a gradient flow. This causes numerous mathematical difficulties in
analyzing asymptotic behaviors of the systems. That’s why studying a
heterogeneous system is much harder than a homogeneous system. Precisely, when
the gradient flow is slightly perturbed in the sense that

$\dot{\mathcal{X}}=-\kappa\nabla_{\mathcal{X}}V(\mathcal{X})+\varepsilon
f(\mathcal{X}),\quad\mathcal{X}(t)\in\mathcal{M},\quad t\geq 0,$ (1.1)

where perturbation term $f(\mathcal{X})$ is given to maintain the positive
invariance of the underlying manifold (see [2] for another perturbation of a
gradient flow). Note that equation (1.1) cannot be written as a gradient flow.
Thus, convergence towards equilibrium would not be guaranteed. In addition,
structure or existence of an equilibrium is not even known. In fact, if
$\kappa$ is sufficiently small, then equilibrium might not exist (see
Proposition 2.3). However, if we adopt our strategy, then we can show that a
solution to (1.1) with some initial frameworks converges to an equilibrium,
and a structure of the equilibrium is revealed.

The purpose of this paper is to analyzing several heterogeneous systems with
our newly developed technique, so-called dimension reduction method which
consists of the following two steps:

$\bullet$ (Step 1): reduce a given heterogeneous system into two subsystems:
one is the heterogeneous Kuramoto system and the other is a homogeneous-like
system.

$\bullet$ (Step 2): use the results for the heterogeneous Kuramoto system,
such as the emergence of phase-locked states to investigate the asymptotic
behavior of the original heterogeneous system.

In this work, we apply this technique to various first-order heterogeneous
aggregation models. As a simple example and motivation for later argument, we
here briefly introduce our method with a complex swarm sphere model in [8] as
a simple example:

$\dot{z}_{j}={\mathrm{i}}H_{j}z_{j}+\frac{\kappa}{N}\sum_{k=1}^{N}(z_{k}-\langle
z_{j},z_{k}\rangle z_{j}),\quad t>0,\quad j\in[N]:=\\{1,2,\cdots,N\\},$

where $z_{j}\in\mathbb{H}\mathbb{S}^{d-1}$ and $H_{j}$ is a $d\times d$
Hermitian matrix. First, we impose the conditions on natural frequencies so
that their degree of freedom is one; that is, natural frequencies are only
determined by one single parameter: for $a_{i}\in\mathbb{R}$ being a scalar,
$H_{j}=a_{j}I_{d}$. Of course, since degree of freedom for a Hermitian matrix
over $\mathbb{R}$ is $\frac{d(d+1)}{2}$, there arises loss in this sense. As
compensate, we are able to associate the Kuramoto model whose natural
frequencies are given as $\\{a_{i}\\}$:

$\dot{\theta}_{j}=a_{j}+\frac{\kappa}{N}\sum_{k=1}^{N}\sin(\theta_{k}-\theta_{j}),\quad
t>0,\quad j\in[N].$ (1.2)

One of the most notable feature of (1.2) is that convergence toward an
equilibrium can be achieved under a large coupling strength regime (see
Proposition 2.1). In other words, asymptotic behavior of a solution to (1.2)
is somehow fully known. Then, we define an auxiliary intermediate variable
$\xi_{j}$ which is defined as

$\xi_{j}(t):=z_{j}(t)e^{-{\mathrm{i}}\theta_{j}(t)}.$

After deriving a temporal evolution of $\xi_{j}$, we will find a sufficient
condition such that all $\xi_{j}$ converges to a common stationary point, say,
$\xi^{\infty}$. By virtue of good convergence property for (1.2), we combine
two convergences to show that $z_{j}$ converges to
$\xi^{\infty}e^{{\mathrm{i}}\theta_{j}^{\infty}}$ whose modulus is turn out to
be one.

Our main result of this paper is developing a novel method, namely, dimension
reduction method in the above to study the asymptotic behavior of
heterogeneous (or nonidentical) aggregation models. For detailed analysis, we
consider four different aggregation models: the real swarm sphere model, the
Schrödinger–Lohe model, the Lohe Hermitian sphere model and the Lohe matrix
model. The first result is concerned with the swarm sphere model on the real
unit sphere $\mathbb{S}^{d-1}$:

$\dot{x}_{i}=\Omega_{i}x_{i}+\frac{\kappa}{N}\sum_{k=1}^{N}(x_{k}-\langle
x_{i},x_{k}\rangle x_{i}),$ (1.3)

subject to the initial data $x_{i}(0)=x_{i}^{0}\in\mathbb{S}^{d-1}$. Here,
$\Omega_{i}$ is an $d\times d$ skew-symmetric matrix which plays a role of
natural frequency of the $i$-th agent. Then, one can verify that
$\mathbb{S}^{d-1}$ is positively invariant under (1.3). We reformulate it to
the model defined on the complex unit sphere $\mathbb{H}\mathbb{S}^{d-1}$ by
means of the complexification (see Lemma 3.1). It is worthwhile mentioning
that it would not be possible to analyze (1.3), since a skew-symmetric matrix
is more harder to handle compared to a skew-Hermitian matrix. Thus, we find an
equivalence reformulation of (1.3) in $\mathbb{C}$. Then, after associating
the Kuramoto model and using aggregation estimates, one can deduce that a
solution converges to equilibrium (see Theorem 3.1 and Theorem 3.2). It should
be noted that to the best of our knowledge, this is the first mathematically
rigorous result for (1.3) with different $\\{\Omega_{i}\\}$ where convergence
to equilibrium is established.

Second, we deal with two aggregation models on $\mathbb{C}$: the
Schrödinger–Lohe model and the Lohe Hermitian sphere model. Similar to the
previous circumstance, we use the dimension reduction method to show that
solutions to both models converge to the equilibrium. Previously, there are
few available results (e.g., [4]) for the convergence of relative distances,
not the point itself. However in this approach, existence of equilibrium for
each configuration can be shown.

Third, the models on real and complex unit spheres are lastly extended to the
model on the unitary group, so-called the Lohe matrix model:

${\mathrm{i}}\dot{U}_{i}U_{i}^{\dagger}=H_{i}+\frac{{\mathrm{i}}\kappa}{2N}\sum_{k=1}^{N}(U_{k}U_{i}^{\dagger}-U_{i}U_{k}^{\dagger}),\quad
t>0,\quad i\in[N],$

where $H_{i}$ is a $d\times d$ Hermitian matrix. We assume that there exist a
common Hermitian matrix $H$ and scalars $\\{a_{i}\\}$ such that
$H_{i}=H-a_{i}I_{d}$. We again apply the dimension reduction method to show
that each element tends to a constant unitary matrix not necessarily the same
under a strong coupling regime. Similar to the aforementioned argument, there
is still a loss of degree of freedom; as a reward, we instead have fully
analyzed the asymptotic behavior. It should be mentioned that the authors in
[7] have intensively studied the Lohe matrix model by using orbital stability
estimate. In this way, they can show that existence of limit for relative
composite matrices $\\{U_{i}U_{j}^{\dagger}\\}$. However, that for each
$\\{U_{i}\\}$ is still unknown. We extend this previous results by associating
additional Kuramoto model with the help of the dimension reduction method.

The rest of this paper is organized as follows. In Section 2, we review
previous results in relevant literature that will be used in later sections,
and provide gradient flow formulation of the aggregation models. In Section 3,
the swarm sphere model on the real unit sphere is considered. In order to
prepare the basic setting, we reformulate the swarm sphere model on
$\mathbb{R}$ to the equivalent model but defined on $\mathbb{C}$. Then, we
introduce our dimension reduction method to show that convergence toward
equilibrium has been established. The Schrödinger–Lohe model and the Lohe
Hermitian sphere model on $\mathbb{C}$ in Section 4 and the Lohe matrix model
on the unitary group in Section 5 are discussed. By applying the same method,
we obtain the same results in the previous section. Finally, Section 6 is
devoted to a brief summary of the main results and some future direction.

Notation (i) For real vectors $x,y\in\mathbb{R}^{d}$, we write

$\langle x,y\rangle:=\sum_{i=1}^{d}x^{i}y^{i},\quad|x|:=\sqrt{\langle
x,x\rangle}.$

For complex valued vectors $w,z\in\mathbb{C}^{d}$, we write

$\langle
w,z\rangle_{\mathbb{C}}:=\sum_{i=1}^{d}w^{i}\bar{z}^{i},\quad|w|:=\sqrt{\langle
w,w\rangle_{\mathbb{C}}}.$

However, if there is no confusion and ambiguity, we simply write
$\langle\cdot,\cdot\rangle_{\mathbb{C}}=\langle\cdot,\cdot\rangle$.

(ii) For matrices, we denote $\mathcal{M}_{d}(\mathbb{R})$ and
$\mathcal{M}_{d}(\mathbb{C})$ by the sets of all $d\times d$ matrices with
real and complex entries, respectively. In addition,
$\textup{Sym}_{d}(\mathbb{R})$ is the set of all $d\times d$ symmetric
matrices, $\textup{Skew}_{d}(\mathbb{R})$ is the set of all $d\times d$ skew-
symmetric matrices, and $\textup{Skew}_{d}(\mathbb{C})$ is the set of all
$d\times d$ skew-Hermitian matrices. Regarding the norm of matrices, we use
the Frobenius norm: for $A\in\mathcal{M}_{d}(\mathbb{C})$,

$\|A\|_{\textup{F}}=\sqrt{\textup{tr}(AA^{\dagger})}.$

##  2\. Preliminaries

In this section, we review the state-of-art results of first-order aggregation
models on the real and complex spheres and the unitary group. For the purpose
of this article, we restrict ourselves to heterogeneous oscillators.

###  2.1. Kuramoto model

First, we recall basic conservation law. If we sum the relation (1.2) with
respect to $j\in[N]$, then one can easily find

$\sum_{j=1}^{N}\dot{\theta}_{j}=\sum_{j=1}^{N}a_{j}.$

Hence, if the total sums of initial data and natural frequencies are zero,
i.e.,

$\sum_{j=1}^{N}\theta_{j}(0)=\sum_{j=1}^{N}a_{j}=0,$ (2.1)

then the zero sum is conserved along the flow.

$\sum_{j=1}^{N}\theta_{j}(t)=0,\quad t>0.$

In what follows, when the Kuramoto model is considered, we always assume
(2.1). Although the Kuramoto model has been intensively and extensively
investigated in literature, however for the concise presentation of the paper,
we only introduce the results without proofs that will be used for our later
argument.

######  Proposition 2.1.

Let $\\{\theta_{j}\\}$ be a solution to the following Cauchy problem:

$\begin{cases}\displaystyle\dot{\theta}_{j}=a_{j}+\frac{\kappa}{N}\sum_{k=1}^{N}\sin(\theta_{k}-\theta_{j}),\quad
t>0,\\\ \theta_{j}(0)=\theta_{j}^{0}\in\mathbb{R},\quad j\in[N],\end{cases}$

with initial data and natural frequencies satisfying (2.1), and
$\mathcal{D}(a)$ be the maximal diameter for natural frequencies:

$\mathcal{D}(a):=\max_{1\leq i,j\leq N}|a_{i}-a_{j}|.$

Then, the following assertions hold.

(i) ([3]) If $\mathcal{D}(\Theta^{0})<\pi$, then the maximal diameter
satisfies

$\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{D}(\Theta)\leq\mathcal{D}(a)-\kappa\mathcal{D}(\Theta),\quad
t>0.$

Thus, if initial diameter satisfy
$\mathcal{D}(\Theta^{0})<\frac{\mathcal{D}(a)}{\kappa}<\pi$, then

$\mathcal{D}(\Theta(t))<\frac{\mathcal{D}(a)}{\kappa},\quad t>0.$

(ii) ([6]) If the coupling strength and initial data satisfy

$R_{0}:=\sqrt{\frac{1}{N^{2}}\sum_{i,j=1}^{N}\cos(\theta_{i}^{0}-\theta_{j}^{0})}>0,\quad\kappa>1.6\frac{\mathcal{D}(a)}{R_{0}},$

then there exists $\\{\theta_{j}^{\infty}\\}$ such that

$\lim_{t\to\infty}\theta_{j}(t)=\theta_{j}^{\infty},\quad j\in[N].$

###  2.2. Real swarm sphere model

In [13], the author has proposed the consensus protocol on the unit sphere,
and later, the author in [12] incorporated the natural frequencies into the
swarming model on the unit sphere:

$\begin{cases}\displaystyle\dot{x}_{i}=\Omega_{i}x_{i}+\frac{\kappa}{N}\sum_{k=1}^{N}(x_{k}-\langle
x_{i},x_{k}\rangle x_{i}),\quad t>0,\\\ \displaystyle
x_{i}(0)=x_{i}^{0}\in\mathbb{S}^{d-1},\quad i\in[N].\end{cases}$ (2.2)

Here, $\Omega_{i}$ is a $d\times d$ skew-symmetric matrix, $\kappa$ is a
uniform coupling strength, and $\langle\cdot,\cdot\rangle$ is the standard
$\ell^{2}$ inner produce in $\mathbb{R}^{d}$.

