# Emergence of synchronization in Kuramoto model with frustration under
general network topology

Tingting Zhu ${\ddagger}$<EMAIL_ADDRESS>
###### Abstract.

In this paper, we will study the emergent behavior of Kuramoto model with
frustration on a general digraph containing a spanning tree. We provide a
sufficient condition for the emergence of asymptotical synchronization if the
initial data is confined in half circle. As lack of uniform coercivity in
general digraph, we apply the node decomposition criteria in [25] to capture a
clear hierarchical structure, which successfully yields the dissipation
mechanism of phase diameter and a small invariant set after finite time. Then
the dissipation of frequency diameter will be clear, which eventually leads to
the synchronization.

###### Key words and phrases:

Kuramoto model, frustration, general digraph, spanning tree, hypo-coercivity,
synchronization

######  Mathematics Subject Classification:

34D06, 34C15,92B25, 70F99.

‡ Key Laboratory of Applied Mathematics and Artificial Intelligence Mechanism,
Hefei University, Hefei 230601, Anhui, China

The author appreciated the guidance and modification of Xiongtao Zhang which
helped to enhance the previous manuscript.

##  1\. Introduction

Synchronized behavior in complex systems is ubiquitous and has been
extensively investigated in various academic communities such as physics,
biology, engineering [2, 7, 29, 37, 36, 39, 40, 42], etc. Recently,
sychronization mechanism has been applied in control of robot systems and
power systems [12, 13, 34]. The rigorous mathematical treatment of
synchronization phenomena was started by two pioneers Winfree [43] and
Kuramoto [27, 28] several decades ago, who introduced different types of
first-order systems of ordinary differential equations to describe the
synchronous behaviors. These models contain rich emergent behaviors such as
synchronization, partially phase-lcoking and nonlinear stability, etc., and
have been extensively studied in both theoretical and numerical level [1, 3,
5, 11, 14, 17, 18, 19, 21, 26, 32, 39].

In this paper, we address the synchronous problem of Kuramoto model on a
general graph under the effect of frustration. To fix the idea, we consider a
digraph $\mathcal{G}=(\mathcal{V},\mathcal{E})$ consisting of a finite set
$\mathcal{V}=\\{1,\ldots,N\\}$ of vertices and a set
$\mathcal{E}\subset\mathcal{V}\times\mathcal{V}$ of directed arcs. We assume
that Kuramoto oscillators are located at vertices and interact with each other
via the underlying network topology. For each vertex $i$, we denote the set of
its neighbors by $\mathcal{N}_{i}$, which is the set of vertices that directly
influence vertex $i$. Now, let $\theta_{i}=\theta_{i}(t)$ be the phase of the
Kuramoto oscillator at vertex $i$, and define the $(0,1)$-adjacency matrix
$(\chi_{ij})$ as follows:

$\chi_{ij}=\begin{cases}\displaystyle 1\quad\mbox{if the $j$th oscillator
influences the $i$th oscillator},\\\ \displaystyle
0\quad\mbox{otherwise}.\end{cases}$

Then, the set of neighbors of $i$-th oscillator is actually
$\mathcal{N}_{i}:=\\{j:\chi_{ij}>0\\}$. In this setting, the Kuramoto model
with frustration on a general network is governed by the following ODE system:

$\begin{cases}\displaystyle\dot{\theta}_{i}(t)=\Omega_{i}+K\underset{j\in\mathcal{N}_{i}}{\sum}\sin(\theta_{j}(t)-\theta_{i}(t)+\alpha),\quad
t>0,\quad i\in\mathcal{V},\\\
\displaystyle\theta_{i}(0)=\theta_{i0},\end{cases}$ (1.1)

where $\Omega_{i},K,N$ and $\alpha\in(0,\frac{\pi}{2})$ are the natural
frequency of the $i$th oscillator, coupling strength, the number of
oscillators and the uniform frustration between oscillators, respectively. For
the case of nonpositive frustration, we can reformulate such a system into
(1.1) form by taking $\hat{\theta}_{i}=-\theta_{i}$ for $i=1,2,\ldots,N$. Note
that the well-posedness of system (1.1) is guaranteed by the standard Cauchy-
Lipschitz theory since the vector field on the R.H.S of (1.1) is analytic.

Comparing to the original Kuramoto model, there are two additional structures,
i.e., frustration and general digraph. The frustration was introduced by
Sakaguchi and Kuramoto [38], due to the observation that a pair of strongly
coupled oscillators eventually oscillate with a common frequency that deviates
from the average of their natural frequencies. On the other hand, the original
all-to-all symmetric network is an ideal setting, thus it is natural to
further consider general digraph case. Therefore, the frustration model with
general digraph is more realistic in some sense. Moreover, these two
structures also lead to richer phenomenon. For instance, the author in [6]
observed that the frustration is common in disordered interactions, and the
author in [44] found that frustration can induce the desynchronization through
varying the value of $\alpha$ in numerical simulations. For more information,
please refer to [4, 10, 16, 23, 24, 25, 30, 33, 35, 41].

However, mathematically, for the Kuramoto model, the frustration and general
digraph structures generate a lot of difficulties in rigorous analysis. For
instance, the conservation law and gradient flow structure are lost, and thus
the asymptotic states and dissipation mechanism become non-trivial. For all-
to-all and symmetric case with frustration, in [15], the authors provided
sufficient frameworks leading to complete synchronization under the effect of
uniform frustration. In their work, they required initial configuration to be
confined in half circle. Furthermore, the authors in [31] dealt with the
stability and uniqueness of emergent phase-locked states. In particular, the
authors in [22] exploited order parameter approach to study the identical
Kuramoto oscillators with frustration. They showed that an initial
configuration whose order parameter is bounded below will evolve to the
complete phase synchronization or the bipolar state exponentially fast. On the
other hand, for non-all-to-all case without frustration, the authors in [9]
lifted the Kuramoto model to second-order system such that the second-order
formulation enjoys several similar mathematical structures to that of Cucker-
Smale flocking model [8]. But this method only works when the size of initial
phases is less than a quarter circle, as we know the cosine function becomes
negative if $\frac{\pi}{2}<\theta<\pi$. To the best knowledge of the authors,
there is few work on the Kuramoto model over general digraph with frustration.
The authors in [20] studied the Kuramoto model with frustrations on a complete
graph which is a small perturbation of all-to-all network, and provided
synchronization estimates in half circle.

Our interest in this paper is studying the system (1.1) with uniform
frustration on a general digraph. As far as the authors know, when the
ensemble is distributed in half circle, the dissipation structure of the
Kuramoto model with general digragh is still unclear. The main difficulties
comes from the loss of uniform coercive inequality, which is due to the non-
all-to-all and non-symmetric interactions. Thus we cannot expect to capture
the dissipation from Gronwall-type inequality of phase diameter. For example,
the time derivative of the phase diameter may be zero at some time for general
digraph case. To this end, we switch to follow similar idea in [25] to gain
the dissipation through hypo-coercivity. Different from [25] which deals with
the Cucker-Smale model on a general digraph, the interactions in Kuramoto
model requires more delicate estimates due to the lack of monotonicity of sine
function in half circle. Eventually, we have the following main theorem..

######  Theorem 1.1.

Suppose the network topology $(\chi_{ij})$ contains a spanning tree,
$D^{\infty}$ is a given positive constant such that
$D^{\infty}<\frac{\pi}{2}$, and all the oscillators are initially confined in
half circle, i.e.,

$D(\theta(0))<\pi.$

Then for sufficient large coupling strength $K$ and small frustration
$\alpha$, there exists a finite time $t_{*}>0$ such that

$D(\theta(t))\leq D^{\infty},\quad\forall t\in[t_{*},\infty).$

######  Remark 1.1.

Theorem 1.1 claims that all oscillators confined initially in half circle will
enter a small region after some finite time. It is natural to ask how large
$K$ and how small $\alpha$ we need to guarantee the Theorem 1.1. In fact,
according to the proof in later sections, we have the following explicit
constraints on $K$ and $\alpha$,

$\displaystyle\tan\alpha<\frac{1}{\left(1+\frac{(d+1)\zeta}{\zeta-D(\theta(0))}\right)2Nc}\frac{\beta^{d+1}D^{\infty}}{[4(2N+1)c]^{d}},\quad
D^{\infty}+\alpha<\frac{\pi}{2},$ (1.2) $\displaystyle
1>\left(1+\frac{(d+1)\zeta}{\zeta-D(\theta(0))}\right)\frac{c[4(2N+1)c]^{d}}{\beta^{d+1}D^{\infty}}\left(\frac{D(\Omega)}{K\cos\alpha}+\frac{2N\sin\alpha}{\cos\alpha}\right),$

where $d$ is the number of general nodes which is smaller than $N$ (see
Definition 2.3), $D(\Omega)$ is the diameter of natural frequency, and the
other parameters $\zeta$, $\gamma$, $\eta$, $\beta$ and $c$ are positive
constants which satisfy the following properties,

$\displaystyle
D(\theta(0))<\zeta<\gamma<\pi,\quad\eta>\max\left\\{\frac{1}{\sin\gamma},\frac{2}{1-\frac{\zeta}{\gamma}}\right\\},$
(1.3) $\displaystyle\beta=1-\frac{2}{\eta},\quad
c=\frac{\left(\sum_{j=1}^{N-1}\eta^{j}A(2N,j)+1\right)\gamma}{\sin\gamma},$

where $A(2N,j)$ denotes the permutation. It’s obvious that we can find
admissible parameters satisfying (1.3) since $D(\theta(0))<\pi$. Once the
parameters are fixed, we immediately conclude (1.2) holds for small $\alpha$
and large $K$.

######  Remark 1.2.

After $t_{*}$, all oscillators are confined in a small region less than
$\frac{\pi}{2}$, and Kuramoto model (1.1) will be equivalent to Cucker-Smale
type model with frustration (see (4.36)). Therefore, we can directly apply the
methods and results in [9] to conclude the emergence of frequency
synchronization for large coupling and small frustration (see Corollary 4.1).
Therefore, to guarantee the emergence of synchronization, it suffices to show
the detailed proof of Theorem 1.1.

The rest of the paper is organized as follows. In Section 2, we recall some
concepts on the network topology and provide an a priori local-in-time
estimate on the phase diameter of whole ensemble with frustration. In Section
3, we consider a strong connected ensemble with frustration for which the
initial phases are distributed in a half circle. We show that for large
coupling strength and small frustration, the phase diameter plus a phase shift
is uniformly bounded by a small value after some finite time. In Section 4, we
study the general network with a spanning tree structure under the effect of
uniform frustration. We use the inductive argument and show that Kuramoto
oscillators will concentrate into a small region less than a quarter circle in
finite time, which eventually leads to the emergence of synchronization
exponentially fast. Section 5 is devoted to a brief summary.

##  2\. Preliminaries

In this section, we first introduce some fundamental concepts such as spanning
tree and node decomposition of a general network (1.1). Then, we will provide
some necessary notations and an a priori estimate that will be frequently used
in later sections.

###  2.1. Directed graph

Let the network topology be registered by the neighbor set $\mathcal{N}_{i}$
which consists of all neighbors of the $i$th oscillator. Then, for a given set
of $\\{\mathcal{N}_{i}\\}_{i=1}^{N}$ in system (1.1), we have the following
definition.

######  Definition 2.1.

  1. (1)

The Kuramoto digraph $\mathcal{G}=(\mathcal{V},\mathcal{E})$ associated to
(1.1) consists of a finite set $\mathcal{V}=\\{1,2,\ldots,N\\}$ of vertices,
and a set $\mathcal{E}\subset\mathcal{V}\times\mathcal{V}$ of arcs with
ordered pair $(j,i)\in\mathcal{E}$ if $j\in\mathcal{N}_{i}$.

  2. (2)

A path in $\mathcal{G}$ from $i_{1}$ to $i_{k}$ is a sequence
$i_{1},i_{2},\ldots,i_{k}$ such that

$i_{s}\in\mathcal{N}_{i_{s+1}}\quad\mbox{for}\ 1\leq s\leq k-1.$

If there exists a path from $j$ to $i$, then vertex $i$ is said to be
reachable from vertex $j$.

  3. (3)

The Kuramoto digraph contains a spanning tree if we can find a vertex such
that any other vertex of $\mathcal{G}$ is reachable from it.

In order to guarantee the emergence of synchronization, we will always assume
the existence of a spanning tree throughout the paper. Now we recall the
concepts of root and general root introduced in [25]. Let $l,k\in\mathbb{N}$
with $1\leq l\leq k\leq N$, and let $C_{l,k}=(c_{l},c_{l+1},\ldots,c_{k})$ be
a vector in $\mathbb{R}^{k-l+1}$ such that

$c_{i}\geq 0,\quad l\leq i\leq k\quad\mbox{and}\quad\sum_{i=l}^{k}c_{i}=1.$

For an ensembel of $N$-oscillators with phases $\\{\theta_{i}\\}_{i=1}^{N}$,
we set $\mathcal{L}_{l}^{k}(C_{l,k})$ to be a convex combination of
$\\{\theta_{i}\\}_{i=l}^{k}$ with the coefficient $C_{l,k}$:

$\mathcal{L}_{l}^{k}(C_{l,k}):=\sum_{i=l}^{k}c_{i}\theta_{i}.$

Note that each $\theta_{i}$ is a convex combination of itself, and
particularly $\theta_{N}=\mathcal{L}_{N}^{N}(1)$ and
$\theta_{1}=\mathcal{L}_{1}^{1}(1)$.

######  Definition 2.2.

(Root and general root)

  1. (1)

We say $\theta_{k}$ is a root if it is not affected by the rest oscillators,
i.e., $j\notin\mathcal{N}_{k}$ for any
$j\in\\{1,2,\ldots,N\\}\setminus\\{k\\}$.

  2. (2)

We say $\mathcal{L}_{l}^{k}(C_{l,k})$ is a general root if
$\mathcal{L}_{l}^{k}(C_{l,k})$ is not affected by the rest oscillators, i.e.,
for any $i\in\\{l,l+1,\ldots,k\\}$ and
$j\in\\{1,2,\ldots,N\\}\setminus\\{l,l+1,\ldots,k\\}$, we have
$j\notin\mathcal{N}_{i}$.

######  Lemma 2.1.

[25] The following assertions hold.

  1. (1)

If the network contains a spanning tree, then there is at most one root.

  2. (2)

Assume the network contains a spanning tree. If $\mathcal{L}_{k}^{N}(C_{k,N})$
is a general root, then $\mathcal{L}_{1}^{l}(C_{1,l})$ is not a general root
for each $l\in\\{1,2,\ldots,k-1\\}$.

###  2.2. Node decomposition

In this part, we will recall the concept of maximum node introduced in [25].
Then, we can follow node decomposition introduced in [25] to represent the
whole graph $\mathcal{G}$ (or say vertex set $\mathcal{V}$) as a disjoint
union of a sequence of nodes. The key point is that the node decomposition
shows a hierarchical structure, then we can exploit this advantage to apply
the induction principle. Let
$\mathcal{G}=(\mathcal{V},\mathcal{E}),\mathcal{V}_{1}\subset\mathcal{V}$, and
a subgraph $\mathcal{G}_{1}=(\mathcal{V}_{1},\mathcal{E}_{1})$ is the digraph
with vertex set $\mathcal{V}_{1}$ and arc set $\mathcal{E}_{1}$ which consists
of the arcs in $\mathcal{G}$ connecting agents in $\mathcal{V}_{1}$. For a
given digraph $\mathcal{G}=(\mathcal{V},\mathcal{E})$, we will identify a
subgraph $\mathcal{G}_{1}=(\mathcal{V}_{1},\mathcal{E}_{1})$ with its vertex
set $\mathcal{V}_{1}$ for convenience. Now we first present the definition of
nodes below.

######  Definition 2.3.

[25] (Node) Let $\mathcal{G}$ be a digraph. A subset $\mathcal{G}_{1}$ of
vertices is called a node if it is strongly connected, i.e., for any subset
$\mathcal{G}_{2}$ of $\mathcal{G}_{1}$, $\mathcal{G}_{2}$ is affected by
$\mathcal{G}_{1}\setminus\mathcal{G}_{2}$. Moreover, if $\mathcal{G}_{1}$ is
not affected by $\mathcal{G}\setminus\mathcal{G}_{1}$, we say
$\mathcal{G}_{1}$ is a maximum node.

Intuitively, a node can be understood through a way that a set of oscillators
can be viewed as a ”large” oscillator. The concept of node can be exploited to
simplify the structure of the digraph, which indeed helps us to catch the
attraction effect more clearly in the network topology.

######  Lemma 2.2.

[25] Any digraph $\mathcal{G}$ contains at least one maximum node. A digraph
$\mathcal{G}$ contains a unique maximum node if and only if $\mathcal{G}$ has
a spanning tree.

######  Lemma 2.3.

[25](Node decomposition) Let $\mathcal{G}$ be any digraph. Then we can
decompose $\mathcal{G}$ to be a union as
$\mathcal{G}=\bigcup_{i=0}^{d}(\bigcup_{j=1}^{k_{i}}\mathcal{G}_{i}^{j})$ such
that

  1. (1)

$\mathcal{G}_{0}^{j}$ are the maximum nodes of $\mathcal{G}$, where $1\leq
j\leq k_{0}$.

  2. (2)

For any $p,q$ where $1\leq p\leq d$ and $1\leq q\leq k_{p}$,
$\mathcal{G}_{p}^{q}$ are the maximum nodes of
$\mathcal{G}\setminus(\bigcup_{i=0}^{p-1}(\bigcup_{j=1}^{k_{i}}\mathcal{G}_{i}^{j}))$.

######  Remark 2.1.

Lemma 2.3 shows a clear hierarchical structure on a general digraph. For the
convenience of later analysis, we make some comments on important notations
and properties that are used throughout the paper.

  1. (1)

From the definition of maximum node, for $1\leq q\neq q^{\prime}\leq k_{p}$,
we see that $\mathcal{G}_{p}^{q}$ and $\mathcal{G}_{p}^{q^{\prime}}$ do not
affect each other. Actually, $\mathcal{G}_{p}^{q}$ will only be affected by
$\mathcal{G}_{0}$ and $\mathcal{G}_{i}^{j}$, where $1\leq i\leq p-1,\ 1\leq
j\leq k_{i}$. Thus in the proof of our main theorem (see Theorem 1.1), without
loss of generality, we may assume $k_{i}=1$ for all $1\leq i\leq d$. Hence,
the decomposition can be further simplified and expressed by

$\mathcal{G}=\bigcup_{i=0}^{d}\mathcal{G}_{i},$

where $\mathcal{G}_{p}$ is a maximum node of
$\mathcal{G}\setminus(\bigcup_{i=0}^{p-1}\mathcal{G}_{i})$.

  2. (2)

For a given oscillator $\theta_{i}^{k+1}\in\mathcal{G}_{k+1}$, we denote by
$\bigcup_{j=0}^{k+1}\mathcal{N}_{i}^{k+1}(j)$ the set of neighbors of
$\theta_{i}^{k+1}$, where $\mathcal{N}_{i}^{k+1}(j)$ represents the neighbors
of $\theta_{i}^{k+1}$ in $\mathcal{G}_{j}$. Note that the node decomposition
and spanning tree structure in $\mathcal{G}$ guarantee that
$\bigcup_{j=0}^{k}\mathcal{N}_{i}^{k+1}(j)\neq\emptyset$.

###  2.3. Notations and local estimates

In this part, for notational simplicity, we introduce some notations, such as
the extreme phase, phase diameter of $\mathcal{G}$ and the first $k+1$ nodes,
natural frequency diameter, and cardinality of subdigraph:

$\displaystyle\theta_{M}=\max_{1\leq k\leq N}\\{\theta_{k}\\}=\max_{0\leq
i\leq d}\max_{1\leq j\leq
N_{i}}\\{\theta_{j}^{i}\\},\quad\theta_{m}=\min_{1\leq k\leq
N}\\{\theta_{k}\\}=\min_{0\leq i\leq d}\min_{1\leq j\leq
N_{i}}\\{\theta_{j}^{i}\\},$ $\displaystyle
D(\theta)=\theta_{M}-\theta_{m},\quad D_{k}(\theta)=\max_{0\leq i\leq
k}\max_{1\leq j\leq N_{i}}\\{\theta_{j}^{i}\\}-\min_{0\leq i\leq k}\min_{1\leq
j\leq N_{i}}\\{\theta_{j}^{i}\\},$ $\displaystyle\Omega_{M}=\max_{0\leq i\leq
d}\max_{1\leq j\leq N_{i}}\\{\Omega_{j}^{i}\\},\quad\Omega_{m}=\min_{0\leq
i\leq d}\min_{1\leq j\leq N_{i}}\\{\Omega_{j}^{i}\\},\quad
D(\Omega)=\Omega_{M}-\Omega_{m},$ $\displaystyle N_{i}=|\mathcal{G}_{i}|,\quad
S_{k}=\sum_{i=0}^{k}N_{i},\quad 0\leq k\leq d,\quad\sum_{i=0}^{d}N_{i}=N.$

Finally, we provide an a priori local-in-time estimate on the phase diameter
to finish this section, which states that all oscillators can be confined in
half circle in short time.

######  Lemma 2.4.

Let $\theta_{i}$ be a solution to system (1.1) and suppose the initial phase
diameter satisfies

$D(\theta(0))<\zeta<\gamma<\pi,$

then there exists a finite time $\bar{t}>0$ such that the phase diameter of
whole ensemble remains less than $\zeta$ before $\bar{t}$, i.e.,

$D(\theta(t))<\zeta,\quad
t\in[0,\bar{t})\quad\text{where}\quad\bar{t}=\frac{\zeta-D(\theta(0))}{D(\Omega)+2NK\sin\alpha}.$

###### Proof.

From system (1.1), we see that the dynamics of extreme phases is given by the
following equations

$\dot{\theta}_{M}=\Omega_{M}+K\sum_{j\in\mathcal{N}_{M}}\sin(\theta_{j}-\theta_{M}+\alpha),\qquad\dot{\theta}_{m}=\Omega_{m}+K\sum_{j\in\mathcal{N}_{m}}\sin(\theta_{j}-\theta_{m}+\alpha).$

We combine the above equations to estimate the dynamics of phase diameter

$\displaystyle\dot{D}(\theta)$
$\displaystyle=\dot{\theta}_{M}-\dot{\theta}_{m}$ (2.1)
$\displaystyle=\Omega_{M}-\Omega_{m}+K\sum_{j\in\mathcal{N}_{M}}\sin(\theta_{j}-\theta_{M}+\alpha)-K\sum_{j\in\mathcal{N}_{m}}\sin(\theta_{j}-\theta_{m}+\alpha)$
$\displaystyle\leq
D(\Omega)+K\sum_{j\in\mathcal{N}_{M}}\left[\sin(\theta_{j}-\theta_{M})\cos\alpha+\cos(\theta_{j}-\theta_{M})\sin\alpha\right]$
$\displaystyle-K\sum_{j\in\mathcal{N}_{m}}\left[\sin(\theta_{j}-\theta_{m})\cos\alpha+\cos(\theta_{j}-\theta_{m})\sin\alpha\right]$
$\displaystyle=D(\Omega)+K\cos\alpha\left(\sum_{j\in\mathcal{N}_{M}}\sin(\theta_{j}-\theta_{M})-\sum_{j\in\mathcal{N}_{m}}\sin(\theta_{j}-\theta_{m})\right)$
$\displaystyle+K\sin\alpha\left(\sum_{j\in\mathcal{N}_{M}}\cos(\theta_{j}-\theta_{M})-\sum_{j\in\mathcal{N}_{m}}\cos(\theta_{j}-\theta_{m})\right),$

where we use the formula

$\sin(x+y)=\sin x\cos y+\cos x\sin y$

When the phase diameter satisfies $D(\theta(t))\leq\zeta<\pi$, it is obvious
that

$\sin(\theta_{j}-\theta_{M})\leq 0,\
j\in\mathcal{N}_{M}\quad\text{and}\quad\sin(\theta_{j}-\theta_{m})\geq 0,\
j\in\mathcal{N}_{m}.$

Then we see from (2.1) that

$\dot{D}(\theta)\leq D(\Omega)+2NK\sin\alpha.$ (2.2)

That is to say, when $D(\theta(t))\leq\zeta$, the growth of phase diameter is
no greater than the linear growth with positive slope
$D(\Omega)+2NK\sin\alpha$. Now we integrate on both sides of (2.2) from $0$ to
$t$ to have

$D(\theta(t))\leq D(\theta(0))+(D(\Omega)+2NK\sin\alpha)t.$

Therefore, it yields that there exists a finite time $\bar{t}>0$ such that

$D(\theta(t))<\zeta,\quad\forall\ t\in[0,\bar{t}),$

where $\bar{t}$ is given as below

$\bar{t}=\frac{\zeta-D(\theta(0))}{D(\Omega)+2NK\sin\alpha}.$

∎

##  3\. Strong connected case

We will first study the special case, i.e., the network is strongly connected.
Without loss of generality, we denote by $\mathcal{G}_{0}$ the strong
connected digraph. According to Definition 2.3, Lemma 2.2 and Lemma 2.3, the
strong connected network consists of only one maximum node. Then in the
present special case, we will show the emergence of complete synchronization.
We now introduce an algorithm to construct a proper convex combination of
oscillators, which can involve the dissipation from interaction of general
network. More precisely, the algorithm for $\mathcal{G}_{0}$ consists of the
following three steps:

Step 1. For any given time $t$, we reorder the oscillator indexes to make the
oscillator phases from minimum to maximum. More specifically, by relabeling
the agents at time $t$, we set

$\theta_{1}^{0}(t)\leq\theta_{2}^{0}(t)\leq\ldots\leq\theta_{N_{0}}^{0}(t).$
(3.1)

In order to introduce the following steps, we first provide the process of
iterations for $\bar{\mathcal{L}}_{k}^{N_{0}}(\bar{C}_{k,N_{0}})$ and
$\underline{\mathcal{L}}_{1}^{l}(\underline{C}_{1,l})$ as follows:

$\bullet$($\mathcal{A}_{1}$): If
$\bar{\mathcal{L}}_{k}^{N_{0}}(\bar{C}_{k,N_{0}})$ is not a general root, then
we construct

$\bar{\mathcal{L}}_{k-1}^{N_{0}}(\bar{C}_{k-1,N_{0}})=\frac{\bar{a}_{k-1}\bar{\mathcal{L}}_{k}^{N_{0}}(\bar{C}_{k,N_{0}})+\theta^{0}_{k-1}}{\bar{a}_{k-1}+1}.$

$\bullet$($\mathcal{A}_{2}$): If
$\underline{\mathcal{L}}_{1}^{l}(\underline{C}_{1,l})$ is not a general root,
then we construct

$\underline{\mathcal{L}}_{1}^{l+1}(\underline{C}_{1,l+1})=\frac{\underline{a}_{l+1}\underline{\mathcal{L}}_{1}^{l}(\underline{C}_{1,l})+\theta^{0}_{l+1}}{\underline{a}_{l+1}+1}$

Step 2. According to the strong connectivity of $\mathcal{G}_{0}$, we
immediately know that $\bar{\mathcal{L}}_{1}^{N_{0}}(\bar{C}_{1,N_{0}})$ is a
general root, and $\bar{\mathcal{L}}_{k}^{N_{0}}(\bar{C}_{k,N_{0}})$ is not a
general root for $k>1$. Therefore, we may start from $\theta^{0}_{N_{0}}$ and
follow the process $\mathcal{A}_{1}$ to construct
$\bar{\mathcal{L}}_{k}^{N_{0}}(\bar{C}_{k,N_{0}})$ until $k=1$.

Step 3. Similarly, we know that
$\underline{\mathcal{L}}_{1}^{N_{0}}(\underline{C}_{1,N_{0}})$ is a general
root and $\underline{\mathcal{L}}_{1}^{l}(\underline{C}_{1,l})$ is not a
general root for $l<N_{0}$. Therefore, we may start from $\theta^{0}_{1}$ and
follow the process $\mathcal{A}_{2}$ until $l=N_{0}$.

It is worth emphasizing that the order of the oscillators may change along
time $t$, but at each time $t$, the above algorithm does work. For
convenience, the algorithm from Step 1 to Step 3 will be referred to as
Algorithm $\mathcal{A}$. Then, based on Algorithm $\mathcal{A}$, we will
provide a priori estimates on a monotone property about the sine function,
which will be crucially used later.

######  Lemma 3.1.

Let $\theta_{i}=\\{\theta^{0}_{i}\\}$ be a solution to system (1.1) with srong
connected network $\mathcal{G}_{0}$. Moreover at time $t$, we also assume that
the oscillators are well-ordered as (3.1), and the phase diameter and quantity
$\eta$ satisfiy the following conditions:

$D_{0}(\theta(t))<\gamma,\quad\eta>\max\left\\{\frac{1}{\sin\gamma},\frac{2}{1-\frac{\zeta}{\gamma}}\right\\},$

where $\zeta,\gamma$ are given in the condition (1.3). Then at time $t$, the
following assertions hold

$\begin{cases}\displaystyle\sum_{i=n}^{N_{0}}(\eta^{i-n}\underset{j\leq
i}{\min_{j\in\mathcal{N}^{0}_{i}(0)}}\sin(\theta_{j}^{0}-\theta_{i}^{0}))\leq\sin(\theta^{0}_{\bar{k}_{n}}-\theta^{0}_{N_{0}}),\
\bar{k}_{n}=\min_{j\in\cup_{i=n}^{N_{0}}\mathcal{N}^{0}_{i}(0)}j,\ 1\leq n\leq
N_{0},\\\ \displaystyle\sum_{i=1}^{n}(\eta^{n-i}\underset{j\geq
i}{\max_{j\in\mathcal{N}^{0}_{i}(0)}}\sin(\theta_{j}^{0}-\theta_{i}^{0}))\geq\sin(\theta^{0}_{\underline{k}_{n}}-\theta_{1}^{0}),\
\underline{k}_{n}=\max_{j\in\cup_{i=1}^{n}\mathcal{N}^{0}_{i}(0)}j,\ 1\leq
n\leq N_{0}.\end{cases}$

###### Proof.

For the proof of this lemma, please see [45] for details.
∎

Recall the strongly connected ensemble $\mathcal{G}_{0}$, and denote by
$\theta_{i}^{0}\ (i=1,2,\ldots,N_{0})$ the members in $\mathcal{G}_{0}$. For
the oscillators in $\mathcal{G}_{0}$, based on a priori estimates in Lemma
3.1, we will design proper coefficients of convex combination which helps us
to capture the dissipation structure. Now we assume that at time $t$, the
oscillators in $\mathcal{G}_{0}$ are well-ordered as follows,

$\theta^{0}_{1}(t)\leq\theta^{0}_{2}(t)\leq\ldots\leq\theta^{0}_{N_{0}}(t).$

Then we apply the process $\mathcal{A}_{1}$ from $\theta^{0}_{N_{0}}$ to
$\theta^{0}_{1}$ and the process $\mathcal{A}_{2}$ from $\theta^{0}_{1}$ to
$\theta^{0}_{N_{0}}$ to respectively construct

$\displaystyle\bar{\mathcal{L}}_{k-1}^{N_{0}}(\bar{C}_{k-1,N_{0}})\
\mbox{with}\ \bar{a}^{0}_{N_{0}}=0,\
\bar{a}^{0}_{k-1}=\eta(2N_{0}-k+2)(\bar{a}^{0}_{k}+1),\quad 2\leq k\leq
N_{0},$ (3.2)
$\displaystyle\underline{\mathcal{L}}_{1}^{k+1}(\underline{C}_{1,k+1})\
\mbox{with}\ \underline{a}^{0}_{1}=0,\
\underline{a}^{0}_{k+1}=\eta(k+1+N_{0})(\underline{a}^{0}_{k}+1),\quad 1\leq
k\leq N_{0}-1,$

where $N_{0}$ is the cardinality of $\mathcal{G}_{0}$ and $\eta$ is given in
the condition (1.3). By induction criteria, we can deduce explict expressions
about the constructed coefficients:

$\displaystyle\bar{a}^{0}_{k-1}=\sum_{j=1}^{N_{0}-k+1}\eta^{j}A(2N_{0}-k+2,j),\quad
2\leq k\leq N_{0},$ (3.3)
$\displaystyle\underline{a}^{0}_{k+1}=\sum_{j=1}^{k}\eta^{j}A(k+1+N_{0},j),\quad
1\leq k\leq N_{0}-1.$

Note that $\bar{a}^{0}_{N_{0}+1-i}=\underline{a}^{0}_{i},\ i=1,2\ldots,N_{0}$.
And we set

$\bar{\theta}^{0}_{k}:=\bar{\mathcal{L}}_{k}^{N_{0}}(\bar{C}_{k,N_{0}}),\quad\underline{\theta}^{0}_{k}:=\underline{\mathcal{L}}_{1}^{k}(\underline{C}_{1,k}),\quad
1\leq k\leq N_{0}.$ (3.4)

We define a non-negative quantity
$Q^{0}=\bar{\theta}_{0}-\underline{\theta}_{0}$ where
$\bar{\theta}_{0}=\bar{\theta}^{0}_{1}$ and
$\underline{\theta}_{0}=\underline{\theta}^{0}_{N_{0}}$. Note that $Q^{0}(t)$
is Lipschitz continuous with respect to $t$. We then establish the comparison
relation between $Q^{0}$ and the phase diameter $D_{0}(\theta)$ of
$\mathcal{G}_{0}$ in the following lemma.