######  Proposition 2.2.

(i) ([1]) Suppose that the coupling strength and initial data satisfy

$\displaystyle\kappa>4(d+1)\mathcal{D}(\Omega),\quad\min_{1\leq i\leq
N}\langle x_{i}^{0},x_{c}^{0}\rangle>4(d+1)\mathcal{D}(\Omega),$
$\displaystyle\rho\in(\alpha_{-},\alpha_{+}),\quad\alpha_{\pm}:=\frac{\kappa\pm\sqrt{\kappa^{2}-2\kappa(d+1)N\mathcal{D}(\Omega)}}{2\kappa},$

and let $\\{x_{i}\\}$ be a solution to (2.2). Then, we have

$\lim_{\kappa\to\infty}\liminf_{t\to\infty}\rho(t)=1.$

(ii) ([14]) We say that an equilibrium is referred to as dispersed if there is
no open hemisphere containing the set $\\{x_{1},\cdots,x_{N}\\}$. Then, any
dispersed equilibrium for (2.2) is (exponentially) unstable if the frequencies
are small in the sense that

$\left(\sum_{i=1}^{N}\|\Omega_{i}\|_{\textup{F}}^{2}\right)^{\frac{1}{2}}<\frac{\kappa}{N+1}\left(N-1-\cos\frac{\pi}{N}\right)\left(1-\cos\frac{\pi}{N}\right).$

We also refer the reader to [17] for similar results in the above when the
underlying graph becomes a digraph containing a directed spanning tree.

######  Remark 2.1.

In fact, as far as the authors know, there are no mathematically available
results including those in Proposition 2.2 relevant to the asymptotic
formation of locked states for (2.2).

###  2.3. Complex swarm sphere model

In [11], the author proposed a coupled system of nonlinear Schrödinger
equation, called the Schrödinger–Lohe model which reads as

$\begin{cases}\displaystyle{\mathrm{i}}\partial_{t}\psi_{j}=-\frac{1}{2}\Delta\psi_{j}+V_{j}\psi_{j}+\frac{{\mathrm{i}}\kappa}{N}\sum_{k=1}^{N}(\psi_{k}-\langle\psi_{j},\psi_{k}\rangle\psi_{j}),\quad
t>0,\\\ \displaystyle\psi_{j}(0,x)=\psi_{j}^{0}(x)\in
L^{2}(\mathbb{R}^{d}),\quad j\in[N].\end{cases}$ (2.3)

Here, $V_{j}$ is an one-body external potential acted on $j$-th node, $\kappa$
is a uniform coupling strength and $\langle\cdot,\cdot\rangle$ is the standard
inner product in $L^{2}(\mathbb{R}^{d})$.

######  Proposition 2.3.

[4] Suppose that system parameters and initial data satisfy

$V_{j}(x)=V(x)+\nu_{j},\quad\kappa>4\max_{1\leq i,j\leq
N}|\nu_{i}-\nu_{j}|,\quad\max_{1\leq i,j\leq
N}|1-\langle\psi_{i}^{0},\psi_{j}^{0}\rangle|\ll 1,$

and let $\\{\psi_{j}\\}$ be a solution to (2.3). Then, we have

$\lim_{t\to\infty}\langle\psi_{i},\psi_{j}\rangle(t)\quad\textup{exists for
$i,j\in[N]$.}$

Recently, the authors in [5] extended the swarm sphere model (2.2) defined on
$\mathbb{S}^{d-1}$ to the model on the complex (or Hermitian) sphere denoted
by $\mathbb{H}\mathbb{S}^{d-1}$:

$\begin{cases}\displaystyle\dot{z}_{j}=\Omega_{j}z_{j}+\frac{\kappa_{0}}{N}\sum_{k=1}^{N}\langle(z_{k}-\langle
z_{j},z_{k}\rangle z_{j})+\frac{\kappa_{1}}{N}\sum_{k=1}^{N}(\langle
z_{k},z_{j}\rangle-\langle z_{j},z_{k}\rangle)z_{j},\quad t>0,\\\
\displaystyle z_{j}(0)=z_{j}^{0}\in\mathbb{H}\mathbb{S}^{d-1},\quad
j\in[N].\end{cases}$ (2.4)

We also refer the reader to [8] for a new swarming model on the complex swarm
sphere.

######  Remark 2.2.

In fact, if one performs similar method used in Proposition 2.3 for (2.4) with
a specific condition, say, $\Omega_{j}=\Omega+{\mathrm{i}}\nu_{j}I_{d}$, then
a similar result in Proposition 2.3 can be easily obtained without any effort.

###  2.4. Lohe matrix model

In [12], the authors introduced a first-order aggregation model on the unitary
group of degree $d$ denoted as $\mathbf{U}(d)$:

$\begin{cases}\displaystyle{\mathrm{i}}\dot{U}_{j}U_{j}^{\dagger}=H_{j}+\frac{{\mathrm{i}}\kappa}{2N}\sum_{k=1}^{N}(U_{k}U_{j}^{\dagger}-U_{j}U_{k}^{\dagger}),\quad
t>0,\\\ U_{j}(0)=U_{j}^{0}\in\mathbf{U}(d),\quad j\in[N].\end{cases}$ (2.5)

Here, $H_{j}$ is a $d\times d$ Hermitian matrix that plays a role of the
natural frequency, $\kappa$ is a uniform coupling strength and $\dagger$
denotes the Hermitian conjugate.

######  Proposition 2.4.

[7] Suppose that system parameters satisfy

$\kappa>\frac{54}{17}\max_{1\leq i,j\leq
N}\|H_{i}-H_{j}\|_{\textup{F}},\quad\max_{1\leq i,j\leq
N}\|U_{i}^{0}-U_{j}^{0}\|_{\textup{F}}\ll 1,$

and let $\\{U_{j}\\}$ be a solution to (2.5). Then, we have

$\lim_{t\to\infty}U_{i}U_{j}^{\dagger}(t)\quad\textup{exists for
$i,j\in[N]$.}$

######  Remark 2.3.

In Proposition 2.4, the authors guaranteed the limit of
$U_{i}U_{j}^{\dagger}$; however, the limit of each $U_{j}$ was unknown at that
time. As will be shown later, we provide the limit of $U_{j}$.

##  3\. Convergence toward equilibrium on the real unit sphere

In this section, we study the asymptotic behavior of the real swarm sphere
model

$\begin{cases}\vspace{0.1cm}\displaystyle\dot{x}_{i}=\Omega_{i}x_{i}+\frac{\kappa}{N}\sum_{k=1}^{N}(x_{k}-\langle
x_{i},x_{k}\rangle x_{i}),\quad t>0,\\\ \displaystyle
x_{i}(0)=x_{i}^{0}\in\mathbb{S}^{d-1},\quad i\in[N].\end{cases}$ (3.1)

###  3.1. Complexification

In what follows, we find an equivalent formulation of (3.1) by means of
‘complexification.’ We first consider the case of an even dimension. For a
given $\Omega_{i}\in\textup{Skew}_{2d}(\mathbb{R})$, we decompose $\Omega_{i}$
into

$\Omega_{i}=\begin{pmatrix}A_{i}&B_{i}\\\
-B_{i}^{\top}&C_{i}\end{pmatrix},\quad
A_{i},C_{i}\in\textup{Skew}_{d}(\mathbb{R}),\quad
B_{i}\in\mathcal{M}_{d}(\mathbb{R}).$

In addition for $x_{i}\in\mathbb{R}^{2d}$, we associate a $d$-dimensional
complex vector $\omega_{i}\in\mathbb{C}^{d}$ defined as

$x_{i}=:(y_{i},z_{i})\in\mathbb{R}^{d}\times\mathbb{R}^{d},\quad\omega_{i}:=y_{i}+{\mathrm{i}}z_{i}\in\mathbb{C}^{d}.$

Then, one can easily check

$|\omega_{i}|^{2}=|y_{i}|^{2}+|z_{i}|^{2}=|x_{i}|^{2}.$

######  Lemma 3.1.

Let $(x_{i},\Omega_{i})$ be a solution to (3.1) that can be decomposed into

$x_{i}=(y_{i},z_{i})\in\mathbb{R}^{d}\times\mathbb{R}^{d},\quad\Omega_{i}=\begin{pmatrix}A_{i}&B_{i}\\\
-B_{i}^{\top}&C_{i}\end{pmatrix}\in\textup{Skew}_{2d}(\mathbb{R}),\quad
i\in[N].$ (3.2)

Suppose that the composition matrices $A_{i},B_{i}$ and $C_{i}$ satisfy

$A_{i}=C_{i}\in\textup{Skew}_{d}(\mathbb{R}),\quad
B_{i}\in\textup{Sym}_{d}(\mathbb{R}),\quad i\in[N].$ (3.3)

Then, the swarm sphere model (3.1) is equivalent to the following model on
$\mathbb{C}^{d}$: for $\omega_{i}=y_{i}+{\mathrm{i}}z_{i}$,

$\dot{\omega}_{i}=\Xi_{i}\omega_{i}+\frac{\kappa}{N}\sum_{k=1}^{N}(\omega_{k}-\textup{Re}[\langle\omega_{i},\omega_{k}\rangle]\omega_{i}),\quad\Xi_{i}:=A_{i}-{\mathrm{i}}B_{i}.$
(3.4)

Moreover, the unit modulus is conserved along flow (3.4). Hence, (3.4) is
defined on $\mathbb{H}\mathbb{S}^{d-1}$.

###### Proof.

We divide (1.3) as two subsystems: a free flow and an interaction flow.

(i) For a free flow part, since $x_{i}$ and $\Omega_{i}$ satisfy (3.2), we
rewrite the free flow as

$\dot{x}_{i}=\Omega_{i}x_{i}\quad\Longleftrightarrow\quad\begin{pmatrix}\dot{y}_{i}\\\
\dot{z}_{i}\end{pmatrix}=\begin{pmatrix}A_{i}y_{i}+B_{i}z_{i}\\\
-B_{i}^{\top}y_{i}+C_{i}z_{i}\end{pmatrix}.$

Then, $\dot{\omega}_{i}$ satisfies

$\dot{\omega}_{i}=\dot{y}_{i}+{\mathrm{i}}\dot{z}_{i}=(A_{i}y_{i}+B_{i}z_{i})+{\mathrm{i}}(-B_{i}^{\top}y_{i}+C_{i}z_{i})=(A_{i}-{\mathrm{i}}B_{i}^{\top})y_{i}+(C_{i}-{\mathrm{i}}B_{i})({\mathrm{i}}z_{i}).$

Since $A_{i},B_{i}$ and $C_{i}$ satisfy (3.3), a free flow
$\dot{x}_{i}=\Omega_{i}x_{i}$ in $\mathbb{R}^{2d}$ is equivalent to
$\dot{\omega}_{i}=\Xi_{i}\omega_{i}$ in $\mathbb{C}^{d}$ with a skew-Hermitian
matrix $\Xi_{i}=A_{i}-{\mathrm{i}}B_{i}$.

(ii) On the other hand for an interaction flow, we observe

$\displaystyle\langle\omega_{i},\omega_{k}\rangle_{\mathbb{C}}=\langle
y_{i}+{\mathrm{i}}z_{i},y_{k}+{\mathrm{i}}z_{k}\rangle_{\mathbb{C}}=\langle
y_{i},y_{k}\rangle+\langle z_{i},z_{k}\rangle+{\mathrm{i}}(\langle
z_{i},y_{k}\rangle-\langle y_{i},z_{k}\rangle).$

Since the relation $\langle x_{i},x_{k}\rangle=\langle
y_{i},y_{k}\rangle+\langle z_{i},z_{k}\rangle$ holds, we find the relation
between the complex and real inner products:

$\langle
x_{i},x_{k}\rangle=\textup{Re}[\langle\omega_{i},\omega_{k}\rangle_{\mathbb{C}}].$

Finally for the last assertion, the unit modulus property direct follows from
the equivalence relation between (3.1) and (3.4):

$\displaystyle\frac{{\textup{d}}}{{\textup{d}t}}|\omega_{i}|^{2}$
$\displaystyle=\langle\dot{\omega}_{i},\omega_{i}\rangle+\langle\omega_{i},\dot{\omega}_{i}\rangle=\langle(A_{i}-{\mathrm{i}}B_{i})\omega_{i},\omega_{i}\rangle+\langle\omega_{i},(A_{i}-{\mathrm{i}}B_{i})\omega_{i}\rangle$
$\displaystyle\hskip
14.22636pt+\frac{\kappa}{N}\sum_{k=1}^{N}\Big{(}\langle\omega_{k},\omega_{i}\rangle-\textup{Re}[\langle\omega_{i},\omega_{k}\rangle]|\omega_{i}|^{2}+\langle\omega_{i},\omega_{k}\rangle-\textup{Re}[\langle\omega_{i},\omega_{k}\rangle]|\omega_{i}|^{2}\Big{)}$
$\displaystyle=\langle
A_{i}\omega_{i},\omega_{i}\rangle+\langle\omega_{i},A_{i}\omega_{i}\rangle+\langle-{\mathrm{i}}B_{i}\omega_{i},\omega_{i}\rangle+\langle\omega_{i},-{\mathrm{i}}B_{i}\omega_{i}\rangle$
$\displaystyle\hskip
14.22636pt+\frac{2\kappa}{N}\sum_{k=1}^{N}\textup{Re}[\langle\omega_{i},\omega_{k}\rangle](1-|\omega_{i}|^{2})$
$\displaystyle=\langle(A_{i}+A_{i}^{\top})\omega_{i},\omega_{i}\rangle+\langle{\mathrm{i}}(B_{i}^{\top}-B_{i})\omega_{i},\omega_{i}\rangle+\frac{2\kappa}{N}\sum_{k=1}^{N}\textup{Re}[\langle\omega_{i},\omega_{k}\rangle](1-|\omega_{i}|^{2})$
$\displaystyle=\frac{2\kappa}{N}\sum_{k=1}^{N}\textup{Re}[\langle\omega_{i},\omega_{k}\rangle](1-|\omega_{i}|^{2}).$