######  Lemma 3.2.

Let $\theta_{i}=\\{\theta^{0}_{i}\\}$ be a solution to system (1.1) with
strong connected digraph $\mathcal{G}_{0}$. Assume that for the group
$\mathcal{G}_{0}$, the coefficients $\bar{a}_{k}^{0}$’s and
$\underline{a}_{k}^{0}$’s satisfy the scheme (3.2). Then at each time $t$, we
have the following relation

$\beta D_{0}(\theta(t))\leq Q^{0}(t)\leq
D_{0}(\theta(t)),\quad\beta=1-\frac{2}{\eta},$

where $\eta$ satisfies the condition (1.3).

###### Proof.

As we choose the same design for coefficients of convex combination as that in
[45], the proof of this lemma is same as that in [45], please see [45] for
details.
∎

From Lemma 3.2, we see that the quantity $Q^{0}$ can control the phase
diameter $D_{0}(\theta)$, which play a key role in analysing the bound of
phase diameter. Based on Algorithm $\mathcal{A}$ and Lemma 3.1, we first study
the dynamics of the constructed quantity $Q^{0}$.

######  Lemma 3.3.

Let $\theta_{i}=\\{\theta^{0}_{i}\\}$ be the solution to system (1.1) with
strong connected digraph $\mathcal{G}_{0}$. Moreover, for a given sufficiently
small $D^{\infty}<\min\left\\{\frac{\pi}{2},\zeta\right\\}$, assume the
following conditions hold,

$\displaystyle
D_{0}(\theta(0))<\zeta<\gamma<\pi,\quad\eta>\max\left\\{\frac{1}{\sin\gamma},\frac{2}{1-\frac{\zeta}{\gamma}}\right\\},$
(3.5)
$\displaystyle\tan\alpha<\frac{1}{\left(1+\frac{\zeta}{\zeta-D(\theta(0))}\right)2N_{0}c}\beta
D^{\infty},$ $\displaystyle D^{\infty}+\alpha<\frac{\pi}{2},\quad
K>\left(1+\frac{\zeta}{\zeta-
D_{0}(\theta(0)}\right)\frac{(D(\Omega)+2N_{0}K\sin\alpha)c}{\cos\alpha}\frac{1}{\beta
D^{\infty}},$

where $\zeta,\gamma$ are constants and

$c=\frac{\left(\sum_{j=1}^{N_{0}-1}\eta^{j}A(2N_{0},j)+1\right)\gamma}{\sin\gamma}.$

Then the phase diameter of the graph $\mathcal{G}_{0}$ is uniformly bounded by
$\gamma$:

$D_{0}(\theta(t))<\gamma,\quad t\in[0,+\infty),$

and the dynamics of $Q^{0}(t)$ is controlled by the following differential
inequality

$\dot{Q}^{0}(t)\leq
D(\Omega)+2N_{0}K\sin\alpha-\frac{K\cos\alpha}{c}Q^{0}(t),\quad
t\in[0,+\infty).$

###### Proof.

The proof is similar to [45] under the assumption that the frustration
$\alpha$ is sufficiently small. However, due to the presence of frustration,
there are some slight differences in the process of analysis, thus we put the
detailed proof in the Appendix A.
∎

Lemma 3.3 states that the phase diameter of the digraph $\mathcal{G}_{0}$
remains less than $\pi$ and provides the dynamics of $Q^{0}$. We next exploit
the dynamics of $Q^{0}$ and find some finite time such that the phase diameter
of the digraph $\mathcal{G}_{0}$ is uniformly bounded by a small value after
the time.

######  Lemma 3.4.

Let $\theta_{i}=\\{\theta^{0}_{i}\\}$ be a solution to system (1.1) with
strong connected digraph $\mathcal{G}_{0}$, and suppose the assumptions in
Lemma 3.3 hold. Then there exists time $t_{0}\geq 0$ such that

$D_{0}(\theta(t))\leq D^{\infty},\quad\mbox{for}\ t\in[t_{0},+\infty),$

where $t_{0}$ can be estimated as below and bounded by $\bar{t}$ given in
Lemma 2.4

$t_{0}<\frac{\zeta}{\frac{K\cos\alpha}{c}\beta
D^{\infty}-(D(\Omega)+2N_{0}K\sin\alpha)}<\bar{t}.$ (3.6)

###### Proof.

From Lemma 3.3, we see that the dynamics of quantity $Q^{0}(t)$ is governed by
the following inequality

$\dot{Q}^{0}(t)\leq
D(\Omega)+2N_{0}K\sin\alpha-\frac{K\cos\alpha}{c}Q^{0}(t),\quad
t\in[0,+\infty).$ (3.7)

We next show that there exists some time $t_{0}$ such that the quantity
$Q^{0}$ in (3.7) is uniformly bounded after $t_{0}$. There are two cases we
consider separately.

$\diamond$ Case 1. We first consider the case that $Q^{0}(0)>\beta
D^{\infty}$. When $Q^{0}(t)\in[\beta D^{\infty},Q^{0}(0)]$, from (A.14), we
have

$\displaystyle\dot{Q}^{0}(t)$ $\displaystyle\leq
D(\Omega)+2N_{0}K\sin\alpha-\frac{K\cos\alpha}{c}Q^{0}(t)$ (3.8)
$\displaystyle\leq D(\Omega)+2N_{0}K\sin\alpha-\frac{K\cos\alpha}{c}\beta
D^{\infty}<0.$

That is to say, when $Q^{0}(t)$ is located in the interval $[\beta
D^{\infty},Q^{0}(0)]$, $Q^{0}(t)$ will keep decreasing with a rate bounded by
a uniform slope. Then we can define a stopping time $t_{0}$ as follows,

$t_{0}=\inf\\{t\geq 0\ |\ Q^{0}(t)\leq\beta D^{\infty}\\}.$

And based on the definition of $t_{0}$, we see that $Q^{0}$ will decrease
before $t_{0}$ and has the following property at $t_{0}$,

$Q^{0}(t_{0})=\beta D^{\infty}.$ (3.9)

Moerover, from (3.8), it is easy to see that the stopping time $t_{0}$
satisfies the following upper bound estimate,

$t_{0}\leq\frac{Q^{0}(0)-\beta D^{\infty}}{\frac{K\cos\alpha}{c}\beta
D^{\infty}-(D(\Omega)+2N_{0}K\sin\alpha)}.$ (3.10)

Now we study the upper bound of $Q^{0}$ on $[t_{0},+\infty)$. In fact, we can
apply (3.8), (3.9) and the same arguments in (A.12) to derive

$Q^{0}(t)\leq\beta D^{\infty},\ t\in[t_{0},+\infty).$ (3.11)

$\diamond$ Case 2. For another case that $Q^{0}(0)\leq\beta D^{\infty}$. We
can apply the similar analysis in (A.12) to obtain

$Q^{0}(t)\leq\beta D^{\infty},\quad t\in[0,+\infty).$ (3.12)

Then in this case, we directly set $t_{0}=0$.

Therefore, from (3.11), (3.12), and Lemma 3.2, we derive the upper bound of
$D_{0}(\theta)$ on $[t_{0},+\infty)$ as below

$D_{0}(\theta(t))\leq\frac{Q^{0}(t)}{\beta}\leq D^{\infty},\quad\mbox{for}\
t\in[t_{0},+\infty).$ (3.13)

On the other hand, in order to verify (3.6), we further study $t_{0}$ in
(3.13). Combining (3.10) in Case 1 and $t_{0}=0$ in Case 2, we see that

$t_{0}<\frac{\zeta}{\frac{K\cos\alpha}{c}\beta
D^{\infty}-(D(\Omega)+2N_{0}K\sin\alpha)}.$ (3.14)

Here, we use the truth that $Q^{0}(0)\leq D_{0}(\theta(0))<\zeta$. Then from
the assumption about $K$ in (3.5), i.e.,

$K>\left(1+\frac{\zeta}{\zeta-
D_{0}(\theta(0)}\right)\frac{(D(\Omega)+2N_{0}K\sin\alpha)c}{\cos\alpha}\frac{1}{\beta
D^{\infty}},\quad
c=\frac{\left(\sum_{j=1}^{N_{0}-1}\eta^{j}A(2N_{0},j)+1\right)\gamma}{\sin\gamma},$

it yields the following estimate about $t_{0}$,

$\displaystyle t_{0}$ $\displaystyle<\frac{\zeta}{(1+\frac{\zeta}{\zeta-
D_{0}(\theta(0))})(D(\Omega)+2N_{0}K\sin\alpha)-(D(\Omega)+2N_{0}K\sin\alpha)}$
(3.15) $\displaystyle=\frac{\zeta-
D_{0}(\theta(0))}{D(\Omega)+2N_{0}K\sin\alpha}=\bar{t}.$

Note that in this special strong connected case, it’s clear that $N_{0}=N$ and
$D_{0}(\theta)=D(\theta)$ in Lemma 2.4.

Thus, combining (3.13), (3.14) and (3.15), we derive the desired results.
∎

##  4\. General network

In this section, we investigate the general network with a spanning tree
structure, and prove our main result Theorem 1.1, which states that
synchronization will emerge for Kuramoto model with frustrations. According to
Definition 2.3 and Lemma 2.2, we see that the digraph $\mathcal{G}$ associated
to system (1.1) has a unique maximum node if it contains a spanning tree
structure. And From Remark 2.1, without loss of generality, we assume
$\mathcal{G}$ is decomposed into a union as
$\mathcal{G}=\bigcup_{i=0}^{d}\mathcal{G}_{i}$, where $\mathcal{G}_{p}$ is a
maximum node of $\mathcal{G}\setminus(\bigcup_{i=0}^{p-1}\mathcal{G}_{i})$.

We have studied the situation $d=0$ in Section 3, and we showed that the phase
diameter of the digraph $\mathcal{G}_{0}$ is uniformly bounded by a small
value after some finite time, i.e., the oscillators of $\mathcal{G}_{0}$ will
concentrate into a small region of quarter-circle. However, for the case that
$d>0$, $\mathcal{G}_{k}$’s are not maximum nodes in $\mathcal{G}$ for $k\geq
1$. Hence, the methods in Lemma 3.3 and Lemma 3.4 can not be directly
exploited for the situation $d>0$. More precisely, the oscillators in
$\mathcal{G}_{i}$ with $i<k$ perform as an attraction source and affect the
agents in $\mathcal{G}_{k}$. Thus when we study the behavior of agents in
$\mathcal{G}_{k}$, the information from $\mathcal{G}_{i}$ with $i<k$ can not
be ignored.

From Remark 2.1 and node decomposition, the graph $\mathcal{G}$ can be
represented as

$\mathcal{G}=\bigcup_{k=0}^{d}\mathcal{G}_{k},\quad|\mathcal{G}_{k}|=N_{k},$

and we denote the oscillators in $\mathcal{G}_{k}$ by $\theta^{k}_{i}$ with
$1\leq i\leq N_{k}$. Then we assume that at time $t$, the oscillators in each
$\mathcal{G}_{k}$ are well-ordered as below:

$\theta^{k}_{1}(t)\leq\theta^{k}_{2}(t)\leq\ldots\leq\theta^{k}_{N_{k}}(t),\quad
0\leq k\leq d.$ (4.1)

For each subdigraph $\mathcal{G}_{k}$ with $k\geq 0$ which is strongly
connected, we follow the process in Algorithm $\mathcal{A}_{1}$ and
$\mathcal{A}_{2}$ to construct
$\bar{\mathcal{L}}_{l-1}^{N_{k}}(\bar{C}_{l-1,N_{k}})$ and
$\underline{\mathcal{L}}_{1}^{l+1}(\underline{C}_{1,l+1})$ by redesigning the
coefficients $\bar{a}^{k}_{l}$ and $\underline{a}_{l}^{k}$ of convex
combination as below:

$\begin{cases}\displaystyle\bar{\mathcal{L}}_{l-1}^{N_{k}}(\bar{C}_{l-1,N_{k}})\
\mbox{with}\ \bar{a}^{k}_{N_{k}}=0,\
\bar{a}^{k}_{l-1}=\eta(2N-l+2)(\bar{a}^{k}_{l}+1),\quad 2\leq l\leq N_{k},\\\
\displaystyle\underline{\mathcal{L}}_{1}^{l+1}(\underline{C}_{1,l+1})\
\mbox{with}\ \underline{a}^{k}_{1}=0,\
\underline{a}^{k}_{l+1}=\eta(l+1+2N-N_{k})(\underline{a}^{k}_{l}+1),\quad
1\leq l\leq N_{k}-1,\end{cases}$ (4.2)

By induction principle, we deduce that

$\begin{cases}\displaystyle\bar{a}^{k}_{l-1}=\sum_{j=1}^{N_{k}-l+1}\eta^{j}A(2N-l+2,j),\quad
2\leq l\leq N_{k},\\\
\displaystyle\underline{a}^{k}_{l+1}=\sum_{j=1}^{l}\eta^{j}A(l+1+2N-N_{k},j),\quad
1\leq l\leq N_{k}-1.\end{cases}$ (4.3)

Note that $\bar{a}^{k}_{N_{k}+1-i}=\underline{a}^{k}_{i},\ i=1,2\ldots,N_{k}$.
By simple calculation, we have

$\bar{a}^{k}_{1}=\sum^{N_{k}-1}_{j=1}(\eta^{j}A(2N,j)),\quad\bar{a}^{k}_{1}\leq\sum^{N-1}_{j=1}(\eta^{j}A(2N,j)),\quad
0\leq k\leq d.$ (4.4)

And we further introduce the following notations,

$\displaystyle\bar{\theta}^{k}_{l}:=\bar{\mathcal{L}}_{l}^{N_{k}}(\bar{C}_{l,N_{k}}),\quad\underline{\theta}^{k}_{l}:=\underline{\mathcal{L}}_{1}^{l}(\underline{C}_{1,l}),\quad
1\leq l\leq N_{k},\quad 0\leq k\leq d,$ (4.5)
$\displaystyle\bar{\theta}_{k}:=\bar{\mathcal{L}}_{1}^{N_{k}}(\bar{C}_{1,N_{k}}),\quad\underline{\theta}_{k}:=\underline{\mathcal{L}}_{1}^{N_{k}}(\underline{C}_{1,N_{k}}),\quad
0\leq k\leq d,$ (4.6) $\displaystyle Q^{k}(t):=\max_{0\leq i\leq
k}\\{\bar{\theta}_{i}\\}-\min_{0\leq i\leq
k}\\{\underline{\theta}_{i}\\},\quad 0\leq k\leq d.$ (4.7)

Due to the analyticity of the solution, $Q^{k}(t)$ is Lipschitz continuous.
Similar to Section 3, we will first establish the comparison between the
quantity $Q^{k}(t)$ and phase diameter $D_{k}(\theta(t))$ of the first $k+1$
nodes, which plays a crucial role in the later analysis.

######  Lemma 4.1.

Let $\theta_{i}$ be a solution to system (1.1), and assume that the network
contains a spanning tree and for each subdigraph $\mathcal{G}_{k}$, the
coefficients $\bar{a}^{k}_{l}$ and $\underline{a}^{k}_{l}$ of convex
combination in Algorithm $\mathcal{A}$ satisfy the scheme (4.2). Then at each
time $t$, we have the following relation

$\beta D_{k}(\theta(t))\leq Q^{k}(t)\leq D_{k}(\theta(t)),\quad 0\leq k\leq
d,\quad\beta=1-\frac{2}{\eta},$

where $D_{k}(\theta)=\max\limits_{0\leq i\leq k}\max\limits_{1\leq j\leq
N_{i}}\\{\theta_{j}^{i}\\}-\min\limits_{0\leq i\leq k}\min\limits_{1\leq j\leq
N_{i}}\\{\theta_{j}^{i}\\}$ and $\eta$ satisfies the condition (1.3).

###### Proof.

As we adopt the same construction of coefficients of convex combination in
[45] which deals with the Kuramoto model without frustration on a general
network, thus for the detailed proof of this lemma, please see [45]. ∎

Now we are ready to prove our main Theorem 1.1. To this end, we will follow
similar arguments in Section 3 to complete the proof. Actually, we will
investigate the dynamics of the constructed quantity $Q^{k}(t)$ that involves
the influences from $\mathcal{G}_{i}$ with $i<k$, which yields the hypo-
coercivity of the phase diameter. Applying similar arguments in Lemma 3.3 and
Lemma 3.4, we have the following estimates for the first maximal node
$\mathcal{G}_{0}$.

######  Lemma 4.2.

Suppose that the network topology contains a spanning tree, and let
$\theta_{i}$ be a solution to (1.1). Moreover, assume that the initial data
and the quantity $\eta$ satisfy

$D(\theta(0))<\zeta<\gamma<\pi,\quad\eta>\max\left\\{\frac{1}{\sin\gamma},\frac{2}{1-\frac{\zeta}{\gamma}}\right\\},$
(4.8)

where $\zeta,\gamma$ are positive constants. And for a given sufficiently
small $D^{\infty}<\min\\{\frac{\pi}{2},\zeta\\}$, assume the frustration
$\alpha$ and coupling strength $\kappa$ satisfy

$\displaystyle\tan\alpha<\frac{1}{\left(1+\frac{(d+1)\zeta}{\zeta-D(\theta(0))}\right)2Nc}\frac{\beta^{d+1}D^{\infty}}{[4(2N+1)c]^{d}},\quad
D^{\infty}+\alpha<\frac{\pi}{2},$ (4.9) $\displaystyle
K>\left(1+\frac{(d+1)\zeta}{\zeta-D(\theta(0))}\right)\frac{(D(\Omega)+2NK\sin\alpha)c}{\cos\alpha}\frac{[4(2N+1)c]^{d}}{\beta^{d+1}D^{\infty}},$

where $d$ is the number of general nodes and

$c=\frac{(\sum_{j=1}^{N-1}\eta^{j}A(2N,j)+1)\gamma}{\sin\gamma}.$

Then the following two assertions hold for the maximum node $\mathcal{G}_{0}$:

  1. (1)

The dynamics of $Q^{0}(t)$ is governed by the following equation

$\dot{Q}^{0}(t)\leq
D(\Omega)+2NK\sin\alpha-\frac{K\cos\alpha}{c}Q^{0}(t),\quad t\in[0,+\infty),$

  2. (2)

there exists time $t_{0}\geq 0$ such that

$D_{0}(\theta(t))\leq\frac{\beta^{d}D^{\infty}}{[4(2N+1)c]^{d}},\quad\mbox{for}\
t\in[t_{0},+\infty),$

where $t_{0}$ can be estimated as below and bounded by $\bar{t}$ given in
Lemma 2.4

$t_{0}<\frac{\zeta}{\frac{K\cos\alpha}{c}\frac{\beta^{d+1}D^{\infty}}{[4(2N+1)c]^{d}}-(D(\Omega)+2NK\sin\alpha)}<\bar{t}.$

Next, inspiring from Lemma 4.2, we make the following reasonable ansatz for
$Q^{k}(t)$ with $0\leq k\leq d$.

Ansatz:

  1. (1)

The dynamics of quantity $Q^{k}(t)$ in time interval $[0,\infty)$ is governed
by the following differential inequality,

$\displaystyle\dot{Q}^{k}(t)$ $\displaystyle\leq
D(\Omega)+2NK\sin\alpha+(2N+1)K\cos\alpha
D_{k-1}(\theta(t))-\frac{K\cos\alpha}{c}Q^{k}(t),\ t\in[0,+\infty).$ (4.10)

where we assume $D_{-1}(\theta(t))=0$.

  2. (2)

there exists a finite time $t_{k}\geq 0$ such that, the phase diameter
$D_{k}(\theta(t))$ of $\bigcup_{i=0}^{k}\mathcal{G}_{i}$ is uniformly bounded
after $t_{k}$, i.e.,

$D_{k}(\theta(t))\leq\frac{\beta^{d-k}D^{\infty}}{[4(2N+1)c]^{d-k}},\quad\forall\
t\in[t_{k},+\infty),$ (4.11)

where $t_{k}$ subjects to the following estimate,

$t_{k}<\frac{(k+1)\zeta}{\frac{K\cos\alpha}{c}\frac{\beta^{d+1}D^{\infty}}{[4(2N+1)c]^{d}}-(D(\Omega)+2NK\sin\alpha)}<\bar{t}=\frac{\zeta-D(\theta(0))}{D(\Omega)+2NK\sin\alpha}.$
(4.12)

In the subsequence, we will split the proof of the ansatz into two lemmas by
induction criteria. More precisely, based on the results in Lemma 4.2 as the
initial step, we suppose the ansatz holds for $Q^{k}$ and $D_{k}(\theta)$ with
$0\leq k\leq d-1$, and then prove that the ansatz also holds for $Q^{k+1}$ and
$D_{k+1}(\theta)$.

######  Lemma 4.3.

Suppose the assumptions in Lemma 4.2 are fulfilled, and the ansatz in (4.10),
(4.11) and (4.12) holds for some $k$ with $0\leq k\leq d-1$. Then the ansatz
(4.10) holds for $k+1$.

###### Proof.

We will use proof by contradiction criteria to verify the ansatz for
$Q^{k+1}$. To this end, define a set

$\mathcal{B}_{k+1}=\\{T>0\ :\ D_{k+1}(\theta(t))<\gamma,\ \forall\
t\in[0,T)\\}.$

From Lemma 2.4, we see that

$D_{k+1}(\theta(t))\leq D(\theta(t))<\zeta<\gamma,\quad\forall\
t\in[0,\bar{t}).$

It is clear that $\bar{t}\in\mathcal{B}_{k+1}$. Thus the set
$\mathcal{B}_{k+1}$ is not empty. Define $T^{*}=\sup\mathcal{B}_{k+1}$. We
will prove by contradiction that $T^{*}=+\infty$. Suppose not, i.e.,
$T^{*}<+\infty$. It is obvious that

$\bar{t}\leq T^{*},\quad D_{k+1}(\theta(t))<\gamma,\ \forall\
t\in[0,T^{*}),\quad D_{k+1}(\theta(T^{*}))=\gamma.$ (4.13)

As the solution to system (1.1) is analytic, in the finite time interval
$[0,T^{*})$, $\bar{\theta}_{i}$ and $\bar{\theta}_{j}$ either collide finite
times or always stay together. Similar to the analysis in Lemma 3.3, without
loss of generality, we only consider the situation that there is no pair of
$\bar{\theta}_{i}$ and $\bar{\theta}_{j}$ staying together through all period
$[0,T^{*})$. That means the order of $\\{\bar{\theta}_{i}\\}_{i=0}^{k+1}$ will
only exchange finite times in $[0,T^{*})$, so does
$\\{\underline{\theta}_{i}\\}_{i=0}^{k+1}$ . Thus, we divide the time interval
$[0,T^{*})$ into a finite union as below

$[0,T^{*})=\bigcup_{l=1}^{r}J_{l},\quad J_{l}=[t_{l-1},t_{l}).$

such that in each interval $J_{l}$, the orders of both
$\\{\bar{\theta}_{i}\\}_{i=0}^{k+1}$ and
$\\{\underline{\theta}_{i}\\}_{i=0}^{k+1}$ are preseved, and the order of
oscillators in each subdigraph $\mathcal{G}_{i}$ with $0\leq i\leq k+1$ does
not change. In the following, we will show the contradiction via two steps.

$\star$ Step 1. In this step, we first verify the Ansatz (4.10) holds for
$Q^{k+1}$ on $[0,T^{*})$, i.e.,

$\displaystyle\dot{Q}^{k+1}(t)$ $\displaystyle\leq
D(\Omega)+2NK\sin\alpha+(2N+1)K\cos\alpha
D_{k}(\theta(t))-\frac{K\cos\alpha}{c}Q^{k+1}(t),\ \ t\in[0,T^{*}).$ (4.14)

As the proof is slightly different from that in [45] and rather lengthy, we
put the detailed proof in Appendix B.

$\star$ Step 2. In this step, we will study the upper bound of $Q^{k+1}$ in
(4.14) in time interval $[t_{k},T^{*})$, where $t_{k}$ is given in Ansatz
$\eqref{F-2}$ for $D_{k}(\theta)$. For the purposes of discussion, we rewrite
the equation (4.14) as below

$\displaystyle\dot{Q}^{k+1}(t)$
$\displaystyle\leq-\frac{K\cos\alpha}{c}\left(Q^{k+1}(t)-(2N+1)cD_{k}(\theta(t))-\frac{(D(\Omega)+2NK\sin\alpha)c}{K\cos\alpha}\right),\
t\in[0,T^{*}).$ (4.15)

where $c$ is expressed by the following equation

$c=\frac{(\sum_{j=1}^{N-1}\eta^{j}A(2N,j)+1)\gamma}{\sin\gamma}.$ (4.16)

For the term $D_{k}(\theta)$ in (4.15), under the assumption of induction
criteria, the Ansatz (4.11) holds for $D_{k}(\theta)$, i.e., there exists time
$t_{k}$ such that

$D_{k}(\theta(t))\leq\frac{\beta^{d-k}D^{\infty}}{[4(2N+1)c]^{d-k}},\quad\
t\in[t_{k},+\infty),\quad t_{k}<\bar{t}.$ (4.17)

And from the condition (4.9), it is obvious that

$\displaystyle K$
$\displaystyle>\left(1+\frac{(d+1)\zeta}{\zeta-D(\theta(0))}\right)\frac{(D(\Omega)+2NK\sin\alpha)c}{\cos\alpha}\frac{[4(2N+1)c]^{d}}{\beta^{d+1}D^{\infty}}$
$\displaystyle>\frac{(D(\Omega)+2NK\sin\alpha)c}{\cos\alpha}\frac{[4(2N+1)c]^{d}}{\beta^{d+1}D^{\infty}}.$

This directly yields that

$\frac{(D(\Omega)+2NK\sin\alpha)c}{K\cos\alpha}<\frac{\beta^{d+1}D^{\infty}}{[4(2N+1)c]^{d}}<\frac{\beta^{d-k}D^{\infty}}{4^{d-k}[(2N+1)c]^{d-k-1}},$
(4.18)

where $0\leq k\leq d-1,\beta<1,c>1.$ Then for the purposes of analysing the
last two terms in the bracket of (4.15), we add the esimates in (4.17) and
(4.18) to get

$\displaystyle(2N+1)cD_{k}(\theta(t))+\frac{(D(\Omega)+2NK\sin\alpha)c}{K\cos\alpha}$
(4.19)
$\displaystyle\leq(2N+1)c\frac{\beta^{d-k}D^{\infty}}{[4(2N+1)c]^{d-k}}+\frac{\beta^{d-k}D^{\infty}}{4^{d-k}[(2N+1)c]^{d-k-1}}$
$\displaystyle\leq\frac{\beta^{d-k}D^{\infty}}{2[4(2N+1)c]^{d-k-1}}<\frac{\beta^{d-k}D^{\infty}}{[4(2N+1)c]^{d-k-1}},\qquad
t\in[t_{k},+\infty).$

From Lemma 2.4, we have $t_{k}<\bar{t}\leq T^{*}$, thus it makes sense when we
consider the time interval $[t_{k},T^{*})$. Now based on the above estiamte
(4.19), we apply the differential equation (4.15) and study the upper bound of
$Q^{k+1}$ on $[t_{k},T^{*})$. We claim that

$Q^{k+1}(t)\leq\max\left\\{Q^{k+1}(t_{k}),\frac{\beta^{d-k}D^{\infty}}{[4(2N+1)c]^{d-k-1}}\right\\}:=M_{k+1},\quad
t\in[t_{k},T^{*}).$ (4.20)

Suppose not, then there exists some $\tilde{t}\in(t_{k},T^{*})$ such that
$Q^{k+1}(\tilde{t})>M_{k+1}$. We construct a set

$\mathcal{C}_{k+1}:=\\{t_{k}\leq t<\tilde{t}:Q^{k+1}(t)\leq M_{k+1}\\}.$

Since $Q^{k+1}(t_{k})\leq M_{k+1}$, the set $\mathcal{C}_{k+1}$ is not empty.
Define $t^{*}=\sup\mathcal{C}_{k+1}$. Then it is easy to see that

$t^{*}<\tilde{t},\quad Q^{k+1}(t^{*})=M_{k+1},\quad
Q^{k+1}(t)>M_{k+1}\quad\mbox{for}\ t\in(t^{*},\tilde{t}].$ (4.21)

From the construction of $M_{k+1}$, (4.19) and (4.21), it is clear that for
$t\in(t^{*},\tilde{t}]$

$\displaystyle-\frac{K\cos\alpha}{c}\left(Q^{k+1}(t)-(2N+1)cD_{k}(\theta(t))-\frac{(D(\Omega)+2NK\sin\alpha)c}{K\cos\alpha}\right)$
$\displaystyle<-\frac{K\cos\alpha}{c}\left(M_{k+1}-\frac{\beta^{d-k}D^{\infty}}{[4(2N+1)c]^{d-k-1}}\right)\leq
0.$

Wen apply the above inequality and integrate on both sides of $\eqref{F-d20}$
from $t^{*}$ to $\tilde{t}$ to get

$\displaystyle Q^{k+1}(\tilde{t})-M_{k+1}$
$\displaystyle=Q^{k+1}(\tilde{t})-Q^{k+1}(t^{*})$
$\displaystyle\leq-\int_{t^{*}}^{\tilde{t}}\frac{K\cos\alpha}{c}\left(Q^{k+1}(t)-(2N+1)cD_{k}(\theta(t))-\frac{(D(\Omega)+2NK\sin\alpha)c}{K\cos\alpha}\right)dt<0$

which contradicts to the truth $Q^{k+1}(\tilde{t})-M_{k+1}>0$. Thus we
complete the proof of (4.20).

$\star$ Step 3. In this step, we will construct a contradiction to (4.13).
From (4.20), Lemma 2.4 and the fact that

$\frac{\beta^{d-k}D^{\infty}}{[4(2N+1)c]^{d-k-1}}<D^{\infty},\quad
t_{k}<\bar{t},\quad Q^{k+1}(t_{k})\leq D_{k+1}(\theta(t_{k}))\leq
D(\theta(t_{k}))<\zeta,$

we directly obtain

$Q^{k+1}(t)\leq\max\left\\{Q^{k+1}(t_{k}),\frac{\beta^{d-k}D^{\infty}}{[4(2N+1)c]^{d-k-1}}\right\\}<\max\left\\{\zeta,D^{\infty}\right\\}=\zeta,\quad
t\in[t_{k},T^{*}).$

From Lemma 4.1 and the condition (4.8), it yields that

$D_{k+1}(\theta(t))\leq\frac{Q^{k+1}(t)}{\beta}<\frac{\zeta}{\beta}<\gamma,\quad
t\in[t_{k},T^{*}).$

Since $D_{k+1}(\theta(t))$ is continuous, we have

$D_{k+1}(\theta(T^{*}))=\lim_{t\to(T^{*})^{-}}D_{k+1}(\theta(t))\leq\frac{\zeta}{\beta}<\gamma,$

which obviously contradicts to the assumption $D_{k+1}(\theta(T^{*}))=\gamma$
in (4.13).

Thus, we combine all above analysis to conclude that $T^{*}=+\infty$, that is
to say,

$D_{k+1}(\theta(t))<\gamma,\quad\forall\ t\in[0,+\infty).$ (4.22)

Then for any finite time $T>0$, we apply (4.22) and repeat the analysis in
Step 1 to obtain that the differential inequality (4.10) holds for $Q^{k+1}$
on $[0,T)$. Thus we obtain the dynamics of $Q^{k+1}$ in whole time interval as
below:

$\displaystyle\dot{Q}^{k+1}(t)$ $\displaystyle\leq
D(\Omega)+2NK\sin\alpha+(2N+1)K\cos\alpha
D_{k}(\theta(t))-\frac{K\cos\alpha}{c}Q^{k+1}(t),\ t\in[0,+\infty).$ (4.23)

Therefore, we complete the proof of the Ansatz (4.10) for $Q^{k+1}$.
∎

######  Lemma 4.4.

Suppose the conditions in Lemma 4.2 are fulfilled, and the ansatz in (4.10),
(4.11) and (4.12) holds for some $k$ with $0\leq k\leq d-1$. Then the ansatz
(4.11) and (4.12) holds for $k+1$.