Hence, we have

$1-|\omega_{i}(t)|^{2}=(1-|\omega_{i}^{0}|^{2})\exp\left(-\frac{2\kappa}{N}\sum_{k=1}^{N}\textup{Re}[\langle\omega_{i},\omega_{k}\rangle]\right),$

which gives the desired result. ∎

On the other hand for the odd dimension, we will basically extend
$\mathbb{R}^{2d+1}$ to $\mathbb{R}^{2d+2}$ and apply the same argument of an
even dimensional case. Let $x_{i}\in\mathbb{R}^{2d+1}$ and
$\Omega_{i}\in\textup{Skew}_{2d+1}(\mathbb{R})$. Then, we consider an
augmented matrix of $\Omega_{i}$, denoted by $\tilde{\Omega}_{i}$:

$\tilde{\Omega}_{i}=\begin{pmatrix}\Omega_{i}&\mathbf{0}_{2d+1}\\\
\mathbf{0}_{2d+1}^{\top}&0\end{pmatrix}\in\textup{Skew}_{2d+2}(\mathbb{R}).$

Similarly, we set an augmented vector of $x_{i}$ denoted as $\tilde{x}_{i}$:

$\tilde{x}_{i}:=(x_{i},0)\in\mathbb{R}^{2d+2}.$

Since $(x_{i},\Omega_{i})$ satisfies (1.3), we see that
$(\tilde{x}_{i},\tilde{\Omega}_{i})$ satisfies

$\dot{\tilde{x}}_{i}=\tilde{\Omega}_{i}\tilde{x}_{i}+\frac{\kappa}{N}\sum_{k=1}^{N}(\tilde{x}_{k}-\langle\tilde{x}_{i},\tilde{x}_{k}\rangle\tilde{x}_{i}),$

which coincides (1.3) in $\mathbb{R}^{2d+2}$. Then, we denote

$\tilde{x}_{i}=(y_{i},z_{i})\in\mathbb{R}^{d+1}\times\mathbb{R}^{d+1},\quad\tilde{\Omega}_{i}=\begin{pmatrix}A_{i}&B_{i}\\\
-B_{i}^{\top}&C_{i}\end{pmatrix},$

where $A_{i},C_{i}\in\textup{Skew}_{d+1}(\mathbb{R})$ and
$B_{i}\in\mathcal{M}_{d+1}(\mathbb{R})$. Then,
$\omega_{i}=y_{i}+{\mathrm{i}}z_{i}$ satisfies (3.4).

In the following subsections, we use the complexified swarm sphere model (3.4)
instead of the real swarm sphere model (3.1).

###  3.2. Dimension reduction method

In this subsection, we introduce a novel method to investigate the asymptotic
behavior for heterogeneous aggregation models. First, we assume that the
natural frequency matrices satisfy the following zeroth-order approximation
condition: precisely, there exists $a_{j}\in\mathbb{R}$ such that

$\Xi_{j}=\Xi+{\mathrm{i}}a_{j}I_{d},\quad j\in[N].$

Due to the hermitian symmetry, one would set

$\Xi=O_{d}.$

Then, our system reads as

$\dot{w}_{j}=\mathrm{i}a_{j}w_{j}+\frac{\kappa}{N}\sum_{k=1}^{N}\left(w_{k}-\mathrm{Re}\left(\langle
w_{k},w_{j}\rangle\right)w_{j}\right),\quad t>0,\quad j\in[N].$ (3.5)

For $a_{i}$ that appears in (3.5), we associate the Kuramoto model whose
natural frequencies are exactly same as $\\{a_{i}\\}$. To be more specific,
let $\\{\theta_{i}\\}$ be a solution to

$\dot{\theta}_{j}=a_{j}+\frac{\kappa}{N}\sum_{k=1}^{N}\sin(\theta_{k}-\theta_{j}),\quad
t>0,\quad j\in[N],$ (3.6)

together with initial data and natural frequencies satisfying zero sum
conditions:

$\sum_{j=1}^{N}\theta_{j}(0)=0\quad\textup{and}\quad\sum_{j=1}^{N}a_{j}=0.$

Thus, we would say that two models (3.5) and (3.6) are weakly coupled through
natural frequencies. We define a vector

$z_{j}(t):=w_{j}(t)e^{-{\mathrm{i}}\theta_{j}(t)}\in\mathbb{H}\mathbb{S}^{d-1}\subseteq\mathbb{C}^{d},\quad
t>0.$

Below, we find the governing equation for $z_{i}$.

######  Lemma 3.2.

Let $w_{j}$ and $\theta_{j}$ be solutions to (3.5) and (3.6), respectively.
Then, $z_{i}=w_{i}e^{-{\mathrm{i}}\theta_{i}}$ satisfies

$\dot{z}_{j}=\frac{\kappa}{N}\sum_{k=1}^{N}\left(e^{\mathrm{i}(\theta_{k}-\theta_{j})}(z_{k}-z_{j})-\mathrm{Re}(\langle
e^{\mathrm{i}(\theta_{k}-\theta_{j})}(z_{k}-z_{j}),z_{j}\rangle)z_{j}\right),\quad
t>0,\quad j\in[N].$ (3.7)

###### Proof.

By straightforward calculation, one finds the desired dynamics for $z_{i}$:

$\displaystyle\dot{z}_{j}$
$\displaystyle=\mathrm{i}a_{j}z_{j}+\frac{\kappa}{N}\sum_{k=1}^{N}\left(e^{\mathrm{i}(\theta_{k}-\theta_{j})}z_{k}-\mathrm{Re}(\langle
e^{\mathrm{i}(\theta_{k}-\theta_{j})}z_{k},z_{j}\rangle)z_{j}\right)-\mathrm{i}\left(a_{j}+\frac{\kappa}{N}\sum_{k=1}^{N}\sin(\theta_{k}-\theta_{j})\right)z_{j}$
$\displaystyle=\frac{\kappa}{N}\sum_{k=1}^{N}\left(e^{\mathrm{i}(\theta_{k}-\theta_{j})}z_{k}-\left(\mathrm{Re}(\langle
e^{\mathrm{i}(\theta_{k}-\theta_{j})}z_{k},z_{j}\rangle)+\mathrm{i}\sin(\theta_{k}-\theta_{j})\right)z_{j}\right)$
$\displaystyle=\frac{\kappa}{N}\sum_{k=1}^{N}\left(e^{\mathrm{i}(\theta_{k}-\theta_{i})}(z_{k}-z_{i})-\mathrm{Re}(\langle
e^{\mathrm{i}(\theta_{k}-\theta_{i})}(z_{k}-z_{i}),z_{i}\rangle)z_{i}\right).$

∎

######  Remark 3.1.

If we write
$\tilde{z}_{jk}:=e^{{\mathrm{i}}(\theta_{k}-\theta_{j})}(z_{k}-z_{j})$, then
(3.7) is written as

$\dot{z}_{j}=\frac{\kappa}{N}\sum_{k=1}^{N}(\tilde{z}_{jk}-\textup{Re}\langle
z_{j},\tilde{z}_{jk}\rangle z_{j}).$

It is worthwhile mentioning that dynamics of $z_{j}$ does not contain free
flow; instead, natural frequency part $a_{j}$ is encoded into
$\tilde{z}_{jk}$.

For later use, we denote

$h_{ij}:=\langle z_{i},z_{j}\rangle,\quad
R_{ij}:=\textup{Re}h_{ij},\quad\mathcal{D}(\mathcal{R}):=\max_{1\leq i,j\leq
N}|1-R_{ij}|.$

Since our goal is to find a sufficient condition leading to the zero
convergence of $\mathcal{D}(\mathcal{R})$, we derive the dynamics of $R_{ij}$.

######  Lemma 3.3.

Let $\\{z_{j}\\}$ be a solution to (3.7). Then, $R_{ij}$ satisfies

$\displaystyle\begin{aligned}
\frac{{\textup{d}}}{{\textup{d}t}}(1-R_{ij})&=-2\kappa(1-R_{ij})+\frac{\kappa}{N}\sum_{k=1}^{N}(1-R_{ik}+1-R_{kj})(1-R_{ij})\\\
&\hskip
14.22636pt-\frac{\kappa}{N}\sum_{k=1}^{N}\Big{[}(\textup{Re}\langle\mathcal{A},z_{j}\rangle+\textup{Re}\langle\mathcal{B},z_{i}\rangle)(1-R_{ij})+\textup{Re}\langle\mathcal{A}-\mathcal{B},z_{j}-z_{i}\rangle\Big{]},\end{aligned}$
(3.8)

where $\mathcal{A}$ and $\mathcal{B}$ are defined as

$\mathcal{A}:=(e^{\mathrm{i}(\theta_{k}-\theta_{i})}-1)(z_{k}-z_{i}),\quad\mathcal{B}:=(e^{\mathrm{i}(\theta_{k}-\theta_{j})}-1)(z_{k}-z_{j}).$
(3.9)

In addition, the maximal diameter $\mathcal{D}(\mathcal{R})$ satisfies

$\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{D}(\mathcal{R})\leq-\kappa\left(1-4\sin\frac{\mathcal{D}(\Theta)}{2}\right)\mathcal{D}(\mathcal{R})+2\kappa\left(\left|\sin\frac{\mathcal{D}(\Theta)}{2}\right|+\frac{1}{2}\right)\mathcal{D}(\mathcal{R})^{2},\quad
t>0.$ (3.10)

###### Proof.

We observe

$\displaystyle\frac{{\textup{d}}}{{\textup{d}t}}\mathrm{Re}\langle
z_{i},z_{j}\rangle$
$\displaystyle=\frac{\kappa}{N}\sum_{k=1}^{N}\left(\mathrm{Re}\langle
e^{\mathrm{i}(\theta_{k}-\theta_{i})}(z_{k}-z_{i}),z_{j}\rangle-\mathrm{Re}(\langle
e^{\mathrm{i}(\theta_{k}-\theta_{i})}(z_{k}-z_{i}),z_{i}\rangle)\mathrm{Re}\langle
z_{i},z_{j}\rangle\right)$ $\displaystyle\hskip
14.22636pt+\frac{\kappa}{N}\sum_{k=1}^{N}\left(\mathrm{Re}\langle
z_{i},e^{\mathrm{i}(\theta_{k}-\theta_{j})}(z_{k}-z_{j})\rangle-\mathrm{Re}(\langle
e^{\mathrm{i}(\theta_{k}-\theta_{j})}(z_{k}-z_{j}),z_{j}\rangle)\mathrm{Re}\langle
z_{i},z_{j}\rangle\right)$
$\displaystyle=:\frac{\kappa}{N}\sum_{k=1}^{N}(\mathcal{I}_{11}-\mathcal{I}_{12})+\frac{\kappa}{N}\sum_{k=1}^{N}(\mathcal{I}_{13}-\mathcal{I}_{14}).$

Below, we consider $\mathcal{I}_{1k}$ ($k=1,2,3,4$), respectively.