###### Proof.

From Lemma 4.3, we know the dynamic of $Q^{k+1}$ is governed by (4.23). For
the purposes of discussion, we rewrite the differential equation (4.23) and
discuss it on $[t_{k},+\infty)$,

$\dot{Q}^{k+1}(t)\leq-\frac{K\cos\alpha}{c}\left(Q^{k+1}(t)-(2N+1)cD_{k}(\theta(t))-\frac{(D(\Omega)+2NK\sin\alpha)c}{K\cos\alpha}\right),\
t\in[t_{k},+\infty).$ (4.24)

where $c$ is given in (4.16). In the subsequence, we will apply (4.24) to find
a finite time $t_{k+1}$ such that the quantity $Q^{k+1}$ in (4.24) is
uniformly bounded by a small value after $t_{k+1}$. We split into two cases to
discuss.

$\bullet$ Case 1. We first consider the case that
$Q^{k+1}(t_{k})>\frac{\beta^{d-k}D^{\infty}}{[4(2N+1)c]^{d-k-1}}$. In this
case, When
$Q^{k+1}(t)\in[\frac{\beta^{d-k}D^{\infty}}{[4(2N+1)c]^{d-k-1}},Q^{k+1}(t_{k})]$,
we combine (4.19) and (4.24) to have

$\displaystyle\dot{Q}^{k+1}(t)$
$\displaystyle\leq-\frac{K\cos\alpha}{c}\left(\frac{\beta^{d-k}D^{\infty}}{[4(2N+1)c]^{d-k-1}}-\frac{\beta^{d-k}D^{\infty}}{2[4(2N+1)c]^{d-k-1}}\right)$
(4.25)
$\displaystyle=-\frac{K\cos\alpha}{c}\frac{\beta^{d-k}D^{\infty}}{2[4(2N+1)c]^{d-k-1}}<0.$

That is to say, when $Q^{k+1}(t)$ is located in the interval
$[\frac{\beta^{d-k}D^{\infty}}{[4(2N+1)c]^{d-k-1}},Q^{k+1}(t_{k})]$,
$Q^{k+1}(t)$ will keep decreasing with a rate bounded by a uniform slope.
Therefore, we can define a stopping time $t_{k+1}$ as follows,

$t_{k+1}=\inf\left\\{t\geq t_{k}\ |\
Q^{k+1}(t)\leq\frac{\beta^{d-k}D^{\infty}}{[4(2N+1)c]^{d-k-1}}\right\\}.$

Then, based on (4.25) and the definition of $t_{k+1}$, we see that $Q^{k+1}$
will decrease before $t_{k+1}$ and has the following property at $t_{k+1}$,

$Q^{k+1}(t_{k+1})=\frac{\beta^{d-k}D^{\infty}}{(4c)^{d-k-1}}.$ (4.26)

Moreover, from (4.25), it yields that the stopping time $t_{k+1}$ satisfies
the following upper bound estimate,

$t_{k+1}\leq\frac{Q^{k+1}(t_{k})-\frac{\beta^{d-k}D^{\infty}}{[4(2N+1)c]^{d-k-1}}}{\frac{K\cos\alpha}{c}\frac{\beta^{d-k}D^{\infty}}{2[4(2N+1)c]^{d-k-1}}}+t_{k}.$
(4.27)

Now we study the upper bound of $Q^{k+1}$ on $[t_{k+1},+\infty)$. In fact, we
can apply (4.25), (4.26) and the same arguments in (4.20) to derive

$Q^{k+1}(t)\leq\frac{\beta^{d-k}D^{\infty}}{[4(2N+1)c]^{d-k-1}},\quad
t\in[t_{k+1},+\infty).$ (4.28)

On the other hand, in order to verify (4.12), we further study $t_{k+1}$ in
(4.27). For the first part on the right-hand side of (4.27), from Lemma 2.4
and the fact that

$Q^{k+1}(t_{k})\leq D_{k+1}(\theta(t_{k}))\leq
D(\theta(t_{k}))<\zeta,\quad\frac{\beta^{d-k}D^{\infty}}{2[4(2N+1)c]^{d-k-1}}>\frac{\beta^{d+1}D^{\infty}}{[4(2N+1)c]^{d}},$

we have the following estimates

$\frac{Q^{k+1}(t_{k})-\frac{\beta^{d-k}D^{\infty}}{[4(2N+1)c]^{d-k-1}}}{\frac{K\cos\alpha}{c}\frac{\beta^{d-k}D^{\infty}}{2[4(2N+1)c]^{d-k-1}}}<\frac{\zeta}{\frac{K\cos\alpha}{c}\frac{\beta^{d+1}D^{\infty}}{[4(2N+1)c]^{d}}-(D(\Omega)+2NK\sin\alpha)},$
(4.29)

where the denominator on the right-hand side of above inequality is positive
from the conditions about $K$ and $\alpha$ in (4.9). For the term $t_{k}$ in
(4.27), based on the assumption (4.12) for $t_{k}$, we have

$t_{k}<\frac{(k+1)\zeta}{\frac{K\cos\alpha}{c}\frac{\beta^{d+1}D^{\infty}}{[4(2N+1)c]^{d}}-(D(\Omega)+2NK\sin\alpha)}<\bar{t}=\frac{\zeta-D(\theta(0))}{D(\Omega)+2NK\sin\alpha}.$
(4.30)

Thus it yields from (4.27), (4.29) and (4.30) that the time $t_{k+1}$
satisfies

$t_{k+1}<\frac{(k+2)\zeta}{\frac{K\cos\alpha}{c}\frac{\beta^{d+1}D^{\infty}}{[4(2N+1)c]^{d}}-(D(\Omega)+2NK\sin\alpha)}.$
(4.31)

Moreover, from (4.9), it is easy to see that the coupling strength $K$
satisfies the following inequality

$\displaystyle K$
$\displaystyle>\left(1+\frac{(d+1)\zeta}{\zeta-D(\theta(0))}\right)\frac{(D(\Omega)+2NK\sin\alpha)c}{\cos\alpha}\frac{[4(2N+1)c]^{d}}{\beta^{d+1}D^{\infty}}$
(4.32)
$\displaystyle\geq\left(1+\frac{(k+2)\zeta}{\zeta-D(\theta(0))}\right)\frac{(D(\Omega)+2NK\sin\alpha)c}{\cos\alpha}\frac{[4(2N+1)c]^{d}}{\beta^{d+1}D^{\infty}},\quad
0\leq k\leq d-1.$

Thus we combine (4.31) and (4.32) to verify the Ansatz (4.12) for $k+1$ in the
first case, i.e., the time $t_{k+1}$ subjects to the following estimate,

$t_{k+1}<\bar{t}=\frac{\zeta-D(\theta(0))}{D(\Omega)+2NK\sin\alpha}.$ (4.33)

$\bullet$ Case 2. For another case that
$Q^{k+1}(t_{k})\leq\frac{\beta^{d-k}D^{\infty}}{(4c)^{d-k-1}}$. Similar to the
analysis in (4.20), we apply (4.25) to conclude that

$Q^{k+1}(t)\leq\frac{\beta^{d-k}D^{\infty}}{[4(2N+1)c]^{d-k-1}},\quad
t\in[t_{k},+\infty).$ (4.34)

In this case, we directly set $t_{k+1}=t_{k}$. Then, from (4.30), it yields
that the inequalities (4.31) and (4.33) also hold, which finish the
verification of the Ansatz (4.12) in the second case.

Finally, we are ready to verify the ansatz (4.11) for $k+1$. Actually, we can
apply (4.28), (4.34) and Lemma 4.1 to have the upper bound of
$D_{k+1}(\theta)$ on $[t_{k+1},+\infty)$ as below

$\displaystyle
D_{k+1}(\theta(t))\leq\frac{Q^{k+1}(t)}{\beta}\leq\frac{\beta^{d-k-1}D^{\infty}}{[4(2N+1)c]^{d-k-1}},\quad
t\in[t_{k+1},+\infty),$ (4.35)

Then we combine (4.31), (4.33) and (4.35) in Case 1 and similar analysis in
Case 2 to conclude that the Ansatz (4.11) and (4.12) is true for
$D^{k+1}(\theta)$.
∎

Now, we are ready to prove our main result.

Proof of Theorem 1.1: Combining Lemma 4.2, Lemma 4.3 and Lemma 4.4, we apply
inductive criteria to conclude that the Ansatz (4.10) –(4.12) hold for all
$0\leq k\leq d$. Then, it yields from (4.11) that there exists a finite time
$t_{d}\geq 0$ such that

$D(\theta(t))=D_{d}(\theta(t))\leq D^{\infty},\quad\mbox{for}\
t\in[t_{d},+\infty).$

Thus we derive the desired result in Theorem 1.1.

######  Remark 4.1.

For the Kuramoto model with frustration, in Theorem 1.1, we show the phase
diameter of whole ensemble will be uniformly bounded by a small value
$D^{\infty}$ after some finite time. Under the assumption that $\alpha$ is
sufficiently small such that $D^{\infty}+\alpha<\frac{\pi}{2}$, the
interaction function $\cos x$ in the dynamics of frequency is positive after
the finite time. Thus, we can lift (1.1) to the second-order formulation,
which enjoys the similar form to Cucker-Smale model with the interaction
function $\cos x$.

More precisely, we can introduce phase velocity or frequency
$\omega_{i}(t):=\dot{\theta}_{i}(t)$ for each oscillator, and directly
differentiate (1.1) with respect to time $t$ to derive the equivalent second-
order Cucker-Smale type model as below

$\begin{cases}\displaystyle\dot{\theta}_{i}(t)=\omega_{i}(t),\quad t>0,\quad
i=1,2,\ldots,N,\\\
\displaystyle\dot{\omega}_{i}(t)=K\sum_{j\in\mathcal{N}_{i}}\cos(\theta_{j}(t)-\theta_{i}(t)+\alpha)(\omega_{j}(t)-\omega_{i}(t)),\\\
\displaystyle(\theta_{i}(0),\omega_{i}(0))=(\theta_{i}(0),\dot{\theta}_{i}(0)).\end{cases}$
(4.36)

Now for the second-order system (4.36), we apply the results in [9] for
Kuramoto model without frustration on a general digraph and present the
frequency synchronization for Kuramoto model with frustrations.

######  Corollary 4.1.

Let $\theta_{i}$ be a solution to system (4.36) and suppose the assumptions in
Lemma 4.2 are fulfilled. Moreover, assume that there exists time $t_{*}>0$
such that

$D(\theta(t))\leq D^{\infty},\quad\ t\in[t_{*},+\infty),$ (4.37)

where $D^{\infty}<\min\\{\zeta,\frac{\pi}{2}\\}$ is a small positive constant
and $\alpha$ is sufficiently small such that
$D^{\infty}+\alpha<\frac{\pi}{2}$. Then there exist positive constants $C_{1}$
and $C_{2}$ such that

$D(\omega(t))\leq C_{1}e^{-C_{2}(t-t_{*})},\quad t>t_{*},$

where $D(\omega(t))=\max_{1\leq i\leq N}\\{\omega_{i}(t)\\}-\min_{1\leq i\leq
N}\\{\omega_{i}(t)\\}$ is the diameter of phase velocity.

###### Proof.

We can apply Theorem 1.1 and the methods and results in the work of Dong et
al. [9] for Kuramoto model without frustration to yield the emergence of
exponentially fast synchronization in (1.1) and (4.36). As the proof is almost
the same as that in [9], we omit its details.
∎

##  5\. Summary

In this paper, under the effect of frustration, we provide sufficient
frameworks leading to the complete synchronization for the Kuramoto model with
general network containing a spanning tree. To this end, we follow a node
decomposition introduced in [25] and construct hypo-coercive inequalities
through which we can study the upper bounds of phase diameters. When the
initial configuration is confined in a half circle, for sufficiently small
frustration and sufficiently large coupling strength, we show that the
relative differences of Kuramoto oscillators adding a phase shift will be
confined into a small region less than a quarter circle in finite time, thus
we can directly apply the methods and results in [9] to prove that the
complete synchronization emerges exponentially fast.

##  Appendix A proof of Lemma 3.3

We will split the proof into six steps. In the first step, we suppose by
contrary that the phase diameter of $\mathcal{G}_{0}$ is bounded by $\gamma$
in a finite time interval. In the second, third and forth steps, we use
induction criteria to construct the differential inequality of $Q^{0}(t)$ in
the finite time interval. In the last two steps, we exploit the derived
differential inequality of $Q^{0}(t)$ to conclude that phase diameter of
$\mathcal{G}_{0}$ is bounded by $\gamma$ on $[0,+\infty)$, and thus the
differential inequality of $Q^{0}(t)$ obtained in the forth step also holds on
$[0,+\infty)$.

$\bigstar$ Step 1. Define a set

$\mathcal{B}_{0}:=\\{T>0:\ D_{0}(\theta(t))<\gamma,\ \forall\ t\in[0,T)\\}.$

From Lemma 2.4 where $N=N_{0}$ in the present section, the set
$\mathcal{B}_{0}$ is non-empty since

$D_{0}(\theta(t))=D(\theta(t))<\zeta<\gamma,\quad\forall\ t\in[0,\bar{t}),$

which directly yields that $\bar{t}\in\mathcal{B}_{0}$. Define
$T^{*}=\sup\mathcal{B}_{0}$. And we claim that $T^{*}=+\infty$. Suppose not,
i.e., $T^{*}<+\infty$, then we apply the continuity of $D_{0}(\theta(t))$ to
have

$D_{0}(\theta(t))<\gamma,\quad\forall\ t\in[0,T^{*}),\quad
D_{0}(\theta(T^{*}))=\gamma.$ (A.1)

In particular, we have $\bar{t}\leq T^{*}$. The analyticity of the solution to
system (1.1) is guaranteed by the standard Cauchy-Lipschitz theory. Therefore,
in the finite time interval $[0,T^{*})$, any two oscillators either collide
finite times or always stay together. If there are some $\theta_{i}$ and
$\theta_{j}$ always staying together in $[0,T^{*}]$, we can view them as one
oscillator and thus the total number of oscillators that we need to study can
be reduced. This is a more simpler situation, and we can similarly deal with
it. Therefore, we only consider the case that there is no pair of oscillators
staying together in $[0,T^{*})$. For this case, there are only finite many
collisions occurring through $[0,T^{*})$. Thus, we divide the time interval
$[0,T^{*})$ into a finite union as below

$[0,T^{*})=\bigcup_{l=1}^{r}J_{l},\quad J_{l}=[t_{l-1},t_{l}),$

where the end point $t_{l}$ denotes the collision instant. It is easy to see
that there is no collision in the interior of $J_{l}$. Now we pick out any
time interval $J_{l}$ and assume that

$\theta_{1}^{0}(t)\leq\theta^{0}_{2}(t)\leq\ldots\leq\theta^{0}_{N_{0}}(t),\quad
t\in J_{l}.$ (A.2)

$\bigstar$ Step 2. According to the notations in (3.4), we follow the process
$\mathcal{A}_{1}$ and $\mathcal{A}_{2}$ to construct $\bar{\theta}^{0}_{n}$
and $\underline{\theta}^{0}_{n},\ 1\leq n\leq N_{0}$, respecively. We first
study the dynamics of $\bar{\theta}_{N_{0}}^{0}=\theta_{N_{0}}^{0}$,

$\displaystyle\dot{\theta}^{0}_{N_{0}}(t)$
$\displaystyle=\Omega^{0}_{N_{0}}+K\sum_{j\in\mathcal{N}^{0}_{N_{0}}(0)}\sin(\theta_{j}^{0}-\theta^{0}_{N_{0}}+\alpha)$
(A.3)
$\displaystyle\leq\Omega_{M}+K\sum_{j\in\mathcal{N}^{0}_{N_{0}}(0)}\left[\sin(\theta^{0}_{j}-\theta^{0}_{N_{0}})\cos\alpha+\cos(\theta^{0}_{j}-\theta^{0}_{N_{0}})\sin\alpha\right]$
$\displaystyle\leq\Omega_{M}+N_{0}K\sin\alpha+K\cos\alpha\min_{j\in\mathcal{N}^{0}_{N_{0}}(0)}\sin(\theta^{0}_{j}-\theta^{0}_{N_{0}}).$

For the dynamics of $\bar{\theta}^{0}_{N_{0}-1}$, according to the process
$\mathcal{A}_{1}$ and $\bar{a}^{0}_{N_{0}-1}=\eta(N_{0}+2)$ in (3.2), we apply
(A.3) and estimate the derivative of $\bar{\theta}^{0}_{N_{0}-1}$ as follows,