$\bullet$ (Calculation of $\mathcal{I}_{12}+\mathcal{I}_{14}$): we observe

$\displaystyle\mathcal{I}_{11}+\mathcal{I}_{13}$
$\displaystyle=\mathrm{Re}\langle
e^{\mathrm{i}(\theta_{k}-\theta_{i})}(z_{k}-z_{i}),z_{j}\rangle+\mathrm{Re}\langle
z_{i},e^{\mathrm{i}(\theta_{k}-\theta_{j})}(z_{k}-z_{j})\rangle$
$\displaystyle=R_{kj}-R_{ij}+R_{ik}-R_{ij}$ $\displaystyle\hskip
14.22636pt+\mathrm{Re}\langle(e^{\mathrm{i}(\theta_{k}-\theta_{i})}-1)(z_{k}-z_{i}),z_{j}\rangle+\mathrm{Re}\langle
z_{i},(e^{\mathrm{i}(\theta_{k}-\theta_{j})}-1)(z_{k}-z_{j})\rangle.$

$\bullet$ (Calculation of $\mathcal{I}_{11}+\mathcal{I}_{13}$): similarly, one
has

$\displaystyle\mathcal{I}_{12}+\mathcal{I}_{14}$
$\displaystyle=\mathrm{Re}(\langle
e^{\mathrm{i}(\theta_{k}-\theta_{i})}(z_{k}-z_{i}),z_{i}\rangle)\mathrm{Re}\langle
z_{i},z_{j}\rangle+\mathrm{Re}(\langle
e^{\mathrm{i}(\theta_{k}-\theta_{j})}(z_{k}-z_{j}),z_{j}\rangle)\mathrm{Re}\langle
z_{i},z_{j}\rangle$ $\displaystyle=(R_{ik}-1)R_{ij}+(R_{kj}-1)R_{ij}$
$\displaystyle\hskip
14.22636pt+\mathrm{Re}(\langle(e^{\mathrm{i}(\theta_{k}-\theta_{i})}-1)(z_{k}-z_{i}),z_{i}\rangle)\mathrm{Re}\langle
z_{i},z_{j}\rangle+\mathrm{Re}(\langle(e^{\mathrm{i}(\theta_{k}-\theta_{j})}-1)(z_{k}-z_{j}),z_{j}\rangle)\mathrm{Re}\langle
z_{i},z_{j}\rangle$
$\displaystyle=(1-R_{ik}+1-R_{kj})(1-R_{ij})-(1-R_{ik}+1-R_{kj})$
$\displaystyle\hskip
14.22636pt+\mathrm{Re}(\langle(e^{\mathrm{i}(\theta_{k}-\theta_{i})}-1)(z_{k}-z_{i}),z_{i}\rangle)\mathrm{Re}\langle
z_{i},z_{j}\rangle-\mathrm{Re}(\langle(e^{\mathrm{i}(\theta_{k}-\theta_{j})}-1)(z_{k}-z_{j}),z_{j}\rangle)\mathrm{Re}\langle
z_{i},z_{j}\rangle.$

We combine the calculations above to find

$\displaystyle\mathcal{I}_{11}-\mathcal{I}_{12}+\mathcal{I}_{13}-\mathcal{I}_{14}$
$\displaystyle=R_{kj}-R_{ij}+R_{ik}-R_{ij}+(1-R_{ik}+1-R_{kj})-(1-R_{ik}+1-R_{kj})(1-R_{ij})$
$\displaystyle\hskip
14.22636pt+\mathrm{Re}\langle(e^{\mathrm{i}(\theta_{k}-\theta_{i})}-1)(z_{k}-z_{i}),z_{j}\rangle+\mathrm{Re}\langle
z_{i},(e^{\mathrm{i}(\theta_{k}-\theta_{j})}-1)(z_{k}-z_{j})\rangle$
$\displaystyle\hskip
14.22636pt-\mathrm{Re}(\langle(e^{\mathrm{i}(\theta_{k}-\theta_{i})}-1)(z_{k}-z_{i}),z_{i}\rangle)\mathrm{Re}\langle
z_{i},z_{j}\rangle+\mathrm{Re}(\langle(e^{\mathrm{i}(\theta_{k}-\theta_{j})}-1)(z_{k}-z_{j}),z_{j}\rangle)\mathrm{Re}\langle
z_{i},z_{j}\rangle$ $\displaystyle=2(1-R_{ij})-(1-R_{ik}+1-R_{kj})(1-R_{ij})$
$\displaystyle\hskip
14.22636pt+\mathrm{Re}\langle(e^{\mathrm{i}(\theta_{k}-\theta_{i})}-1)(z_{k}-z_{i}),z_{j}\rangle+\mathrm{Re}\langle
z_{i},(e^{\mathrm{i}(\theta_{k}-\theta_{j})}-1)(z_{k}-z_{j})\rangle$
$\displaystyle\hskip
14.22636pt-\mathrm{Re}(\langle(e^{\mathrm{i}(\theta_{k}-\theta_{i})}-1)(z_{k}-z_{i}),z_{i}\rangle)\mathrm{Re}\langle
z_{i},z_{j}\rangle-\mathrm{Re}(\langle(e^{\mathrm{i}(\theta_{k}-\theta_{j})}-1)(z_{k}-z_{j}),z_{j}\rangle)\mathrm{Re}\langle
z_{i},z_{j}\rangle.$

If we recall the definition of $\mathcal{A}$ and $\mathcal{B}$ in (3.9), then
one has

$\displaystyle\mathcal{I}_{11}-\mathcal{I}_{12}+\mathcal{I}_{13}-\mathcal{I}_{14}$
$\displaystyle\hskip
14.22636pt=R_{kj}-R_{ij}+R_{ik}-R_{ij}+(1-R_{ik}+1-R_{kj})-(1-R_{ik}+1-R_{kj})(1-R_{ij})$
$\displaystyle\hskip
28.45274pt+\textup{Re}\langle\mathcal{A},z_{j}\rangle-\textup{Re}\langle\mathcal{A},z_{i}\rangle
R_{ij}+\textup{Re}\langle\mathcal{B},z_{i}\rangle-\textup{Re}\langle\mathcal{B},z_{j}\rangle
R_{ij}$
$\displaystyle=R_{kj}-R_{ij}+R_{ik}-R_{ij}+(1-R_{ik}+1-R_{kj})-(1-R_{ik}+1-R_{kj})(1-R_{ij})$
$\displaystyle\hskip
28.45274pt+(\textup{Re}\langle\mathcal{A},z_{j}\rangle+\textup{Re}\langle\mathcal{B},z_{i}\rangle)(1-R_{ij})+\textup{Re}\langle\mathcal{A}-\mathcal{B},z_{j}-z_{i}\rangle.$

Consequently, we can derive the desired dynamics (3.8) for $R_{ij}$.

For the second assertion, we write

$\displaystyle\mathcal{I}_{15}:=(\textup{Re}\langle\mathcal{A},z_{j}\rangle+\textup{Re}\langle\mathcal{B},z_{i}\rangle)(1-R_{ij}),\quad\mathcal{I}_{16}:=\textup{Re}\langle\mathcal{A}-\mathcal{B},z_{j}-z_{i}\rangle.$

Below, we present estimates for $\mathcal{I}_{15}$ and $\mathcal{I}_{16}$,
respectively.

$\bullet$ (Estimate of $\mathcal{I}_{15}$): we observe

$|e^{{\mathrm{i}}\varphi}-1|\leq 2\left|\sin\frac{\varphi}{2}\right|,$ (3.11)

which yields

$|\mathcal{A}|=|(e^{\mathrm{i}(\theta_{k}-\theta_{i})}-1)(z_{k}-z_{i})|\leq
2\left|\sin\frac{\mathcal{D}(\Theta)}{2}\right|\mathcal{D}(\mathcal{R}),\quad|\mathcal{B}|\leq
2\left|\sin\frac{\mathcal{D}(\Theta)}{2}\right|\mathcal{D}(\mathcal{R}).$

Hence, one finds

$\displaystyle\mathcal{I}_{15}=(\textup{Re}\langle\mathcal{A},z_{j}\rangle+\textup{Re}\langle\mathcal{B},z_{i}\rangle)(1-R_{ij})\leq
4\left|\sin\frac{\mathcal{D}(\Theta)}{2}\right|\mathcal{D}(\mathcal{R})\cdot\frac{\mathcal{D}(\mathcal{R})^{2}}{2}=2\left|\sin\frac{\mathcal{D}(\Theta)}{2}\right|\mathcal{D}(\mathcal{R})^{3}.$

$\bullet$ (Estimate of $\mathcal{I}_{16}$): We see

$\mathcal{A}-\mathcal{B}=(e^{{\mathrm{i}}(\theta_{k}-\theta_{i})}-1)(z_{j}-z_{i})+(e^{{\mathrm{i}}(\theta_{k}-\theta_{i})}-e^{{\mathrm{i}}(\theta_{k}-\theta_{j})})(z_{k}-z_{j}).$

Thus, we obtain

$\displaystyle\mathcal{I}_{16}$
$\displaystyle=\textup{Re}\langle\mathcal{A}-\mathcal{B},z_{j}-z_{i}\rangle=(\cos(\theta_{k}-\theta_{i})-1)|z_{j}-z_{i}|^{2}$
$\displaystyle\hskip
14.22636pt+\textup{Re}\langle(e^{{\mathrm{i}}(\theta_{k}-\theta_{i})}-e^{{\mathrm{i}}(\theta_{k}-\theta_{j})})(z_{k}-z_{j}),z_{j}-z_{i}\rangle$
$\displaystyle\leq
4\sin\frac{\mathcal{D}(\Theta)}{2}\mathcal{D}(\mathcal{R})^{2}.$

Since $1-R_{ij}$ satisfies

$\displaystyle\begin{aligned}
\frac{{\textup{d}}}{{\textup{d}t}}(1-R_{ij})&=-2\kappa(1-R_{ij})+\frac{\kappa}{N}\sum_{k=1}^{N}(1-R_{ik}+1-R_{kj})(1-R_{ij})\\\
&-\frac{\kappa}{N}\sum_{k=1}^{N}\Big{[}(\textup{Re}\langle\mathcal{A},z_{j}\rangle+\textup{Re}\langle\mathcal{B},z_{i}\rangle)(1-R_{ij})+\textup{Re}\langle\mathcal{A}-\mathcal{B},z_{j}-z_{i}\rangle\Big{]}\\\
&=-2\kappa(1-R_{ij})+\frac{\kappa}{N}\sum_{k=1}^{N}(1-R_{ik}+1-R_{kj})(1-R_{ij})\\\
&\hskip
14.22636pt-\frac{\kappa}{N}\sum_{k=1}^{N}\Big{[}\mathcal{I}_{15}+\mathcal{I}_{16}\Big{]},\end{aligned}$
(3.12)

in (3.12), we collect all estimates to derive

$\displaystyle\begin{aligned}
\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{D}(\mathcal{R})&\leq-\kappa\left(1-4\sin\frac{\mathcal{D}(\Theta)}{2}\right)\mathcal{D}(\mathcal{R})+2\kappa\left|\sin\frac{\mathcal{D}(\Theta)}{2}\right|\mathcal{D}(\mathcal{R})^{2}+\frac{\kappa}{2}\mathcal{D}(\mathcal{R})^{3}\\\
&\leq-\kappa\left(1-4\sin\frac{\mathcal{D}(\Theta)}{2}\right)\mathcal{D}(\mathcal{R})+2\kappa\left(\left|\sin\frac{\mathcal{D}(\Theta)}{2}\right|+\frac{1}{2}\right)\mathcal{D}(\mathcal{R})^{2}.\end{aligned}$

∎

We are now ready to provide a sufficient condition under which
$\mathcal{D}(\mathcal{R})$ tends to zero. Furthermore, all $z_{j}$ tends to a
common stationary point.

######  Proposition 3.1.

Suppose that the coupling strength (sufficiently large in the sense below) and
natural frequencies satisfy

$\kappa>2\mathcal{D}(a),\quad\sum_{j=1}^{N}a_{j}=0,$ (3.13)

and that there exists a positive constant $\delta\in(0,\frac{1}{2})$ such that

$\displaystyle\begin{aligned}
\mathcal{D}(\Theta^{0})<\delta<\frac{\mathcal{D}(a)}{\kappa}<\frac{1}{2}\quad\textup{and}\quad\mathcal{D}(\mathcal{R}^{0})<\frac{1-2\delta}{1+\delta}.\end{aligned}$
(3.14)

Let $\\{z_{i}\\}$ be a solution to (3.7) where $\\{\theta_{i}\\}$ is a
solution to (3.6). Then, the maximal diameter $\mathcal{D}(\mathcal{R})$
converges to zero with an exponential convergence rate:

$\lim_{t\to\infty}\mathcal{D}(\mathcal{R}(t))=0.$

In addition, there exists $z^{\infty}\in\mathbb{H}\mathbb{S}^{d-1}$ such that

$\lim_{t\to\infty}z_{i}(t)=z^{\infty},\quad i\in[N].$ (3.15)

###### Proof.

Since we assume (3.13) and $\eqref{C-27}_{1}$, it follows from Proposition 2.1
that inequality (3.10) for $\mathcal{D}(\mathcal{R})$ becomes

$\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{D}(\mathcal{R})\leq-\kappa(1-2\delta)\mathcal{D}(\mathcal{R})+\kappa(1+\delta)\mathcal{D}(\mathcal{R})^{2}.$

Then, initial condition $\eqref{C-27}_{2}$ gives the desired exponential
convergence. For the second assertion, we use the governing dynamics (3.7) for
$z_{i}$ to obtain

$|\dot{z}_{j}|\leq 2\kappa\mathcal{D}(\mathcal{R}),\quad t>0,$

which gives that $\dot{z}_{j}\in L^{1}(\mathbb{R}_{+})$. Thus, one finds

$\lim_{t\to\infty}z_{j}(t)=z_{j}^{0}+\int_{0}^{\infty}\dot{z}_{j}(s){\textup{d}}s=:z_{j}^{\infty}.$

Moreover, since complete aggregation occurs, we observe for $i\neq j$,

$|z_{i}^{\infty}-z_{j}^{\infty}|\leq\lim_{t\to\infty}\Big{(}|z_{i}^{\infty}-z_{i}(t)|+|z_{i}(t)-z_{j}(t)|+|z_{j}(t)-z_{j}^{\infty}|\Big{)}=0.$

Hence, there exists $z^{\infty}$ satisfying (3.15). ∎

In Proposition 3.1, we have investigated the asymptotic behavior of $z_{i}$.
In the following theorem, we use Theorem 3.1 to study the asymptotic behavior
of $w_{i}$ that is governed by (3.5). For this, we write

$\mathcal{D}(W(t)):=\max_{1\leq i,j\leq N}|w_{i}(t)-w_{j}(t)|,\quad t>0.$

######  Theorem 3.1.

Suppose that the coupling strength and initial data satisfy

$\kappa>2\mathcal{D}(a),\quad\sum_{j=1}^{N}a_{j}=0$

and that there exists $\delta\in(0,\beta)$ such that

$0<\delta<\frac{\mathcal{D}(a)}{\kappa}<\beta,\quad\mathcal{D}(W^{0})<\sqrt{\frac{2(1-2\delta)}{1+\delta}}-2\sin\frac{\delta}{2},$
(3.16)

where $\beta\approx 0.434$ is a unique positive root for
$\sqrt{\frac{2(1-2s)}{1+s}}=2\sin\frac{s}{2}$. Let $\\{w_{j}\\}$ be a solution
to (3.5). Then, there exists $w_{j}^{\infty}\in\mathbb{H}\mathbb{S}^{d-1}$
such that

$\lim_{t\to\infty}w_{j}(t)=w_{j}^{\infty},\quad j\in[N].$

In addition, $\\{w_{j}^{\infty}\\}$ satisfy

$|\langle w_{i}^{\infty},w_{j}^{\infty}\rangle|=1,\quad i,j\in[N].$

###### Proof.

In order to adapt Proposition 3.1, we use the initial frameworks in (3.13) and
(3.14) to see

$\displaystyle|z_{i}^{0}-z_{j}^{0}|=|w_{i}^{0}-e^{{\mathrm{i}}(\theta_{i}^{0}-\theta_{j}^{0})}w_{j}^{0}|\leq|w_{i}^{0}-w_{j}^{0}|+|e^{{\mathrm{i}}(\theta_{i}^{0}-\theta_{j}^{0})}|-1|,$

which yields

$\displaystyle\sqrt{2\mathcal{D}(\mathcal{R}^{0})}\leq\mathcal{D}(W^{0})+2\sin\frac{\mathcal{D}(\Theta^{0})}{2}<\mathcal{D}(W^{0})+2\sin\frac{\delta}{2}.$

Thus, if $\mathcal{D}(W^{0})$ satisfy (3.16), then initial conditions for
$\mathcal{D}(\mathcal{R}^{0})$ and $\mathcal{D}(\Theta^{0})$ are fulfilled.
Consequently, it follows from Theorem 3.1 and Proposition 2.1 that there
exists $z^{\infty}$ and $\theta^{\infty}$ such that

$\lim_{t\to\infty}z_{j}(t)=z^{\infty},\quad\lim_{t\to\infty}\theta_{j}(t)=\theta_{j}^{\infty}.$

Hence, we have

$\lim_{t\to\infty}w_{j}(t)=z^{\infty}e^{{\mathrm{i}}\theta_{j}^{\infty}}=:w_{j}^{\infty}.$

Moreover,

$\langle w_{i}^{\infty},w_{j}^{\infty}\rangle=\langle
z^{\infty}e^{{\mathrm{i}}\theta_{i}^{\infty}},z^{\infty}e^{{\mathrm{i}}\theta_{j}^{\infty}}\rangle=e^{{\mathrm{i}}(\theta_{i}^{\infty}-\theta_{j}^{\infty})}$

which has a unit modulus. ∎

######  Remark 3.2.

We mention that $w_{i}(t)$ is asymptotically decomposed into two parts:
$z^{\infty}$ as a common axis part and $e^{{\mathrm{i}}\theta_{i}^{\infty}}$
as its own rotation part. Heuristically, we would say that the asymptotic
behavior of a high dimensional object $w_{i}(t)$ is determined by a one-
dimensional object $\theta_{i}^{\infty}$.

In fact, we have studied the asymptotic behavior of $w_{i}$ on
$\mathbb{H}\mathbb{S}^{d-1}$ by using auxiliary variables $z_{i}$ and
$\theta_{i}$. However, we are essentially concerned with the asymptotic
behavior of (3.1) where a solution $\\{x_{i}\\}$ to (3.1) lies on
$\mathbb{S}^{d-1}$. Thus, our last step in this section is dedicated to
representing Theorem 3.1 in terms of $x_{i}$.

######  Theorem 3.2.

Consider the real swarm sphere model (3.1) on even dimension $2d$ whose
solution is written as
$x_{j}=(x_{j}^{1},\cdots,x_{j}^{2d})\in\mathbb{S}^{2d-1}$. Suppose that
initial data and system parameters satisfy the following conditions:

$\displaystyle\textup{(i)}~{}~{}\kappa>2\mathcal{D}(a),\quad
0<\delta<\frac{\mathcal{D}(a)}{\kappa}<\beta.$
$\displaystyle\textup{(ii)}~{}~{}\Omega_{j}=\begin{pmatrix}A&B\\\
-B&A\end{pmatrix}+a_{j}\begin{pmatrix}O_{d}&I_{d}\\\
-I_{d}&O_{d}\end{pmatrix},\quad\forall~{}j\in[N],$
$\displaystyle\textup{(iii)}~{}~{}\max_{1\leq i,j\leq
N}\Big{(}|(x_{i}^{1,0},\cdots,x_{i}^{d,0})-(x_{j}^{1,0},\cdots,x_{j}^{d,0})|+|(x_{i}^{d+1,0},\cdots,x_{i}^{2d,0})-(x_{j}^{d+1,0},\cdots,x_{j}^{2d,0})|\Big{)}$
$\displaystyle\hskip
28.45274pt<\sqrt{\frac{2(1-2\delta)}{1+\delta}}-2\sin\frac{\delta}{2},$

where $A$ and $B$ are $d\times d$ skew-symmetric and symmetric matrices,
respectively, and $\delta,\mathcal{D}(a),\beta$ are defined in Theorem 3.1.
Then, there exists $x_{i}^{\infty}\in\mathbb{S}^{2d-1}$ such that

$\lim_{t\to\infty}x_{j}(t)=x_{j}^{\infty},\quad i\in[N].$

In addition, if we decompose
$x_{j}^{\infty}=(y_{j}^{\infty},z_{j}^{\infty})\in\mathbb{R}^{d}\times\mathbb{R}^{d}$,
then we have

$|\langle x_{i}^{\infty},x_{j}^{\infty}\rangle|^{2}+|\langle
z_{i}^{\infty},y_{j}^{\infty}\rangle-\langle
y_{i}^{\infty},z_{j}^{\infty}\rangle|^{2}=1,\quad i,j\in[N].$

######  Remark 3.3.

If we observe

$\displaystyle|x_{i}^{0}-x_{j}^{0}|<|(x_{i}^{1,0},\cdots,x_{i}^{d,0})-(x_{j}^{1,0},\cdots,x_{j}^{d,0})|+|(x_{i}^{d+1,0},\cdots,x_{i}^{2d,0})-(x_{j}^{d+1,0},\cdots,x_{j}^{2d,0})|,$

then the assumption on initial data in Theorem 3.2(iii) merely becomes

$\max_{1\leq i,j\leq N}|x_{i}^{0}-x_{j}^{0}|\ll 1.$

###  3.3. Numeric simulations

In this subsection, we conduct numerical simulations to support our
theoretical results presented in Theorem 3.1. For the numerical method, the
Runge–Kutta fourth-order method is employed to discretize system (3.5) with
the time step $\Delta t=0.01$:

$\dot{w}_{j}=\mathrm{i}a_{j}w_{j}+\frac{\kappa}{N}\sum_{k=1}^{N}\left(w_{k}-\mathrm{Re}\left(\langle
w_{k},w_{j}\rangle\right)w_{j}\right),\quad t>0,\quad j\in[N].$ (3.17)

We fix the following parameters

$\displaystyle d=2,\quad
N=4,\quad(a_{1},a_{2},a_{3},a_{4})=\left(-0.2831,-0.0196,0.0708,0.2318\right),$
(3.18)

where $\\{a_{i}\\}$ are chosen to satisfy zero-average with increasing order,
i.e., $a_{1}<a_{2}<a_{3}<a_{4}$. In addition, initial data
$\\{w_{i}^{0}\\}_{i=1}^{4}$ are randomly chosen from
$\mathbb{H}\mathbb{S}^{1}\subseteq\mathbb{C}^{2}$.

$\displaystyle w_{1}^{0}=(0.3895-0.9178i,-0.0770+0.0004i),\quad
w_{2}^{0}=(-0.5190+0.4832i,-0.5974-0.3746i),$ $\displaystyle
w_{3}^{0}=(-0.2123-0.8137i,-0.5232+0.1381i),\quad
w_{4}^{0}=(0.1684+0.3132i,0.6192-0.7001i).$

(a) Evolution of $\dot{w}_{i}$

(b) Evolution of $\log|\dot{w}_{i}|$

(c) Evolution of $|\langle w_{i},w_{j}\rangle|$

(d) Evolution of geodesic distances

Figure 1. Numeric simulations of system (3.17) with parameters (3.18).

In Figure 1(a) and Figure 1(b), we see that $\dot{w}_{i}$ converges to zero
exponentially and this shows that each $w_{i}$ converges to definite value.
From the numerical simulation, we find the final configurations for $w_{i}$:

$\displaystyle w_{1}^{\infty}=(-0.5002-0.4755i,-0.3890-0.6102i),\quad
w_{2}^{\infty}=(-0.3537-0.5926i,-0.2104-0.6924i),$ $\displaystyle
w_{3}^{\infty}=(-0.2977-0.6226i,-0.1458-0.7089i),\quad
w_{4}^{\infty}=(-0.1906-0.6633i,-0.0265-0.7232i),$

and direct calculation yields

$|\langle
w_{i}^{\infty},w_{j}^{\infty}\rangle|=1\quad\forall~{}i,j\in\\{1,2,3,4\\}.$

In Figure 1(c), we also provide temporal evolutions for $|\langle
w_{i},w_{j}\rangle|$, and this coincides with the calculations above. On the
other hand, we define a $4\times 4$ matrix
$\Delta\Theta^{\infty}=(\Delta\Theta^{\infty}_{ij})_{i,j=1}^{4}$ where
$\Delta\Theta^{\infty}_{ij}$ is a geodesic distance between $w_{i}^{\infty}$
and $w_{j}^{\infty}$ on the hermitian sphere
$\mathbb{HS}^{1}\simeq\mathbb{S}^{3}$, i.e.,

$\Delta\Theta^{\infty}_{ij}=\arccos\left(1-\frac{|w_{i}^{\infty}-w_{j}^{\infty}|^{2}}{2}\right),\quad\forall~{}i,j\in\\{1,2,3,4\\}.$

Since $\Delta\Theta^{\infty}$ is calculated as

$\Delta\Theta^{\infty}=\begin{bmatrix}0&0.2726&0.3647&0.5310\\\
0.2726&0&0.0922&0.2584\\\ 0.3647&0.0922&0&0.1662\\\
0.5310&0.2584&0.1662&0\end{bmatrix},$

one can easily check

$\displaystyle\begin{cases}\Delta\Theta^{\infty}_{13}&=\Delta\Theta^{\infty}_{12}+\Delta\Theta^{\infty}_{23},\\\
\Delta\Theta^{\infty}_{24}&=\Delta\Theta^{\infty}_{23}+\Delta\Theta^{\infty}_{34},\\\
\Delta\Theta^{\infty}_{14}&=\Delta\Theta^{\infty}_{12}+\Delta\Theta^{\infty}_{23}+\Delta\Theta^{\infty}_{34}.\end{cases}$

This shows that $w_{1}^{\infty}$, $w_{2}^{\infty}$, $w_{3}^{\infty}$, and
$w_{4}^{\infty}$ are on the same geodesic in order of $\\{a_{i}\\}$.
Furthermore, we can check that $\Theta=(\theta_{i})_{i=1}^{N}$ defined by

$\theta_{i}=\varphi+\Theta_{1i}\quad\forall~{}i\in\\{1,2,3,4\\},$ (3.19)

is a steady state of system (3.6). Finally, we conclude that (3.19) gives the
relationship between (3.5) and (3.6).

##  4\. Convergence toward equilibrium on the complex unit sphere

In this section, we investigate the detailed asymptotic behavior of the
Schrödinger–Lohe model and the Lohe Hermitian model whose states are
represented in general by complex numbers. We continue to apply our dimension
reduction method.