$\displaystyle\dot{\bar{\theta}}^{0}_{N_{0}-1}$
$\displaystyle=\frac{d}{dt}\left(\frac{\bar{a}^{0}_{N_{0}-1}\theta^{0}_{N_{0}}+\theta^{0}_{N_{0}-1}}{\bar{a}^{0}_{N_{0}-1}+1}\right)=\frac{\bar{a}^{0}_{N_{0}-1}}{\bar{a}^{0}_{N_{0}-1}+1}\dot{\theta}^{0}_{N_{0}}+\frac{1}{\bar{a}^{0}_{N_{0}-1}+1}\dot{\theta}^{0}_{N_{0}-1}$
(A.4)
$\displaystyle\leq\frac{\bar{a}^{0}_{N_{0}-1}}{\bar{a}^{0}_{N_{0}-1}+1}\left(\Omega_{M}+N_{0}K\sin\alpha+K\cos\alpha\min_{j\in\mathcal{N}^{0}_{N_{0}}(0)}\sin(\theta^{0}_{j}-\theta^{0}_{N_{0}})\right)$
$\displaystyle+\frac{1}{\bar{a}^{0}_{N_{0}-1}+1}\left(\Omega^{0}_{N_{0}-1}+K\sum_{j\in\mathcal{N}^{0}_{N_{0}-1}(0)}\sin(\theta^{0}_{j}-\theta^{0}_{N_{0}-1}+\alpha)\right)$
$\displaystyle\leq\Omega_{M}+K\cos\alpha\frac{1}{\bar{a}^{0}_{N_{0}-1}+1}\eta(N_{0}+2)\min_{j\in\mathcal{N}^{0}_{N_{0}}(0)}\sin(\theta^{0}_{j}-\theta^{0}_{N_{0}})$
$\displaystyle+\frac{\bar{a}^{0}_{N_{0}-1}}{\bar{a}^{0}_{N_{0}-1}+1}N_{0}K\sin\alpha+K\cos\alpha\frac{1}{\bar{a}^{0}_{N_{0}-1}+1}\sum_{j\in\mathcal{N}^{0}_{N_{0}-1}(0)}\sin(\theta^{0}_{j}-\theta^{0}_{N_{0}-1})$
$\displaystyle+K\sin\alpha\frac{1}{\bar{a}^{0}_{N_{0}-1}+1}\sum_{j\in\mathcal{N}^{0}_{N_{0}-1}(0)}\cos(\theta^{0}_{j}-\theta^{0}_{N_{0}-1})$
$\displaystyle\leq\Omega_{M}+K\cos\alpha\frac{1}{\bar{a}^{0}_{N_{0}-1}+1}2\eta\min_{j\in\mathcal{N}^{0}_{N_{0}}(0)}\sin(\theta^{0}_{j}-\theta^{0}_{N_{0}})+\frac{\bar{a}^{0}_{N_{0}-1}}{\bar{a}^{0}_{N_{0}-1}+1}N_{0}K\sin\alpha$
$\displaystyle+K\cos\alpha\frac{1}{\bar{a}^{0}_{N_{0}-1}+1}\left(\underset{j\leq
N_{0}-1}{\sum_{j\in\mathcal{N}^{0}_{N_{0}-1}(0)}}\sin(\theta^{0}_{j}-\theta^{0}_{N_{0}-1})+\sin(\theta^{0}_{N_{0}}-\theta^{0}_{N_{0}-1})\right)$
$\displaystyle+\frac{1}{\bar{a}^{0}_{N_{0}-1}+1}N_{0}K\sin\alpha$
$\displaystyle\leq\Omega_{M}+K\cos\alpha\frac{1}{\bar{a}^{0}_{N_{0}-1}+1}\eta\min_{j\in\mathcal{N}^{0}_{N_{0}}(0)}\sin(\theta^{0}_{j}-\theta^{0}_{N_{0}})$
$\displaystyle+K\cos\alpha\frac{1}{\bar{a}^{0}_{N_{0}-1}+1}\underset{j\leq
N_{0}-1}{\min_{j\in\mathcal{N}^{0}_{N_{0}-1}(0)}}\sin(\theta^{0}_{j}-\theta^{0}_{N_{0}-1})$
$\displaystyle+K\cos\alpha\frac{1}{\bar{a}^{0}_{N_{0}-1}+1}\underbrace{\left(\eta\min_{j\in\mathcal{N}^{0}_{N_{0}}(0)}\sin(\theta^{0}_{j}-\theta^{0}_{N_{0}})+\sin(\theta^{0}_{N_{0}}-\theta^{0}_{N_{0}-1})\right)}_{\mathcal{I}_{1}}+N_{0}K\sin\alpha,$

where we use

$\displaystyle|\sum_{j\in\mathcal{N}^{0}_{N_{0}-1}(0)}\cos(\theta^{0}_{j}-\theta^{0}_{N_{0}-1})|\leq
N_{0},\quad K\cos\alpha\frac{1}{\bar{a}^{0}_{N_{0}-1}+1}\eta
N_{0}\min_{j\in\mathcal{N}^{0}_{N_{0}}(0)}\sin(\theta^{0}_{j}-\theta^{0}_{N_{0}})\leq
0,$ $\displaystyle\underset{j\leq
N_{0}-1}{\sum_{j\in\mathcal{N}^{0}_{N_{0}-1}(0)}}\sin(\theta^{0}_{j}-\theta^{0}_{N_{0}-1})\leq\underset{j\leq
N_{0}-1}{\min_{j\in\mathcal{N}^{0}_{N_{0}-1}(0)}}\sin(\theta^{0}_{j}-\theta^{0}_{N_{0}-1}).$

Next we show the term $\mathcal{I}_{1}$ is non-positive. We only consider the
situation $\gamma>\frac{\pi}{2}$, and the case $\gamma\leq\frac{\pi}{2}$ can
be similarly dealt with. It is clear that

$\min_{j\in\mathcal{N}^{0}_{N_{0}}(0)}\sin(\theta^{0}_{j}-\theta^{0}_{N_{0}})\leq\sin(\theta^{0}_{\bar{k}_{N_{0}}}-\theta^{0}_{N_{0}})\quad\text{where}\
\bar{k}_{N_{0}}=\min_{j\in\mathcal{N}^{0}_{N_{0}}(0)}j.$

Note that $\bar{k}_{N_{0}}\leq N_{0}-1$ since
$\bar{\mathcal{L}}^{N_{0}}_{N_{0}}(\bar{C}_{N_{0},N_{0}})$ is not a general
root. Therefore, if
$0\leq\theta^{0}_{N_{0}}(t)-\theta^{0}_{\bar{k}_{N_{0}}}(t)\leq\frac{\pi}{2}$,
we immediately obtain that

$0\leq\theta^{0}_{N_{0}}(t)-\theta^{0}_{N_{0}-1}(t)\leq\theta^{0}_{N_{0}}(t)-\theta^{0}_{\bar{k}_{N_{0}}}(t)\leq\frac{\pi}{2},$

which implies that

$\mathcal{I}_{1}\leq\eta\sin(\theta^{0}_{\bar{k}_{N_{0}}}-\theta^{0}_{N_{0}})+\sin(\theta^{0}_{N_{0}}-\theta^{0}_{N_{0}-1})\leq\sin(\theta^{0}_{\bar{k}_{N_{0}}}-\theta^{0}_{N_{0}})+\sin(\theta^{0}_{N_{0}}-\theta^{0}_{N_{0}-1})\leq
0.$

On the other hand, if
$\frac{\pi}{2}<\theta^{0}_{N_{0}}(t)-\theta^{0}_{\bar{k}_{N_{0}}}(t)<\gamma$,
we use the fact

$\eta>\frac{1}{\sin\gamma}\quad\mbox{and}\quad\sin(\theta^{0}_{N_{0}}(t)-\theta^{0}_{\bar{k}_{N_{0}}}(t))>\sin\gamma,$

to conclude that
$\eta\sin(\theta^{0}_{\bar{k}_{N_{0}}}-\theta^{0}_{N_{0}})\leq-1$. Hence, in
this case, we still obtain that

$\mathcal{I}_{1}\leq\eta\sin(\theta^{0}_{\bar{k}_{N_{0}}}-\theta^{0}_{N_{0}})+\sin(\theta^{0}_{N_{0}}-\theta^{0}_{N_{0}-1})\leq-1+1\leq
0.$

Thus, for $t\in J_{l}$, we combine above analysis to conclude that

$\mathcal{I}_{1}=\eta\min_{j\in\mathcal{N}^{0}_{N_{0}}(0)}\sin(\theta^{0}_{j}-\theta^{0}_{N_{0}})+\sin(\theta^{0}_{N_{0}}-\theta^{0}_{N_{0}-1})\leq
0.$ (A.5)

Then combining (A.4) and (A.5), we derive that

$\displaystyle\dot{\bar{\theta}}^{0}_{N_{0}-1}$
$\displaystyle\leq\Omega_{M}+N_{0}K\sin\alpha$ (A.6)
$\displaystyle+K\cos\alpha\frac{1}{\bar{a}^{0}_{N_{0}-1}+1}\left(\eta\min_{j\in\mathcal{N}^{0}_{N_{0}}(0)}\sin(\theta^{0}_{j}-\theta^{0}_{N_{0}})+\underset{j\leq
N_{0}-1}{\min_{j\in\mathcal{N}^{0}_{N_{0}-1}(0)}}\sin(\theta^{0}_{j}-\theta^{0}_{N_{0}-1})\right).$

$\bigstar$ Step 3. Now we apply the induction principle to cope with
$\bar{\theta}^{0}_{n}$ in (3.4), which is construced in the iteration process
$\mathcal{A}_{1}$. We will prove for $1\leq n\leq N_{0}$ that,

$\dot{\bar{\theta}}^{0}_{n}(t)\leq\Omega_{M}+N_{0}K\sin\alpha+K\cos\alpha\frac{1}{\bar{a}^{0}_{n}+1}\sum_{i=n}^{N_{0}}\eta^{i-n}\underset{j\leq
i}{\min_{j\in\mathcal{N}^{0}_{i}(0)}}\sin(\theta^{0}_{j}(t)-\theta^{0}_{i}(t))$
(A.7)

In fact, it is known that (A.7) already holds for $n=N_{0},N_{0}-1$ from (A.3)
and (A.6). Then by induction criteria, suppose (A.7) holds for $n$, Next we
verify that (A.7) still holds for $n-1$. According to the Algorithm
$\mathcal{A}_{1}$ and similar calculations in (A.4), the dynamics of the
quantity $\bar{\theta}^{0}_{n-1}(t)$ subjects to the following estimates

$\displaystyle\dot{\bar{\theta}}^{0}_{n-1}$
$\displaystyle\leq\Omega_{M}+N_{0}K\sin\alpha$ (A.8)
$\displaystyle+K\cos\alpha\frac{1}{\bar{a}^{0}_{n-1}+1}\eta\sum_{i=n}^{N_{0}}\eta^{i-n}\underset{j\leq
i}{\min_{j\in\mathcal{N}^{0}_{i}(0)}}\sin(\theta^{0}_{j}-\theta^{0}_{i})$
$\displaystyle+K\cos\alpha\frac{1}{\bar{a}^{0}_{n-1}+1}\underset{j\leq
n-1}{\min_{j\in\mathcal{N}_{n-1}^{0}(0)}}\sin(\theta^{0}_{j}-\theta^{0}_{n-1})$
$\displaystyle+K\cos\alpha\frac{1}{\bar{a}^{0}_{n-1}+1}$
$\displaystyle\times\left(\underbrace{\eta(N_{0}-n+1)\sum_{i=n}^{N_{0}}\eta^{i-n}\underset{j\leq
i}{\min_{j\in\mathcal{N}^{0}_{i}(0)}}\sin(\theta^{0}_{j}-\theta^{0}_{i})+\underset{j>n-1}{\sum_{j\in\mathcal{N}^{0}_{n-1}(0)}}\sin(\theta^{0}_{j}-\theta^{0}_{n-1})}_{\mathcal{I}_{2}}\right),$

Moreover, we can prove the term $\mathcal{I}_{2}$ is non-positive. As the
proof is very similar as that in the previous step, we omit the details and
directly claim that $\mathcal{I}_{2}\leq 0$, which together with (A.8)
verifies (A.7).

$\bigstar$ Step 4. Now, we set $n=1$ in (A.7) and apply Lemma 3.1 to have

$\displaystyle\dot{\bar{\theta}}^{0}_{1}$
$\displaystyle\leq\Omega_{M}+N_{0}K\sin\alpha+K\cos\alpha\frac{1}{\bar{a}^{0}_{1}+1}\sum_{i=1}^{N_{0}}\eta^{i-1}\underset{j\leq
i}{\min_{j\in\mathcal{N}^{0}_{i}(0)}}\sin(\theta^{0}_{j}-\theta^{0}_{i})$
(A.9)
$\displaystyle\leq\Omega_{M}+N_{0}K\sin\alpha+K\cos\alpha\frac{1}{\bar{a}^{0}_{1}+1}\sin(\theta^{0}_{\bar{k}_{1}}-\theta^{0}_{N_{0}})$
$\displaystyle=\Omega_{M}+N_{0}K\sin\alpha+K\cos\alpha\frac{1}{\bar{a}^{0}_{1}+1}\sin(\theta^{0}_{1}-\theta^{0}_{N_{0}}),$

where $\bar{k}_{1}=\min_{j\in\bigcup_{i=1}^{N_{0}}\mathcal{N}^{0}_{i}(0)}j=1$
due to the strong connectivity of $\mathcal{G}_{0}$. Similarly, we can follow
the process $\mathcal{A}_{2}$ to construct $\underline{\theta}^{0}_{k}$ in
(3.4) until $k=N_{0}$. Then, we can apply the similar argument in (A.7) to
obtain that,

$\displaystyle\frac{d}{dt}\underline{\theta}^{0}_{N_{0}}(t)$
$\displaystyle\geq\Omega_{m}-N_{0}K\sin\alpha+K\cos\alpha\frac{1}{\bar{a}^{0}_{1}+1}\sin(\theta^{0}_{N_{0}}-\theta^{0}_{1}),$
(A.10)

Then we recall the notations $\bar{\theta}_{0}=\bar{\theta}^{0}_{1}$ and
$\underline{\theta}_{0}=\underline{\theta}^{0}_{N_{0}}$, and combine (A.9) and
(A.10) to obtain that

$\displaystyle\dot{Q}^{0}(t)$
$\displaystyle=\frac{d}{dt}(\bar{\theta}_{0}-\underline{\theta}_{0})\leq
D(\Omega)+2N_{0}K\sin\alpha-K\cos\alpha\frac{2}{\bar{a}^{0}_{1}+1}\sin(\theta^{0}_{N_{0}}-\theta^{0}_{1})$
$\displaystyle\leq
D(\Omega)+2N_{0}K\sin\alpha-K\cos\alpha\frac{1}{\sum_{j=1}^{N_{0}-1}\eta^{j}A(2N_{0},j)+1}\sin(\theta^{0}_{N_{0}}-\theta^{0}_{1})$

where we use the property

$\bar{a}^{0}_{1}=\sum_{j=1}^{N_{0}-1}\eta^{j}A(2N_{0},j).$

As the function $\frac{\sin x}{x}$ is monotonically decreasing in $(0,\pi]$,
we apply (A.1) to obtain that

$\sin(\theta^{0}_{N_{0}}-\theta^{0}_{1})\geq\frac{\sin\gamma}{\gamma}(\theta^{0}_{N_{0}}-\theta^{0}_{1}).$

Moreover, due to the fact
$Q^{0}(t)\leq\theta^{0}_{N_{0}}(t)-\theta^{0}_{1}(t)$, we have

$\displaystyle\dot{Q}^{0}(t)$ $\displaystyle\leq
D(\Omega)+2N_{0}K\sin\alpha-K\cos\alpha\frac{1}{\sum_{j=1}^{N_{0}-1}\eta^{j}A(2N_{0},j)+1}\frac{\sin\gamma}{\gamma}(\theta^{0}_{N_{0}}-\theta^{0}_{1})$
(A.11) $\displaystyle\leq
D(\Omega)+2N_{0}K\sin\alpha-K\cos\alpha\frac{1}{\sum_{j=1}^{N_{0}-1}\eta^{j}A(2N_{0},j)+1}\frac{\sin\gamma}{\gamma}Q^{0}(t),\quad
t\in J_{l}.$

Note that the constructed quantity
$Q^{0}(t)=\bar{\theta}_{0}(t)-\underline{\theta}_{0}(t)$ is Lipschitz
continuous on $[0,T^{*})$. Moreover, the above analysis does not depend on the
time interval $J_{l},\ l=1,2,\ldots,r$, thus the differential inequality
(A.11) holds almost everywhere on $[0,T^{*})$.