###  4.1. Schrödinger–Lohe model

In this subsection, we consider the heterogeneous Schrödinger–Lohe model with
a specific structure:

$V_{j}(x)=V(x)+a_{j},\quad a_{j}\in\mathbb{R},\quad j\in[N].$

In other words, external potentials are given as a (small) perturbation
$\\{a_{j}\\}$ of a common potential $V(x)$. Furthermore, as in the previous
section, we assume that the perturbation $a_{j}$ has zero average. Then, our
model reads as

${\mathrm{i}}\partial_{t}\psi_{j}=\left(-\frac{1}{2}\Delta+V\right)\psi_{j}+a_{j}\psi_{j}+\frac{{\mathrm{i}}\kappa}{N}\sum_{k=1}^{N}(\psi_{k}-\langle\psi_{j},\psi_{k}\rangle\psi_{j}).$

Since $e^{-{\mathrm{i}}(-\frac{1}{2}\Delta+V)}$ is a unitary operator, it
follows from the solution splitting property that we may assume
$\psi_{j}=\psi_{j}(t)$ satisfying

$\dot{\psi}_{j}={\mathrm{i}}a_{j}\psi_{j}+\frac{\kappa}{N}\sum_{k=1}^{N}(\psi_{k}-\langle\psi_{j},\psi_{k}\rangle\psi_{j}).$
(4.1)

Although $\psi_{j}(t)\in\mathbb{C}$, it can be extend to $\mathbb{C}^{d}$
without any effort. In this case, the inner product is given as the standard
one in $\mathbb{C}^{d}$. As discussed in the previous section, we associate
the Kuramoto model whose natural frequencies are determined as $\\{a_{j}\\}$
with zero-average natural frequency:

$\dot{\theta}_{j}=a_{j}+\frac{2\kappa}{N}\sum_{k=1}^{N}\sin(\theta_{k}-\theta_{j}).$
(4.2)

It is worthwhile mentioning that the coupling strength in (4.2) is doubled
compared to the (original) Kuramoto (see Remark 4.1). We define

$\xi_{j}(t):=\psi_{j}(t)e^{-{\mathrm{i}}\theta_{j}(t)}.$

######  Lemma 4.1.

Let $\psi_{j}$ and $\theta_{j}$ be solutions to (4.1) and (4.2), respectively.
Then, $\xi_{j}=\psi_{j}e^{-{\mathrm{i}}\theta_{j}}$ satisfies

$\dot{\xi}_{j}=\frac{\kappa}{N}\sum_{k=1}^{N}\Big{[}e^{{\mathrm{i}}(\theta_{k}-\theta_{j})}(\xi_{k}-\xi_{j})-\langle\xi_{j},e^{{\mathrm{i}}(\theta_{k}-\theta_{j})}(\xi_{k}-\xi_{j})\rangle\xi_{j}\Big{]},\quad
t>0,\quad j\in[N].$ (4.3)

In addition, temporal evolution of $h_{ij}:=\langle\xi_{i},\xi_{j}\rangle$ is
given by

$\displaystyle\begin{aligned}
\dot{h}_{ij}&=\frac{\kappa}{N}\sum_{k=1}^{N}\Big{[}e^{{\mathrm{i}}(\theta_{k}-\theta_{i})}(h_{kj}-h_{ij})+e^{{\mathrm{i}}(\theta_{i}-\theta_{k})}(1-h_{ik})h_{ij}\\\
&\hskip
56.9055pt+e^{{\mathrm{i}}(\theta_{j}-\theta_{k})}(h_{ik}-h_{ij})+e^{{\mathrm{i}}(\theta_{k}-\theta_{j})}(1-h_{kj})h_{ij}\Big{]}.\end{aligned}$
(4.4)

###### Proof.

It directly follows from straightforward calculation. ∎

We denote the maximal diameter

$\mathcal{D}(\mathcal{H}(t)):=\max_{1\leq i,j\leq N}|1-h_{ij}(t)|,\quad
h_{ij}(t)=\langle\xi_{i},\xi_{j}\rangle(t),\quad t>0.$

Since the following relation holds due to the unit modulus property of
$\xi_{i}$,

$|\xi_{i}-\xi_{j}|^{2}=2(1-\textup{Re}\langle\xi_{i},\xi_{j}\rangle)\leq
2|1-h_{ij}|,$

it suffices to focus on $1-h_{ij}$ instead of the relative distance
$|\xi_{i}-\xi_{j}|^{2}$. Below, we derive a differential inequality for
$\mathcal{D}(\mathcal{H})$.

######  Lemma 4.2.

Let $\\{\xi_{i}\\}$ be a solution to (4.3). Then, the maximal diameter
$\mathcal{D}(\mathcal{H})$ satisfies

$\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{D}(\mathcal{H})\leq-2\kappa\left(1-6\sin\frac{\mathcal{D}(\Theta)}{2}\right)\mathcal{D}(\mathcal{H})+2\kappa\mathcal{D}(\mathcal{H})^{2},\quad
t>0.$ (4.5)

###### Proof.

We use (4.4) to find

$\displaystyle\begin{aligned}
\frac{{\textup{d}}}{{\textup{d}t}}(1-h_{ij})&=-2\kappa(1-h_{ij})+\frac{\kappa}{N}\sum_{k=1}^{N}(1-h_{ik}+1-h_{kj})(1-h_{ij})\\\
&\hskip
14.22636pt-\frac{\kappa}{N}\sum_{k=1}^{N}\Big{[}(e^{{\mathrm{i}}(\theta_{k}-\theta_{i})}-1)(h_{kj}-h_{ij})+(e^{{\mathrm{i}}(\theta_{j}-\theta_{k})}-1)(h_{ik}-h_{ij})\\\
&\hskip
62.59596pt+(e^{{\mathrm{i}}(\theta_{i}-\theta_{k})}-1)(1-h_{ik})h_{ij}+(e^{{\mathrm{i}}(\theta_{k}-\theta_{j})}-1)(1-h_{kj})h_{ij}\Big{]}.\end{aligned}$
(4.6)

Then, (4.6) and (3.11) give

$\displaystyle\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{D}(\mathcal{H})$
$\displaystyle\leq-2\kappa\mathcal{D}(\mathcal{H})+2\kappa\mathcal{D}(\mathcal{H})^{2}+12\kappa\sin\frac{\mathcal{D}(\Theta)}{2}\mathcal{D}(\mathcal{H})$
$\displaystyle=-2\kappa\left(1-6\sin\frac{\mathcal{D}(\Theta)}{2}\right)\mathcal{D}(\mathcal{H})+2\kappa\mathcal{D}(\mathcal{H})^{2}$

∎

We are now ready to provide a sufficient condition leading to the zero
convergence of $\mathcal{D}(\mathcal{H})$ under a large coupling strength
regime.

######  Proposition 4.1.

Suppose that the coupling strength (sufficiently large in the sense below) and
natural frequencies satisfy

$\kappa>\frac{3}{2}\mathcal{D}(a),\quad\sum_{i=1}^{N}a_{i}=0,$

and that there exists a positive constant $\delta\in(0,\frac{1}{3})$ such that

$\mathcal{D}(\Theta^{0})<\delta<\frac{\mathcal{D}(a)}{2\kappa}<\frac{1}{3}\quad\textup{and}\quad\mathcal{D}(\mathcal{H}^{0})<1-3\delta.$
(4.7)

Let $\\{\xi_{j}\\}$ and $\\{\theta_{j}\\}$ be solutions to (4.3) and (4.2),
respectively. Then, the maximal diameter converges to zero exponentially:

$\lim_{t\to\infty}\mathcal{D}(\mathcal{H}(t))=0.$

Moreover, there exists $\xi^{\infty}\in\mathbb{H}\mathbb{S}^{d-1}$ such that

$\lim_{t\to\infty}\xi_{j}(t)=\xi^{\infty},\quad j\in[N].$

###### Proof.

Since initial data satisfy $\eqref{D-11}_{1}$, we use Proposition 2.1 to see

$\mathcal{D}(\Theta(t))<\delta,\quad t>0.$

Then, (4.5) becomes

$\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{D}(\mathcal{H})\leq-2\kappa(1-3\delta)\mathcal{D}(\mathcal{H})+2\kappa\mathcal{D}(\mathcal{H})^{2},\quad
t>0.$

Again, initial condition $\eqref{D-11}_{2}$ gives the desired exponential
decay of $\mathcal{D}(\mathcal{H})$ toward zero. For the second assertion, we
observe from (4.3)

$|\dot{\xi}_{j}|\leq 2\sqrt{2}\kappa\sqrt{\mathcal{D}(\mathcal{H})},$

which gives $|\dot{\xi}_{j}|\in L^{1}(\mathbb{R}_{+})$. Thus, we have

$\lim_{t\to\infty}\xi_{j}(t)=\xi_{j}^{0}+\int_{0}^{\infty}\dot{\xi}_{j}(s){\textup{d}}s=:\xi_{j}^{\infty}.$

Since $\mathcal{D}(\mathcal{H})$ tends to zero, we can easily deduce that all
$\\{\xi_{i}^{\infty}\\}$ should be same. This shows the existence of the
desired common vector $\xi^{\infty}$. ∎

In Proposition 4.1, we have shown that all $\xi_{i}$ collapse to a common
stationary point $\xi^{\infty}$ under some initial framework. We are now
concerned with the asymptotic behavior of (4.1), which is our main interest in
this subsection. For this, we denote

$\mathcal{D}(\Psi(t)):=\max_{1\leq i,j\leq
N}|1-\langle\psi_{i},\psi_{j}\rangle(t)|,\quad t>0.$

######  Theorem 4.1.

Suppose that the coupling strength and natural frequencies satisfy

$\kappa>\frac{3}{2}\mathcal{D}(a),\quad\sum_{i=1}^{N}a_{i}=0,$

and that there exists $\delta\in(0,\beta)$ such that

$0<\delta<\frac{\mathcal{D}(a)}{\kappa}<\beta,\quad\mathcal{D}(\Psi^{0})<1-3\delta-2\sin\frac{\delta}{2},$
(4.8)

where $\beta\approx 0.250$ is a unique positive root for
$1-3s=2\sin\frac{s}{2}$. Let $\\{\psi_{j}\\}$ be a solution to (4.1). Then,
there exists $\psi_{j}^{\infty}\in\mathbb{H}\mathbb{S}^{d-1}$ such that

$\lim_{t\to\infty}\psi_{j}(t)=\psi_{j}^{\infty},\quad j\in[N].$

In addition, such $\\{\psi_{j}^{\infty}\\}$ satisfy

$|\langle\psi_{i}^{\infty},\psi_{j}^{\infty}\rangle|=1,\quad i,j\in[N].$

###### Proof.

In order to use Proposition 4.1, we observe

$|1-\langle\xi_{i}^{0},\xi_{j}^{0}\rangle|=|1-\langle\psi_{i}^{0},\psi_{j}^{0}\rangle+\langle\psi_{i}^{0},\psi_{j}^{0}\rangle(1-e^{{\mathrm{i}}(\theta_{j}^{0}-\theta_{i}^{0})})|\leq\mathcal{D}(\Psi^{0})+2\sin\frac{\mathcal{D}(\Theta^{0})}{2}.$
(4.9)

Since initial diameter $\mathcal{D}(\Psi^{0})$ satisfies (4.8), initial
conditions for $\mathcal{D}(\mathcal{H}^{0})$ and $\mathcal{D}(\Theta^{0})$ in
(4.7) hold. Thus, we use Proposition 4.1 and Proposition 2.1 to conclude that
there exist $\xi^{\infty}\in\mathbb{H}\mathbb{S}^{d-1}$ and
$\theta_{j}^{\infty}$ such that

$\lim_{t\to\infty}\psi_{j}(t)=\xi^{\infty}e^{{\mathrm{i}}\theta_{j}^{\infty}}=:\psi_{j}^{\infty}$

which shows that $\langle\psi_{i}^{\infty},\psi_{j}^{\infty}\rangle$ has unit
modulus:

$\langle\psi_{i}^{\infty},\psi_{j}^{\infty}\rangle=\langle\xi^{\infty}e^{{\mathrm{i}}\theta_{i}^{\infty}},\xi^{\infty}e^{{\mathrm{i}}\theta_{j}^{\infty}}\rangle=e^{{\mathrm{i}}(\theta_{i}^{\infty}-\theta_{j}^{\infty})}.$

∎

######  Remark 4.1.

We here mention that the Kuramoto model (4.2) with doubled coupling strength
was formally derived from (4.1) by using the ansatz:

$\psi_{j}(t)=e^{{\mathrm{i}}\theta_{j}(t)}.$

Similar to Remark 3.2, all information for the natural frequency part of (4.1)
have been encoded to the Kuramoto model. Thus, we would say that the
asymptotic behavior of high-dimensional quantity $\psi_{j}$ can be determined
by the one of one-dimensional quantity $\theta_{j}$.

######  Remark 4.2.

In fact, the authors of [4] have shown that $\langle\psi_{i},\psi_{j}\rangle$
converges to a definite value. However, there is no further information on the
convergent values. In addition, the existence of a limit for $\psi_{j}$ was
not provided. By applying the dimension reduction method, we are here able to
show that such convergent values have unit modulus.