$\bigstar$ Step 5. Next we study the upper bound of $Q^{0}(t)$ in the period
$[0,T^{*})$. Define

$M_{0}=\max\left\\{Q^{0}(0),\beta D^{\infty}\right\\}.$

We claim that

$Q^{0}(t)\leq M_{0}\quad\mbox{for all}\ t\in[0,T^{*}).$ (A.12)

Suppose not, then there exists some $\tilde{t}\in[0,T^{*})$ such that
$Q^{0}(\tilde{t})>M_{0}$. We construct a set

$\mathcal{C}_{0}:=\\{t<\tilde{t}\ |\ Q^{0}(t)\leq M_{0}\\}.$

Since $0\in\mathcal{C}_{0}$, the set $\mathcal{C}_{0}$ is not empty. Then we
denote $t^{*}=\sup\mathcal{C}_{0}$. It is easy to see that

$t^{*}<\tilde{t},\quad Q^{0}(t^{*})=M_{0},\quad Q^{0}(t)>M_{0}\quad\mbox{for}\
t\in(t^{*},\tilde{t}].$ (A.13)

For a given sufficiently small $D^{\infty}<\min\\{\frac{\pi}{2},\zeta\\}$,
based on the assumptions about the frustration and the coupling strength in
(3.5), it is clear that

$K>\left(1+\frac{\zeta}{\zeta-D(\theta(0)}\right)\frac{(D(\Omega)+2N_{0}K\sin\alpha)c}{\cos\alpha}\frac{1}{\beta
D^{\infty}}>\frac{(D(\Omega)+2N_{0}K\sin\alpha)c}{\cos\alpha}\frac{1}{\beta
D^{\infty}}$ (A.14)

where

$c=\frac{\left(\sum_{j=1}^{N_{0}-1}\eta^{j}A(2N_{0},j)+1\right)\gamma}{\sin\gamma}.$

Thus combing the construction of $M_{0}$, (A.13) and (A.14), we obtain that
for $t\in(t^{*},\tilde{t}]$, the following estimate holds ,

$\displaystyle
D(\Omega)+2N_{0}K\sin\alpha-K\cos\alpha\frac{1}{\sum_{j=1}^{N_{0}-1}\eta^{j}A(2N_{0},j)+1}\frac{\sin\gamma}{\gamma}Q^{0}(t)$
$\displaystyle<D(\Omega)+2N_{0}K\sin\alpha-K\cos\alpha\frac{1}{\sum_{j=1}^{N_{0}-1}\eta^{j}A(2N_{0},j)+1}\frac{\sin\gamma}{\gamma}\beta
D^{\infty}<0.$

Then, we apply the above inequality and integrate on the both sides of (A.11)
from $t^{*}$ to $\tilde{t}$ to get

$\displaystyle Q^{0}(\tilde{t})-M_{0}$
$\displaystyle=Q^{0}(\tilde{t})-Q^{0}(t^{*})$
$\displaystyle\leq\int_{t^{*}}^{\tilde{t}}\left(D(\Omega)+2N_{0}K\sin\alpha-K\cos\alpha\frac{1}{\sum_{j=1}^{N_{0}-1}\eta^{j}A(2N_{0},j)+1}\frac{\sin\gamma}{\gamma}Q^{0}(t)\right)dt<0,$

which obviously contradicts to the fact $Q^{0}(\tilde{t})-M_{0}>0$, and
verifies (A.12).

$\bigstar$ Step 6. Now we are ready to show the contradiction to (A.1), which
implies that $T^{*}=+\infty$. In fact, from the fact that
$\beta<1,D^{\infty}<\zeta$ and $Q^{0}(0)\leq D_{0}(\theta(0))<\zeta$, we see

$Q^{0}(t)\leq M_{0}=\max\left\\{Q^{0}(0),\beta D^{\infty}\right\\}<\zeta,\quad
t\in[0,T^{*}).$

Then we apply the relation $\beta D_{0}(\theta(t))\leq Q^{0}(t)$ given in
Lemma 3.2 and the assumption $\eta>\frac{2}{1-\frac{\zeta}{\gamma}}$ in (3.5)
to obtain that

$D_{0}(\theta(t))\leq\frac{Q^{0}(t)}{\beta}<\frac{\zeta}{\beta}<\gamma,\quad
t\in[0,T^{*})\quad\mbox{where}\ \beta=1-\frac{2}{\eta}.$

As $D_{0}(\theta(t))$ is continuous, we have

$D_{0}(\theta(T^{*}))=\lim_{t\to(T^{*})^{-}}D_{0}(\theta(t))\leq\frac{\zeta}{\beta}<\gamma,$

which contradicts to the situation that $D_{0}(\theta(T^{*}))=\gamma$ in
(A.1). Therefore, we conclude that $T^{*}=+\infty$, which implies that

$D_{0}(\theta(t))<\gamma,\quad\mbox{for all}\ t\in[0,+\infty).$ (A.15)

Then for any finite time $T>0$, we apply (A.15) and repeat the same argument
in the second, third, forth steps to obtain the dynamics of $Q^{0}(t)$ in
(A.11) holds on $[0,T)$. Thus we obtain the following differential inequality
of $Q^{0}$ on the whole time interval:

$\dot{Q}^{0}(t)\leq
D(\Omega)+2N_{0}K\sin\alpha-K\cos\alpha\frac{1}{\sum_{j=1}^{N_{0}-1}\eta^{j}A(2N_{0},j)+1}\frac{\sin\gamma}{\gamma}Q^{0}(t),\
t\in[0,+\infty).$

∎

##  Appendix B proof of step 1 in lemma 4.3

We will show the detailed proof of Step 1 in Lemma 4.3. Now we pick out any
interval $J_{l}$ with $1\leq l\leq r$, where the orders of both
$\\{\bar{\theta}_{i}\\}_{i=0}^{k+1}$ and
$\\{\underline{\theta}_{i}\\}_{i=0}^{k+1}$ are preseved and the order of
oscillators in each subdigraph $\mathcal{G}_{i}$ with $0\leq i\leq k+1$ will
not change in each time interval. Then, we consider four cases depending on
the possibility of relative position between $\bigcup_{i=0}^{k}\mathcal{G}$
and $\mathcal{G}_{k+1}$.

(a) Case 1

(b) Case 2

(c) Case 3

(d) Case 4

Figure 1. The four cases

Figure 1 above shows the four possible relations between
$\bigcup_{i=0}^{k}\mathcal{G}$ and $\mathcal{G}_{k+1}$ at any time $t$. Case
$1$ and Case $4$ are similar and relative simple, while the analysis on Case
$2$ and Case $3$ are similar but much more complicated. Therefore, we will
only show the detailed proof of Case $2$ for simplicity. In this case, we have
from Figure 1 that

$\max_{0\leq i\leq
k+1}\\{\bar{\theta}_{i}\\}=\bar{\theta}_{k+1},\quad\min_{0\leq i\leq
k+1}\\{\underline{\theta}_{i}\\}=\underline{\theta}_{k+1}\quad\mbox{for $t\in
J_{l}$}.$

Without loss of generality, we assume that

$\theta^{k+1}_{1}\leq\theta^{k+1}_{2}\leq\dots\leq\theta^{k+1}_{N_{k+1}},\quad\mbox{for
$t\in J_{l}$}.$

$\bigstar$ Step 1. Similar to (A.7), we claim that for $1\leq n\leq N_{k+1}$,
the following inequalities hold

$\displaystyle\frac{d}{dt}\bar{\theta}_{n}^{k+1}(t)$
$\displaystyle\leq\Omega_{M}+S_{k+1}K\sin\alpha+S_{k}K\cos\alpha
D_{k}(\theta(t))$ (B.1)
$\displaystyle+K\cos\alpha\frac{1}{\bar{a}^{k+1}_{n}+1}\sum_{i=n}^{N_{k+1}}\left(\eta^{i-n}\underset{j\leq
i}{\min_{j\in\mathcal{N}_{i}^{k+1}(k+1)}}\sin(\theta^{k+1}_{j}(t)-\theta^{k+1}_{i}(t))\right).$

where $S_{k}=\sum_{i=0}^{k}N_{i}$. In the following, we will prove the claim
(B.1) via induction principle.

$\bigstar$ Step 1.1. As an initial step, we first verify that (B.1) holds for
$n=N_{k+1}$. In fact, we have

$\displaystyle\frac{d}{dt}\bar{\theta}^{k+1}_{N_{k+1}}$
$\displaystyle=\frac{d}{dt}\theta^{k+1}_{N_{k+1}}$ (B.2)
$\displaystyle=\Omega^{k+1}_{N_{k+1}}+K\cos\alpha\sum_{j\in\mathcal{N}^{k+1}_{N_{k+1}}(k+1)}\sin(\theta^{k+1}_{j}-\theta^{k+1}_{N_{k+1}})$
$\displaystyle+K\cos\alpha\sum_{l=0}^{k}\sum_{j\in\mathcal{N}^{k+1}_{N_{k+1}}(l)}\sin(\theta^{l}_{j}-\theta^{k+1}_{N_{k+1}})+K\sin\alpha\sum_{l=0}^{k+1}\sum_{j\in\mathcal{N}^{k+1}_{N_{k+1}}(l)}\cos(\theta^{l}_{j}-\theta^{k+1}_{N_{k+1}})$
$\displaystyle\leq\Omega_{M}+S_{k+1}K\sin\alpha$
$\displaystyle+K\cos\alpha\underbrace{\sum_{j\in\mathcal{N}^{k+1}_{N_{k+1}}(k+1)}\sin(\theta^{k+1}_{j}-\theta^{k+1}_{N_{k+1}})}_{\mathcal{I}_{11}}+K\cos\alpha\underbrace{\sum_{l=0}^{k}\sum_{j\in\mathcal{N}^{k+1}_{N_{k+1}}(l)}\sin(\theta^{l}_{j}-\theta^{k+1}_{N_{k+1}})}_{\mathcal{I}_{12}},$

where we use

$|\sum_{l=0}^{k+1}\sum_{j\in\mathcal{N}^{k+1}_{N_{k+1}}(l)}\cos(\theta^{l}_{j}-\theta^{k+1}_{N_{k+1}})|\leq\sum_{l=0}^{k+1}N_{l}=S_{k+1}.$

$\diamond$ Estimates on $\mathbf{\mathcal{I}_{11}}$ in (B.2). We know that
$\theta^{k+1}_{N_{k+1}}$ is the largest phase among $\mathcal{G}_{k+1}$, and
all the oscillators in $\bigcup_{i=0}^{k+1}\mathcal{G}_{i}$ are confined in
half circle before $T^{*}$. Therefore, it is clear that

$\sin(\theta^{k+1}_{j}-\theta^{k+1}_{N_{k+1}})\leq 0,\quad\mbox{for}\
j\in\mathcal{N}_{N_{k+1}}^{k+1}(k+1).$

Then we immediately have

$\mathcal{I}_{11}=\sum_{j\in\mathcal{N}_{N_{k+1}}^{k+1}(k+1)}\sin(\theta^{k+1}_{j}-\theta^{k+1}_{N_{k+1}})\leq\min_{j\in\mathcal{N}^{k+1}_{N_{k+1}}(k+1)}\sin(\theta^{k+1}_{j}-\theta^{k+1}_{N_{k+1}}).$
(B.3)

$\diamond$ Estimates on $\mathbf{\mathcal{I}_{12}}$ in (B.2). For
$\theta^{l}_{j}$ which is the neighbor of $\theta^{k+1}_{N_{k+1}}$ in
$\mathcal{G}_{l}$ with $0\leq l\leq k$, i.e.,
$j\in\mathcal{N}_{N_{k+1}}^{k+1}(l)$, we consider two possible orderings
between $\theta^{l}_{j}$ and $\theta^{k+1}_{N_{k+1}}$:

If $\theta^{l}_{j}\leq\theta^{k+1}_{N_{k+1}}$, we immediately have

$\sin(\theta^{l}_{j}-\theta^{k+1}_{N_{k+1}})\leq 0.$

If $\theta^{l}_{j}>\theta^{k+1}_{N_{k+1}}$, from the fact that

$\theta^{i}_{N_{i}}\geq\bar{\theta}_{i}\geq\underline{\theta}_{i}\geq\theta^{i}_{1},\quad
0\leq i\leq d,$ (B.4)

we immediately obtain

$\theta^{k+1}_{N_{k+1}}\geq\bar{\theta}_{k+1}=\max_{0\leq i\leq
k+1}\\{\bar{\theta}_{i}\\}\geq\max_{0\leq i\leq
k}\\{\bar{\theta}_{i}\\}\geq\min_{0\leq i\leq
k}\\{\underline{\theta}_{i}\\}\geq\min_{0\leq i\leq k}\min_{1\leq j\leq
N_{i}}\\{\theta^{i}_{j}\\}.$ (B.5)

Thus we use the property of $\sin x\leq x,\ x\geq 0$ and (B.5) to get

$\sin(\theta^{l}_{j}-\theta^{k+1}_{N_{k+1}})\leq\theta^{l}_{j}-\theta^{k+1}_{N_{k+1}}\leq\theta^{l}_{j}-\min_{0\leq
i\leq k}\min_{1\leq j\leq N_{i}}\\{\theta^{i}_{j}\\}\leq D_{k}(\theta(t)).$

Therefore, combining the above discussion, we have

$\mathcal{I}_{12}=\sum_{l=0}^{k}\sum_{j\in\mathcal{N}^{k+1}_{N_{k+1}}(l)}\sin(\theta^{l}_{j}-\theta^{k+1}_{N_{k+1}})\leq
S_{k}D_{k}(\theta(t))$ (B.6)

From (B.2), (B.3) and (B.6), it yields that (B.1) holds for $n=N_{k+1}$.

$\bigstar$ Step 1.2. Next, we assume that (B.1) holds for $n$ with $2\leq
n\leq N_{k+1}$, and we will show that (B.1) holds for $n-1$. Following the
process $\mathcal{A}_{1}$ and similar analysis in (B.2), we have

$\displaystyle\dot{\bar{\theta}}^{k+1}_{n-1}$
$\displaystyle\leq\Omega_{M}+\frac{\bar{a}^{k+1}_{n-1}}{\bar{a}^{k+1}_{n-1}+1}S_{k+1}K\sin\alpha+\frac{\bar{a}^{k+1}_{n-1}}{\bar{a}^{k+1}_{n-1}+1}S_{k}K\cos\alpha
D_{k}(\theta(t))+K\sin\alpha\frac{1}{\bar{a}^{k+1}_{n-1}+1}S_{k+1}$ (B.7)
$\displaystyle+K\cos\alpha\frac{1}{\bar{a}^{k+1}_{n-1}+1}\eta(N_{k+1}-n+2+S_{k})\sum_{i=n}^{N_{k+1}}\left(\eta^{i-n}\underset{j\leq
i}{\min_{j\in\mathcal{N}_{i}^{k+1}(k+1)}}\sin(\theta^{k+1}_{j}-\theta^{k+1}_{i})\right)$
$\displaystyle+K\cos\alpha\frac{1}{\bar{a}^{k+1}_{n-1}+1}\underset{j\leq
n-1}{\min_{j\in\mathcal{N}_{n-1}^{k+1}(k+1)}}\sin(\theta^{k+1}_{j}-\theta^{k+1}_{n-1})$
$\displaystyle+K\cos\alpha\frac{1}{\bar{a}^{k+1}_{n-1}+1}\underbrace{\underset{j>n-1}{\sum_{j\in\mathcal{N}_{n-1}^{k+1}(k+1)}}\sin(\theta^{k+1}_{j}-\theta^{k+1}_{n-1})}_{\mathcal{I}_{21}}$
$\displaystyle+K\cos\alpha\frac{1}{\bar{a}^{k+1}_{n-1}+1}\underbrace{\sum_{l=0}^{k}\sum_{j\in\mathcal{N}_{n-1}^{k+1}(l)}\sin(\theta^{l}_{j}-\theta^{k+1}_{n-1})}_{\mathcal{I}_{22}}.$

Next we do some estimates about the terms $\mathcal{I}_{21}$ and
$\mathcal{I}_{22}$ in (B.7) seperately.