###  4.2. Lohe Hermitian sphere model

In this subsection, we turn to the Lohe Hermitian sphere model on
$\mathbb{H}\mathbb{S}^{d-1}$ which reads as

$\dot{z}_{j}={\mathrm{i}}a_{j}z_{j}+\frac{\kappa_{0}}{N}\sum_{k=1}^{N}(z_{k}-\langle
z_{j},z_{k}\rangle z_{j})+\frac{\kappa_{1}}{N}\sum_{k=1}^{N}(\langle
z_{k},z_{j}\rangle-\langle z_{j},z_{k}\rangle)z_{j},\quad t>0,\quad j\in[N].$
(4.10)

As we have done, we couple the Kuramoto model whose natural frequencies are
$\\{a_{j}\\}$ with zero average:

$\dot{\theta}_{j}=a_{j}+\frac{2(\kappa_{0}+\kappa_{1})}{N}\sum_{k=1}^{N}\sin(\theta_{k}-\theta_{j}),$
(4.11)

and define an auxiliary variable

$\xi_{j}(t):=z_{j}(t)e^{-{\mathrm{i}}\theta_{j}(t)}.$

######  Lemma 4.3.

Let $z_{j}$ and $\theta_{j}$ be solutions to (4.10) and (4.11), respectively.
Then, $\xi_{j}=z_{j}e^{-{\mathrm{i}}\theta_{j}}$ satisfies

$\displaystyle\begin{aligned}
\dot{\xi}_{j}&=\frac{\kappa_{0}}{N}\sum_{k=1}^{N}\Big{[}e^{{\mathrm{i}}(\theta_{k}-\theta_{j})}(\xi_{k}-\xi_{j})-\langle\xi_{j},e^{{\mathrm{i}}(\theta_{k}-\theta_{j})}(\xi_{k}-\xi_{j})\rangle\xi_{j}\Big{]}\\\
&\hskip 14.22636pt+\frac{\kappa_{1}}{N}\sum_{k=1}^{N}\Big{[}\langle
e^{{\mathrm{i}}(\theta_{k}-\theta_{j})}(\xi_{k}-\xi_{j}),\xi_{j}\rangle-\langle\xi_{j},e^{{\mathrm{i}}(\theta_{k}-\theta_{j})}(\xi_{k}-\xi_{j})\rangle\Big{]}\xi_{j}.\end{aligned}$
(4.12)

In addition, temporal evolution of $h_{ij}:=\langle\xi_{i},\xi_{j}\rangle$ is
given as

$\displaystyle\begin{aligned}
\dot{h}_{ij}&=\frac{\kappa_{0}}{N}\sum_{k=1}^{N}\Big{[}e^{{\mathrm{i}}(\theta_{k}-\theta_{i})}(h_{kj}-h_{ij})+e^{{\mathrm{i}}(\theta_{i}-\theta_{k})}(1-h_{ik})h_{ij}\\\
&\hskip
56.9055pt+e^{{\mathrm{i}}(\theta_{j}-\theta_{k})}(h_{ik}-h_{ij})+e^{{\mathrm{i}}(\theta_{k}-\theta_{j})}(1-h_{kj})h_{ij}\Big{]}\\\
&\hskip
14.22636pt+\frac{\kappa_{1}}{N}\sum_{k=1}^{N}\Big{[}e^{{\mathrm{i}}(\theta_{k}-\theta_{i})}(h_{ki}-1)h_{ij}-e^{{\mathrm{i}}(\theta_{i}-\theta_{k})}(h_{ik}-1)h_{ij}\\\
&\hskip
71.13188pt+e^{{\mathrm{i}}(\theta_{j}-\theta_{k})}(h_{jk}-1)h_{ij}-e^{{\mathrm{i}}(\theta_{k}-\theta_{j})}(h_{kj}-1)h_{ij}\Big{]}.\end{aligned}$
(4.13)

###### Proof.

The proof directly follows from straightforward calculation. ∎

We denote the maximal diameter

$\mathcal{D}(\mathcal{H}(t)):=\max_{1\leq i,j\leq N}|1-h_{ij}(t)|,\quad
h_{ij}=\langle\xi_{i},\xi_{j}\rangle(t),\quad t>0.$

As for the Schrödinger–Lohe model, we consider the dynamics of the maximal
quantities among $|1-h_{ij}(t)|$ instead of, for instance, relative distance
$|\xi_{i}-\xi_{j}|^{2}$. Below, we find a differential inequality for
$\mathcal{D}(\mathcal{H})$.

######  Lemma 4.4.

Let $\\{\xi_{j}\\}$ be a solution to (4.12). Then, the maximal diameter
$\mathcal{D}(\mathcal{H})$ satisfies

$\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{D}(\mathcal{H})\leq-2\left(\kappa_{0}-2\kappa_{1}-(6\kappa_{0}+4\kappa_{1})\sin\frac{\mathcal{D}(\Theta)}{2}\right)\mathcal{D}(\mathcal{H})+2(\kappa_{0}+2\kappa_{1})\mathcal{D}(\mathcal{H})^{2}.$
(4.14)

###### Proof.

We use (4.13) to find

$\displaystyle\frac{{\textup{d}}}{{\textup{d}t}}(1-h_{ij})$
$\displaystyle=-2\kappa_{0}(1-h_{ij})+\frac{\kappa_{0}}{N}\sum_{k=1}^{N}(1-h_{ik}+1-h_{kj})(1-h_{ij})$
$\displaystyle\hskip
14.22636pt+\frac{\kappa_{1}}{N}\sum_{k=1}^{N}(1-h_{ik}+1-h_{kj}-(1-h_{ki})-(1-h_{jk}))(1-h_{ij})$
$\displaystyle\hskip
14.22636pt+\frac{\kappa_{1}}{N}\sum_{k=1}^{N}(1-h_{ik}+1-h_{kj}-(1-h_{ki})-(1-h_{jk}))$
$\displaystyle\hskip
14.22636pt-\frac{\kappa_{0}}{N}\sum_{k=1}^{N}\Big{[}(e^{{\mathrm{i}}(\theta_{k}-\theta_{i})}-1)(h_{kj}-h_{ij})+(e^{{\mathrm{i}}(\theta_{i}-\theta_{k})}-1)(1-h_{ik})h_{ij}$
$\displaystyle\hskip
71.13188pt+(e^{{\mathrm{i}}(\theta_{j}-\theta_{k})}-1)(h_{ik}-h_{ij})+(e^{{\mathrm{i}}(\theta_{k}-\theta_{j})}-1)(1-h_{kj})h_{ij}\Big{]}$
$\displaystyle\hskip
14.22636pt-\frac{\kappa_{1}}{N}\sum_{k=1}^{N}\Big{[}(e^{{\mathrm{i}}(\theta_{k}-\theta_{i})}-1)(h_{ki}-1)h_{ij}-(e^{{\mathrm{i}}(\theta_{i}-\theta_{k})}-1)(h_{ik}-1)h_{ij}$
$\displaystyle\hskip
71.13188pt+(e^{{\mathrm{i}}(\theta_{j}-\theta_{k})}-1)(h_{jk}-1)h_{ij}-(e^{{\mathrm{i}}(\theta_{k}-\theta_{j})}-1)(h_{kj}-1)h_{ij}\Big{]}.$

Then, we derive the desired inequality:

$\displaystyle\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{D}(\mathcal{H})$
$\displaystyle\leq-2\left(\kappa_{0}-2\kappa_{1}-(6\kappa_{0}+4\kappa_{1})\sin\frac{\mathcal{D}(\Theta)}{2}\right)\mathcal{D}(\mathcal{H})+2(\kappa_{0}+2\kappa_{1})\mathcal{D}(\mathcal{H})^{2}.$

∎

We are now ready to provide a sufficient framework under which the maximal
diameter $\mathcal{D}(\mathcal{H})$ decays to zero.

######  Proposition 4.2.

Suppose that the coupling strength (sufficiently large in the sense below) and
natural frequencies satisfy

$\kappa_{0}>2\kappa_{1},\quad\frac{2(\kappa_{0}-2\kappa_{1})(\kappa_{0}+\kappa_{1})}{3\kappa_{0}+2\kappa_{1}}>\mathcal{D}(a),\quad\sum_{j=1}^{N}a_{j}=0,$

and that there exists a positive constant
$\delta\in(0,\frac{2\kappa_{0}-4\kappa_{1}}{6\kappa_{0}+4\kappa_{1}})$ such
that

$\mathcal{D}(\Theta^{0})<\delta<\frac{\mathcal{D}(\Omega)}{2(\kappa_{0}+\kappa_{1})}<\frac{2\kappa_{0}-4\kappa_{1}}{6\kappa_{0}+4\kappa_{1}}\quad\textup{and}\quad\mathcal{D}(\mathcal{H}^{0})<\frac{\kappa_{0}-2\kappa_{1}-(3\kappa_{0}+2\kappa_{1})\delta}{\kappa_{0}+2\kappa_{1}}.$
(4.15)

Let $\\{\xi_{j}\\}$ and $\\{\theta_{j}\\}$ be solutions to (4.12) and (4.11),
respectively. Then, the maximal diameter converges to zero with an exponential
rate:

$\lim_{t\to\infty}\mathcal{D}(\mathcal{H}(t))=0.$

Moreover, there exists $\xi^{\infty}\in\mathbb{H}\mathbb{S}^{d-1}$ such that

$\lim_{t\to\infty}\xi_{j}(t)=\xi^{\infty},\quad j\in[N].$

###### Proof.

Since initial data satisfy $\eqref{D-59}_{1}$, inequality (4.14) becomes

$\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{D}(\mathcal{H})\leq-2(\kappa_{0}-2\kappa_{1}-(3\kappa_{0}+2\kappa_{1})\delta)\mathcal{D}(\mathcal{H})+2(\kappa_{0}+2\kappa_{1})\mathcal{D}(\mathcal{H})^{2},\quad
t>0.$

Then, initial condition $\eqref{D-59}_{2}$ yields the desired exponential
decay of $\mathcal{D}(\mathcal{H})$. For the second assertion, we notice from
(4.12) that

$|\dot{\xi}_{j}|\leq
2\sqrt{2}(\kappa_{0}+\kappa_{1})\sqrt{\mathcal{D}(\mathcal{H})},$

which yields $|\dot{\xi}_{j}|\in L^{1}(\mathbb{R}_{+})$. Thus,

$\lim_{t\to\infty}\xi_{j}(t)=\xi_{j}^{0}+\int_{0}^{\infty}\dot{\xi}_{j}(s){\textup{d}}s=:\xi_{j}^{\infty},$

and it directly follows from the zero convergence of
$\mathcal{D}(\mathcal{H})$ that all $\\{\xi_{j}^{\infty}\\}$ should be same.
Hence, existence of $\xi^{\infty}$ is shown. ∎

By virtue of Proposition 4.2, we see that all $\\{\xi_{i}\\}$ collapse to a
common stationary point $\xi^{\infty}$ under a well-prepared initial
framework. We are now concerned with the asymptotic behavior of (4.10), which
is our main interest in this subsection. For this, we denote

$\mathcal{D}(\mathcal{Z}(t)):=\max_{1\leq i,j\leq N}|1-\langle
z_{i},z_{j}\rangle(t)|,\quad t>0.$

######  Theorem 4.2.

Suppose that the coupling strength natural frequencies satisfy

$\kappa_{0}>2\kappa_{1},\quad\frac{2(\kappa_{0}-2\kappa_{1})(\kappa_{0}+\kappa_{1})}{3\kappa_{0}+2\kappa_{1}}>\mathcal{D}(a),\quad\sum_{i=1}^{N}a_{i}=0,$

and that there exists $\delta\in(0,\beta)$ such that

$0<\delta<\frac{\mathcal{D}(\Omega)}{2(\kappa_{0}+\kappa_{1})}<\beta,\quad\mathcal{D}(\mathcal{Z}^{0})<\frac{\kappa_{0}-2\kappa_{1}-(3\kappa_{0}+2\kappa_{1})\delta}{\kappa_{0}+2\kappa_{1}}-2\sin\frac{\delta}{2},$
(4.16)

where $\beta=\beta(\rho)$ is a unique positive root for the algebraic
equation:

$\frac{\rho-2-(3\rho+2)s}{\rho+2}=2\sin\frac{s}{2},\quad\rho:=\frac{\kappa_{0}}{\kappa_{1}}>2$

Let $\\{z_{j}\\}$ be a solution to (4.10). Then, there exists
$z_{j}^{\infty}\in\mathbb{H}\mathbb{S}^{d-1}$ such that

$\lim_{t\to\infty}z_{j}(t)=z_{j}^{\infty},\quad j\in[N].$

Moreover, such $\\{z_{j}^{\infty}\\}$ satisfy

$|\langle z_{i}^{\infty},z_{j}^{\infty}\rangle|=1,\quad i,j\in[N].$

###### Proof.

In order to apply Proposition 4.2, we recall (4.9):

$|1-\langle\xi_{i}^{0},\xi_{j}^{0}\rangle|=|1-\langle\psi_{i}^{0},\psi_{j}^{0}\rangle+\langle\psi_{i}^{0},\psi_{j}^{0}\rangle(1-e^{{\mathrm{i}}(\theta_{j}^{0}-\theta_{i}^{0})})|\leq\mathcal{D}(\mathcal{Z}^{0})+2\sin\frac{\mathcal{D}(\Theta^{0})}{2}.$

Since initial data $\mathcal{D}(\mathcal{Z}^{0})$ satisfy (4.16), initial
conditions for $\mathcal{D}(\mathcal{H}^{0})$ and $\mathcal{D}(\Theta^{0})$ in
(4.15) hold. Hence, it follows from Proposition 4.2 and Proposition 2.1 that
there exist $\xi^{\infty}\in\mathbb{H}\mathbb{S}^{d-1}$ and
$e^{{\mathrm{i}}\theta_{j}^{\infty}}$ such that

$\lim_{t\to\infty}z_{j}(t)=\xi^{\infty}e^{{\mathrm{i}}\theta_{j}^{\infty}}=:z_{j}^{\infty},$

which directly yields that $\langle z_{i}^{\infty},z_{j}^{\infty}\rangle$ has
unit modulus:

$\langle
z_{i}^{\infty},z_{j}^{\infty}\rangle=\langle\xi^{\infty}e^{{\mathrm{i}}\theta_{i}^{\infty}},\xi^{\infty}e^{{\mathrm{i}}\theta_{j}^{\infty}}\rangle=e^{{\mathrm{i}}(\theta_{i}^{\infty}-\theta_{j}^{\infty})}.$

∎

######  Remark 4.3.

As in Remark 4.1, the Kuramoto model (4.11) with the coupling strength
$2(\kappa_{0}+\kappa_{1})$ can be formally derived from (4.10) by using the
ansatz:

$z_{j}(t)=e^{{\mathrm{i}}\theta_{j}(t)}.$

##  5\. Convergence toward equilibrium on the unitary group

In this section, we are interested in the detailed asymptotic behavior of
first-order matrix aggregation model, namely the Lohe matrix model:

$\dot{U}_{j}=-{\mathrm{i}}H_{j}U_{j}+\frac{\kappa}{2N}\sum_{k=1}^{N}(U_{k}-U_{j}U_{k}^{\dagger}U_{j}),\quad
t>0,\quad j\in[N].$

In particular, we restrict ourselves to specific natural frequency matrices
satisfying

$H_{j}=-a_{j}I_{d},\quad j\in[N].$ (5.1)

Then, our model reads as

$\dot{U}_{j}={\mathrm{i}}a_{j}U_{j}+\frac{\kappa}{2N}\sum_{k=1}^{N}(U_{k}-U_{j}U_{k}^{\dagger}U_{j}),$
(5.2)

linked with the Kuramoto model by natural frequencies $\\{a_{j}\\}$ with zero
average:

$\dot{\theta}_{j}=a_{j}+\frac{\kappa}{N}\sum_{k=1}^{N}\sin(\theta_{k}-\theta_{j}).$
(5.3)

In order to proceed our dimension reduction method, we introduce

$V_{j}(t):=e^{-{\mathrm{i}}\theta_{j}(t)}U_{j}(t).$

######  Lemma 5.1.

Let $U_{j}$ and $\theta_{j}$ be solutions to (5.2) and (5.3), respectively.
Then, $V_{j}=e^{-{\mathrm{i}}\theta_{j}}U_{j}$ satisfies

$\dot{V}_{j}=\frac{\kappa}{2N}\sum_{k=1}^{N}\Big{[}e^{{\mathrm{i}}(\theta_{k}-\theta_{j})}(V_{k}-V_{j})-e^{{\mathrm{i}}(\theta_{j}-\theta_{k})}V_{j}(V_{k}^{\dagger}-V_{j}^{\dagger})V_{j}\Big{]}.$
(5.4)

In addition, temporal evolution for $V_{i}V_{j}^{\dagger}$ is determined by

$\displaystyle\begin{aligned}
(V_{i}V_{j}^{\dagger})^{\prime}&=\frac{\kappa}{2N}\sum_{k=1}^{N}\Big{[}e^{{\mathrm{i}}(\theta_{k}-\theta_{i})}(V_{k}V_{j}^{\dagger}-V_{i}V_{j}^{\dagger})-e^{{\mathrm{i}}(\theta_{i}-\theta_{k})}V_{i}(V_{k}^{\dagger}-V_{i}^{\dagger})V_{i}V_{j}^{\dagger}\\\
&\hskip
56.9055pt+e^{{\mathrm{i}}(\theta_{j}-\theta_{k})}(V_{i}V_{k}^{\dagger}-V_{i}V_{j}^{\dagger})+e^{{\mathrm{i}}(\theta_{k}-\theta_{j})}V_{i}V_{j}^{\dagger}(V_{k}-V_{i})V_{j}^{\dagger}\Big{]}.\end{aligned}$
(5.5)

###### Proof.

Direct calculation gives the desired result. ∎

We denote the maximal diameter

$\mathcal{D}(\mathcal{V}(t)):=\max_{1\leq i,j\leq
N}\|V_{i}(t)-V_{j}(t)\|_{\textup{F}}=\max_{1\leq i,j\leq
N}\|I_{d}-V_{i}(t)V^{\dagger}_{j}(t)\|_{\textup{F}},\quad t>0.$

Below, we derive a differential inequality for $\mathcal{D}(\mathcal{V})$.

######  Lemma 5.2.

Let $\\{V_{j}\\}$ be a solution to (5.4). Then, the maximal diameter
$\mathcal{D}(\mathcal{V})$ satisfies

$\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{D}(\mathcal{V})\leq-\kappa\left(1-2\sin\frac{\mathcal{D}(\Theta)}{2}\right)\mathcal{D}(\mathcal{V})+\kappa\mathcal{D}(\mathcal{V})^{2},\quad
t>0.$ (5.6)

###### Proof.

We use (5.5) to rewrite

$\displaystyle\frac{{\textup{d}}}{{\textup{d}t}}(I_{d}-V_{i}V_{j}^{\dagger})$
$\displaystyle=\frac{\kappa}{2N}\sum_{k=1}^{N}(-2(I_{d}-V_{i}V_{j}^{\dagger})+(I_{d}-V_{i}V_{j}^{\dagger})(I_{d}-V_{k}V_{j}^{\dagger})+(I_{d}-V_{i}V_{k}^{\dagger})(I_{d}-V_{i}V_{j}^{\dagger}))$
$\displaystyle-\frac{\kappa}{2N}\sum_{k=1}^{N}\Big{[}(e^{{\mathrm{i}}(\theta_{k}-\theta_{i})}-1)(V_{k}V_{j}^{\dagger}-V_{i}V_{j}^{\dagger})-(e^{{\mathrm{i}}(\theta_{i}-\theta_{k})}-1)V_{i}(V_{k}^{\dagger}-V_{i}^{\dagger})V_{i}V_{j}^{\dagger}$
$\displaystyle\hskip
56.9055pt+(e^{{\mathrm{i}}(\theta_{j}-\theta_{k})}-1)(V_{i}V_{k}^{\dagger}-V_{i}V_{j}^{\dagger})+(e^{{\mathrm{i}}(\theta_{k}-\theta_{j})}-1)V_{i}V_{j}^{\dagger}(V_{k}-V_{i})V_{j}^{\dagger}\Big{]}.$

Hence, the desired inequality is derived.

∎

Next, we provide a sufficient condition leading to the zero convergence of
$\mathcal{D}(\mathcal{V})$.

######  Proposition 5.1.

Suppose that coupling strength and natural frequencies satisfy

$\kappa>\mathcal{D}(a),\quad\sum_{j=1}^{N}a_{j}=0,$

and that there exists a positive constant $\delta\in(0,1)$ such that

$\mathcal{D}(\Theta^{0})<\delta<\frac{\mathcal{D}(\Omega)}{\kappa}<1,\quad\mathcal{D}(\mathcal{V}^{0})<1-\delta$
(5.7)

Let $\\{V_{j}\\}$ and $\\{\theta_{j}\\}$ be solutions to (5.4) and (5.3),
respectively. Then, the maximal diameter converges to zero exponentially:

$\lim_{t\to\infty}\mathcal{D}(\mathcal{V}(t))=0.$

Moreover, there exists a unitary matrix $V^{\infty}\in\mathbf{U}(d)$ such that

$\lim_{t\to\infty}V_{j}(t)=V^{\infty},\quad j\in[N].$ (5.8)

###### Proof.

Since initial data $\Theta^{0}$ satisfy $\eqref{E-8}_{1}$, we use Proposition
2.1 to find

$\mathcal{D}(\Theta(t))<\delta,\quad t>0.$

Then, (5.6) becomes

$\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{D}(\mathcal{V})\leq-\kappa(1-\delta)\mathcal{D}(\mathcal{V})+\kappa\mathcal{D}(\mathcal{V})^{2},$

which yields the desired zero convergence of $\mathcal{D}(\mathcal{V})$, since
initial data $\mathcal{V}^{0}$ satisfy condition $\eqref{E-8}_{2}$.

On the other hand for the second assertion, we use (5.4) to attain

$\|\dot{V}_{j}\|_{\textup{F}}\leq\kappa\sqrt{2d}\mathcal{D}(\mathcal{V}),$

which gives $\|\dot{V}_{j}\|_{\textup{F}}\in L^{1}(\mathbb{R}_{+})$. Thus, the
limit of $V_{j}(t)$ exists:

$\lim_{t\to\infty}V_{j}(t)=V_{j}^{0}+\int_{0}^{\infty}\dot{V}_{j}(s){\textup{d}}s=:V_{j}^{\infty}.$

Moreover, since $\mathcal{D}(\mathcal{V})$ tends to zero, all
$\\{V_{j}^{\infty}\\}$ should be same. Hence, $V^{\infty}$ in (5.8) exists. ∎

In Proposition 5.1, we have show that all $V_{j}$ converge to constant unitary
matrix $V^{\infty}\in\mathbf{U}(d)$ under some initial conditions. We are now
concerned with the asymptotic behavior of (5.2) as our main interest in this
section. For this, we write

$\mathcal{D}(\mathcal{U}(t)):=\max_{1\leq i,j\leq
N}\|I_{d}-U_{i}(t)U_{j}(t)^{\dagger}\|_{\textup{F}},\quad t>0.$

######  Theorem 5.1.

Suppose that the coupling strength and natural frequencies satisfy

$\kappa>\mathcal{D}(a),\quad\sum_{j=1}^{N}a_{j}=0,$

and that there exists $\delta\in(0,\beta)$ such that

$0<\delta<\frac{\mathcal{D}(a)}{\kappa}<\beta,\quad\mathcal{D}(\mathcal{U}^{0})<1-\delta-2d\sin\frac{\delta}{2},$
(5.9)

where $\beta=\beta(d)$ is a unique positive root for $1-s=2d\sin\frac{s}{2}$.
Let $\\{U_{j}\\}$ be a solution to (5.2). Then, there exists
$U_{j}^{\infty}\in\mathbf{U}(d)$ such that

$\lim_{t\to\infty}U_{j}(t)=U_{j}^{\infty},\quad j\in[N].$

Moreover, the following relation holds:

$U_{i}^{\infty}U_{j}^{\infty,\dagger}=e^{{\mathrm{i}}(\theta_{i}^{\infty}-\theta_{j}^{\infty})}I_{d},\quad
i,j\in[N].$ (5.10)

###### Proof.

In order to apply Proposition 5.1, we write

$I_{d}-V_{i}^{0}V_{j}^{0,\dagger}=I_{d}-U_{i}^{0}U_{j}^{0,\dagger}+U_{i}^{0}U_{j}^{0,\dagger}(1-e^{{\mathrm{i}}(\theta_{j}^{0}-\theta_{i}^{0})}),$

and this gives

$\mathcal{D}(\mathcal{V}^{0})\leq\mathcal{D}(\mathcal{U}^{0})+2d\sin\frac{\mathcal{D}(\Theta^{0})}{2}.$

Since (5.9) holds, initial conditions for $\mathcal{D}(\mathcal{V}^{0})$ and
$\mathcal{D}(\Theta^{0})$ in (5.7) are fulfilled. Hence, Proposition 5.1
together with Proposition 2.1 gives $V^{\infty}\in\mathbf{U}(d)$ and
$e^{{\mathrm{i}}\theta_{j}^{\infty}}$ such that

$\lim_{t\to\infty}U_{j}(t)=e^{{\mathrm{i}}\theta_{j}^{\infty}}V^{\infty}=:U_{j}^{\infty}\in\mathbf{U}(d).$

Thus, (5.10) directly follows. ∎

######  Remark 5.1.

In [7], the authors studied the Lohe matrix model:

$\dot{U}_{j}={\mathrm{i}}H_{j}U_{j}+\frac{\kappa}{2N}\sum_{k=1}^{N}(U_{k}-U_{j}U_{k}^{\dagger}U_{j}),$

with a general Hermitian matrix $H_{j}$. As mentioned in Section 2.4, the
authors showed that for a large coupling strength, asymptotic phase-locking
emerges. In other words,

$\lim_{t\to\infty}U_{i}(t)U_{j}(t)^{\dagger}\quad\textup{exists.}$

Although limits of $\\{U_{i}U_{j}^{\dagger}\\}$ exist, existence of limit for
each $U_{j}$ would not be guaranteed. Thus, if we let

$X_{i}^{\infty}:=\lim_{t\to\infty}(U_{i}U_{1}^{\dagger})(t),\quad i\in[N],$

then $\\{X_{i}^{\infty}\\}$ should satisfy

$X_{i}^{\infty}\Lambda
X_{i}^{\infty,\dagger}=H_{i}-\frac{{\mathrm{i}}\kappa}{2N}\sum_{k=1}^{N}\Big{(}X_{i}^{\infty}X_{k}^{\infty,\dagger}-X_{k}^{\infty}X_{i}^{\infty,\dagger}\Big{)},$

where $\Lambda$ is an index-independent Hermitian matrix. Our result stated in
Theorem 5.1 says that if $H_{j}$ has a specific structure (5.1), then the
convergence of limit for $U_{j}$ can be exactly identified.

##  6\. Conclusion

In this work, we studied the detailed asymptotic dynamics of first-order
heterogeneous aggregation models, for instance, real and complex swarm sphere
models, the Schrödinger-Lohe model, the Lohe Hermitian sphere model, and the
Lohe matrix model. It should be noted that the aforementioned models can be
regarded as generalized and high-dimensional Kuramoto models and thus can be
reduced to the Kuramoto model by a simple ansatz. Since the existence and
structure of equilibria for the Kuramoto model are well-known, the key idea,
called dimension reduction method, is to decompose the given high-dimensional
system into two subsystems: a heterogeneous Kuramoto model and a homogeneous
modified model. In this manner, we can establish the existence of a solution
to the high-dimensional model. Several numerical examples are provided in
order to support theoretical results. There are still remaining issues. In
fact, we have assumed that the natural frequency has the degree of freedom
$1$. One of the future work is dedicated to generalizing this issue.

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