$\diamond$ Estimates on $\mathbf{\mathcal{I}_{21}}$ in (B.7). Without loss of
generality, we only deal with $\mathcal{I}_{21}$ under the situation
$\gamma>\frac{\pi}{2}$. We first apply the strong connectivity of
$\mathcal{G}_{k+1}$ and Lemma 3.1 to obtain that

$\sum_{i=n}^{N_{k+1}}\left(\eta^{i-n}\underset{j\leq
i}{\min_{j\in\mathcal{N}_{i}^{k+1}(k+1)}}\sin(\theta^{k+1}_{j}(t)-\theta^{k+1}_{i}(t))\right)\leq\sin(\theta^{k+1}_{\bar{k}_{n}}-\theta^{k+1}_{N_{k+1}}),$
(B.8)

where
$\bar{k}_{n}=\min_{j\in\bigcup_{i=n}^{N_{k+1}}\mathcal{N}^{k+1}_{i}(k+1)}j\leq
n-1$. According to (B.8), we consider two cases depending on comparison
between $\theta^{k+1}_{N_{k+1}}-\theta^{k+1}_{\bar{k}_{n}}$ and
$\frac{\pi}{2}$.
(i) For the first case that
$0\leq\theta^{k+1}_{N_{k+1}}-\theta^{k+1}_{\bar{k}_{n}}\leq\frac{\pi}{2}$, we
immediately obtain that for $j\in\mathcal{N}^{k+1}_{n-1}(k+1),\ j>n-1$,

$0\leq\theta^{k+1}_{j}(t)-\theta^{k+1}_{n-1}(t)\leq\theta^{k+1}_{N_{k+1}}(t)-\theta^{k+1}_{n-1}(t)\leq\theta^{k+1}_{N_{k+1}}(t)-\theta^{k+1}_{\bar{k}_{n}}(t)\leq\frac{\pi}{2}.$
(B.9)

Then it yieldst from (B.8), (B.9) and $\eta>2$ that

$\displaystyle\eta(N_{k+1}-n+1)\sum_{i=n}^{N_{k+1}}\left(\eta^{i-n}\underset{j\leq
i}{\min_{j\in\mathcal{N}_{i}^{k+1}(k+1)}}\sin(\theta^{k+1}_{j}(t)-\theta^{k+1}_{i}(t))\right)+\mathcal{I}_{21}$
$\displaystyle\leq\eta(N_{k+1}-n+1)\sin(\theta^{k+1}_{\bar{k}_{n}}-\theta^{k+1}_{N_{k+1}})+\underset{j>n-1}{\sum_{j\in\mathcal{N}_{n-1}^{k+1}(k+1)}}\sin(\theta^{k+1}_{j}-\theta^{k+1}_{n-1})$
$\displaystyle\leq(N_{k+1}-n+1)\sin(\theta^{k+1}_{\bar{k}_{n}}-\theta^{k+1}_{N_{k+1}})+(N_{k+1}-n+1)\sin(\theta^{k+1}_{N_{k+1}}-\theta^{k+1}_{n-1})$
$\displaystyle\leq 0.$

(ii) For the second case that
$\frac{\pi}{2}<\theta^{k+1}_{N_{k+1}}-\theta^{k+1}_{\bar{k}_{n}}<\gamma$, it
is known that

$\eta>\frac{1}{\sin\gamma}\quad\mbox{and}\quad\sin(\theta^{k+1}_{N_{k+1}}-\theta^{k+1}_{\bar{k}_{n}})>\sin\gamma,$
(B.10)

which yields
$\eta\sin(\theta^{k+1}_{\bar{k}_{n}}-\theta^{k+1}_{N_{k+1}})\leq-1$. Thus we
immediately derive that

$\displaystyle\eta(N_{k+1}-n+1)\sum_{i=n}^{N_{k+1}}\left(\eta^{i-n}\underset{j\leq
i}{\min_{j\in\mathcal{N}_{i}^{k+1}(k+1)}}\sin(\theta^{k+1}_{j}(t)-\theta^{k+1}_{i}(t))\right)+\mathcal{I}_{21}$
$\displaystyle\leq\eta(N_{k+1}-n+1)\sin(\theta^{k+1}_{\bar{k}_{n}}-\theta^{k+1}_{N_{k+1}})+\underset{j>n-1}{\sum_{j\in\mathcal{N}_{n-1}^{k+1}(k+1)}}\sin(\theta^{k+1}_{j}-\theta^{k+1}_{n-1})$
$\displaystyle\leq-(N_{k+1}-n+1)+(N_{k+1}-n+1)=0.$

Then, we combine the above arguments in (i) and (ii) to obtain

$\eta(N_{k+1}-n+1)\sum_{i=n}^{N_{k+1}}\left(\eta^{i-n}\underset{j\leq
i}{\min_{j\in\mathcal{N}_{i}^{k+1}(k+1)}}\sin(\theta^{k+1}_{j}-\theta^{k+1}_{i})\right)+\mathcal{I}_{21}\leq
0.$ (B.11)

$\diamond$ Estimates on $\mathbf{\mathcal{I}_{22}}$ in (B.7). For the term
$\mathcal{I}_{22}$, there are three possible relations between
$\theta^{k+1}_{n-1}$ and $\theta^{l}_{j}$ with $0\leq l\leq k$:

(i) If $\theta^{l}_{j}\leq\theta^{k+1}_{n-1}$, we immediately have
$\sin(\theta^{l}_{j}-\theta^{k+1}_{n-1})\leq 0$.

(ii) If $\theta^{k+1}_{n-1}<\theta^{l}_{j}\leq\theta^{k+1}_{N_{k+1}}$, we
consider two cases separately:

(a) For the case that
$0\leq\theta^{k+1}_{N_{k+1}}-\theta^{k+1}_{\bar{k}_{n}}\leq\frac{\pi}{2}$, it
is clear that

$0\leq\theta^{l}_{j}-\theta^{k+1}_{n-1}\leq\theta^{k+1}_{N_{k+1}}-\theta^{k+1}_{n-1}\leq\theta^{k+1}_{N_{k+1}}-\theta^{k+1}_{\bar{k}_{n}}\leq\frac{\pi}{2}.$

Thus from the above inequality and (B.8), we have

$\displaystyle\eta\sum_{i=n}^{N_{k+1}}\left(\eta^{i-n}\underset{j\leq
i}{\min_{j\in\mathcal{N}_{i}^{k+1}(k+1)}}\sin(\theta^{k+1}_{j}(t)-\theta^{k+1}_{i}(t))\right)+\sin(\theta^{l}_{j}-\theta^{k+1}_{n-1})$
$\displaystyle\leq\eta\sin(\theta^{k+1}_{\bar{k}_{n}}-\theta^{k+1}_{N_{k+1}})+\sin(\theta^{l}_{j}-\theta^{k+1}_{n-1})$
$\displaystyle\leq\sin(\theta^{k+1}_{\bar{k}_{n}}-\theta^{k+1}_{N_{k+1}})+\sin(\theta^{k+1}_{N_{k+1}}-\theta^{k+1}_{\bar{k}_{n}})=0.$

(b) For another case that
$\frac{\pi}{2}<\theta^{k+1}_{N_{k+1}}-\theta^{k+1}_{\bar{k}_{n}}<\gamma$, it
is yields from (B.10) that

$\displaystyle\eta\sum_{i=n}^{N_{k+1}}\left(\eta^{i-n}\underset{j\leq
i}{\min_{j\in\mathcal{N}_{i}^{k+1}(k+1)}}\sin(\theta^{k+1}_{j}(t)-\theta^{k+1}_{i}(t))\right)+\sin(\theta^{l}_{j}-\theta^{k+1}_{n-1})$
$\displaystyle\leq\eta\sin(\theta^{k+1}_{\bar{k}_{n}}-\theta^{k+1}_{N_{k+1}})+\sin(\theta^{l}_{j}-\theta^{k+1}_{n-1})$
$\displaystyle\leq-1+1=0$

Hence, combining the above arguments in (a) and (b), we obtain that

$\eta\sum_{i=n}^{N_{k+1}}\left(\eta^{i-n}\underset{j\leq
i}{\min_{j\in\mathcal{N}_{i}^{k+1}(k+1)}}\sin(\theta^{k+1}_{j}(t)-\theta^{k+1}_{i}(t))\right)+\sin(\theta^{l}_{j}-\theta^{k+1}_{n-1})\leq
0.$

(iii) If $\theta^{l}_{j}>\theta^{k+1}_{N_{k+1}}$, we exploit the concave
property of sine function in $[0,\pi]$ to get

$\sin(\theta^{l}_{j}-\theta^{k+1}_{n-1})\leq\sin(\theta^{l}_{j}-\theta^{k+1}_{N_{k+1}})+\sin(\theta^{k+1}_{N_{k+1}}-\theta^{k+1}_{n-1}).$
(B.12)

For the second part on the right-hand side of above inequality (B.12), we
apply the same analysis in (ii) to obtain

$\eta\sum_{i=n}^{N_{k+1}}\left(\eta^{i-n}\underset{j\leq
i}{\min_{j\in\mathcal{N}_{i}^{k+1}(k+1)}}\sin(\theta^{k+1}_{j}-\theta^{k+1}_{i})\right)+\sin(\theta^{k+1}_{N_{k+1}}-\theta^{k+1}_{n-1})\leq
0.$

For the first part on the right-hand side of (B.12), the calculation is the
same as (B.6), thus we have

$\sin(\theta^{l}_{j}-\theta^{k+1}_{N_{k+1}})\leq\theta^{l}_{j}-\theta^{k+1}_{N_{k+1}}\leq\theta^{l}_{j}-\min_{0\leq
i\leq k}\min_{1\leq j\leq N_{i}}\\{\theta^{i}_{j}\\}\leq D_{k}(\theta(t)).$

Therefore, we combine the above estimates to obtain

$\displaystyle\eta S_{k}\sum_{i=n}^{N_{k+1}}\left(\eta^{i-n}\underset{j\leq
i}{\min_{j\in\mathcal{N}_{i}^{k+1}(k+1)}}\sin(\theta^{k+1}_{j}(t)-\theta^{k+1}_{i}(t))\right)+\mathcal{I}_{22}$
(B.13) $\displaystyle\leq\eta
S_{k}\sin(\theta^{k+1}_{\bar{k}_{n}}-\theta^{k+1}_{N_{k+1}})+\sum_{l=0}^{k}\sum_{j\in\mathcal{N}_{n-1}^{k+1}(l)}\sin(\theta^{l}_{j}-\theta^{k+1}_{n-1})$
$\displaystyle\leq S_{k}D_{k}(\theta(t)).$

Then from (B.7), (B.11), and (B.13) , it yields that

$\displaystyle\frac{d}{dt}\bar{\theta}^{k+1}_{n-1}$
$\displaystyle\leq\Omega_{M}+S_{k+1}K\sin\alpha+\frac{\bar{a}^{k+1}_{n-1}}{\bar{a}^{k+1}_{n-1}+1}S_{k}K\cos\alpha
D_{k}(\theta(t))++K\cos\alpha\frac{1}{\bar{a}^{k+1}_{n-1}+1}S_{k}D_{k}(\theta(t))$
$\displaystyle+K\cos\alpha\frac{1}{\bar{a}^{k+1}_{n-1}+1}\eta\sum_{i=n}^{N_{k+1}}\left(\eta^{i-n}\underset{j\leq
i}{\min_{j\in\mathcal{N}_{i}^{k+1}(k+1)}}\sin(\theta^{k+1}_{j}-\theta^{k+1}_{i})\right)$
$\displaystyle+K\cos\alpha\frac{1}{\bar{a}^{k+1}_{n-1}+1}\underset{j\leq
n-1}{\min_{j\in\mathcal{N}_{n-1}^{k+1}(k+1)}}\sin(\theta^{k+1}_{j}-\theta^{k+1}_{n-1})$
$\displaystyle\leq\Omega_{M}+S_{k+1}K\sin\alpha+S_{k}K\cos\alpha
D_{k}(\theta(t))$
$\displaystyle+K\cos\alpha\frac{1}{\bar{a}^{k+1}_{n-1}+1}\sum_{i=n-1}^{N_{k+1}}\left(\eta^{i-(n-1)}\underset{j\leq
i}{\min_{j\in\mathcal{N}_{i}^{k+1}(k+1)}}\sin(\theta^{k+1}_{j}-\theta^{k+1}_{i})\right)$

This means that the claim (B.1) does hold for $n-1$. Therefore, we apply the
inductive criteria to verify the claim (B.1).

$\bigstar$ Step 2. Now we are ready to prove (4.10) on $J_{l}$ for Case 2. In
fact, we apply Lemma 3.1 and the strong connectivity of $\mathcal{G}_{k+1}$ to
have

$\sum_{i=1}^{N_{k+1}}\left(\eta^{i-1}\underset{j\leq
i}{\min_{j\in\mathcal{N}_{i}^{k+1}(k+1)}}\sin(\theta^{k+1}_{j}-\theta^{k+1}_{i})\right)\leq\sin(\theta^{k+1}_{1}-\theta^{k+1}_{N_{k+1}})$

From the notations in (4.5) and (4.6), it is known that

$\bar{\theta}_{1}^{k+1}=\bar{\theta}_{k+1},\quad\underline{\theta}_{N_{k+1}}^{k+1}=\underline{\theta}_{k+1}.$

Thus, we exploit the above inequality and set $n=1$ in (B.1) to obtain

$\displaystyle\frac{d}{dt}\bar{\theta}_{k+1}$
$\displaystyle=\frac{d}{dt}\bar{\theta}^{k+1}_{1}$ (B.14)
$\displaystyle\leq\Omega_{M}+S_{k+1}K\sin\alpha+S_{k}K\cos\alpha
D_{k}(\theta(t))$
$\displaystyle+K\cos\alpha\frac{1}{\bar{a}^{k+1}_{1}+1}\sum_{i=1}^{N_{k+1}}\left(\eta^{i-1}\underset{j\leq
i}{\min_{j\in\mathcal{N}_{i}^{k+1}(k+1)}}\sin(\theta^{k+1}_{j}-\theta^{k+1}_{i})\right)$
$\displaystyle\leq\Omega_{M}+S_{k+1}K\sin\alpha+S_{k}K\cos\alpha
D_{k}(\theta(t))+K\cos\alpha\frac{1}{\bar{a}^{k+1}_{1}+1}\sin(\theta^{k+1}_{1}-\theta^{k+1}_{N_{k+1}})$

We further apply the similar arguments in (B.14) to derive the differential
inequality of $\underline{\theta}_{k+1}$ as below

$\frac{d}{dt}\underline{\theta}_{k+1}\geq\Omega_{m}-S_{k+1}K\sin\alpha-
S_{k}K\cos\alpha
D_{k}(\theta(t))+K\cos\alpha\frac{1}{\bar{a}^{k+1}_{1}+1}\sin(\theta^{k+1}_{N_{k+1}}-\theta^{k+1}_{1}).$
(B.15)

Due to the monotone decreasing property of $\frac{\sin x}{x}$ in $(0,\pi]$ and
from (4.13), it is obvious that

$\sin(\theta^{k+1}_{N_{k+1}}-\theta^{k+1}_{1})\geq\frac{\sin\gamma}{\gamma}(\theta^{k+1}_{N_{k+1}}-\theta^{k+1}_{1}).$

Then we combine the above inequality, (B.14), (B.15) and (4.4) to get

$\displaystyle\dot{Q}^{k+1}(t)$
$\displaystyle=\frac{d}{dt}(\bar{\theta}_{k+1}-\underline{\theta}_{k+1})$
$\displaystyle\leq D(\Omega)+2S_{k+1}K\sin\alpha+2S_{k}K\cos\alpha
D_{k}(\theta(t))$
$\displaystyle-K\cos\alpha\frac{2}{\bar{a}^{k+1}_{1}+1}\sin(\theta^{k+1}_{N_{k+1}}-\theta^{k+1}_{1})$
$\displaystyle\leq D(\Omega)+2S_{k+1}K\sin\alpha+2S_{k}K\cos\alpha
D_{k}(\theta(t))$
$\displaystyle-K\cos\alpha\frac{1}{\bar{a}^{k+1}_{1}+1}\frac{\sin\gamma}{\gamma}(\theta^{k+1}_{N_{k+1}}-\theta^{k+1}_{1})$
$\displaystyle\leq D(\Omega)+2NK\sin\alpha+(2N+1)K\cos\alpha D_{k}(\theta(t))$
$\displaystyle-K\cos\alpha\frac{1}{\sum_{j=1}^{N-1}\eta^{j}A(2N,j)+1}\frac{\sin\gamma}{\gamma}Q^{k+1}(t),\qquad
t\in J_{l},$

where we use the fact that
$Q^{k+1}(t)\leq\theta^{k+1}_{N_{k+1}}(t)-\theta^{k+1}_{1}(t)$ and (4.4).
Eventually, for Case 2, we obtain the dynamics for $Q^{k+1}(t)$ in (4.10) on
$J_{l}$, i.e.,

$\displaystyle\dot{Q}^{k+1}(t)$ $\displaystyle\leq
D(\Omega)+2NK\sin\alpha+(2N+1)K\cos\alpha
D_{k}(\theta(t))-\frac{K\cos\alpha}{c}Q^{k+1}(t),\ t\in[0,T^{*}).\newline $

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