Synchronization Patterns in Networks of Kuramoto Oscillators:
A Geometric Approach for Analysis and Control

arXiv:1709.06193v1 [math.OC] 18 Sep 2017

Lorenzo Tiberi, Chiara Favaretto, Mario Innocenti, Danielle S. Bassett, and Fabio Pasqualetti
Abstract— Synchronization is crucial for the correct functionality of many natural and man-made complex systems. In
this work we characterize the formation of synchronization
patterns in networks of Kuramoto oscillators. Specifically, we
reveal conditions on the network weights and structure and on
the oscillators’ natural frequencies that allow the phases of a
group of oscillators to evolve cohesively, yet independently from
the phases of oscillators in different clusters. Our conditions
are applicable to general directed and weighted networks of
heterogeneous oscillators. Surprisingly, although the oscillators
exhibit nonlinear dynamics, our approach relies entirely on
tools from linear algebra and graph theory. Further, we develop
a control mechanism to determine the smallest (as measured
by the Frobenius norm) network perturbation to ensure the
formation of a desired synchronization pattern. Our procedure
allows us to constrain the set of edges that can be modified, thus
enforcing the sparsity structure of the network perturbation.
The results are validated through a set of numerical examples.

I. I NTRODUCTION
Synchronization of coupled oscillators is everywhere in
nature [1], [2], [3] and in several man-made systems, including power grids [4] and computer networks [5]. While some
systems require complete synchronization among all the parts
to function properly [6], [7], others rely on cluster or partial
synchronization [8], where subsets of nodes exhibit coherent
behaviors that remain independent from the evolution of
other oscillators in the network. For example, while partial
synchronization patterns have been observed in healthy individuals [9], complete synchronization in neural systems is
associated with degenerative diseases including Parkinson’s
and Huntington’s diseases [10], [11], and epilepsy [12].
Cluster synchronization has received attention only recently,
and several fundamental questions remain unanswered, including the characterization of the network features enabling
the formation of a desired pattern, and the development of
control mechanisms to enforce the emergence of clusters.
In this paper we focus on networks of Kuramoto oscillators
[13], and we characterize intrinsic and topological conditions
that ensure the formation of desired clusters of oscillators.
This material is based upon work supported by NSF award
#BCS-1430279 and ARO award 71603NSYIP. Lorenzo Tiberi and
Fabio Pasqualetti are with the Mechanical Engineering Department,
University
of
California
at
Riverside,
ltiberi@ucr.edu,
fabiopas@engr.ucr.edu.
Chiara
Favaretto
is
with
the
Department of Information Engineering, University of Padova,
chiara.favaretto.2@phd.unipd.it. Danielle S. Bassett is
with the Departments of Bioengineering and Electrical and Systems
Engineering, University of Pennsylvania, dsb@seas.upenn.edu. Mario
Innocenti is with the Department of Electrical Systems and Automation,
University of Pisa, mario.innocenti@unipi.it.

Our network model is motivated by a large body of literature showing broad applicability of the Kuramoto model to
virtually all systems that exhibit synchronization properties.
Although Kuramoto networks exhibit nonlinear dynamics,
we adopt tools from linear algebra and graph theory to
characterize network conditions enabling the formation of
a given synchronization pattern. Further, we design a control
mechanism to perturb (a subset of) the network weights so
as to enforce or prevent desired synchronization patterns.
Related work Complete synchronization in networks of
Kuramoto oscillators has been extensively studied, e.g.,
see [14], [15]. It has been shown that synchronization of
all nodes emerges when the coupling strength among the
agents is sufficiently larger than the heterogeneity of the
oscillators’ natural frequencies. Partial synchronization and
pattern formation have received considerably less attention,
with the literature being composed of only few recent works.
In [16] it is shown how symmetry of the interconnections
may lead to partial synchronization. Methods based on graph
symmetry have also been used to find all possible clusters
in networks of Laplacian-coupled oscillators [17]. The relationship between clusterization and network topology has
been studied in [18] for unweighted interconnections. In [19],
the emergence and the stability of groups of synchronized
agents within a network has been studied for different
classes of dynamics, like delay-coupled laser models and
neuronal spiking models. Here, the approach of master
stability function has been used to characterize the results.
In [20], [21] the idea is put forth to study an approximate
notion of cluster synchronization in Kuramoto networks
via tools from linear systems theory. It is quantitatively
shown how cluster synchronization depends on strong intracluster and weak inter-cluster connections, similarity with
respect to the natural frequencies of the oscillators within
each cluster, and heterogeneity of the natural frequencies of
coupled oscillators belonging to different subnetworks. With
respect to this work, we focus on an exact notion of cluster
synchronization, identify necessary and sufficient conditions
on the network weights and oscillators’ natural frequencies
for the emergence of a desired synchronization pattern, and
exploit our analysis to design a structural control algorithm
for the formation of a desired synchronization pattern.
The work that is closer to this paper is [22], where the
authors relate cluster synchronization to the notion of an
external equitable partition in a graph. In fact, the notion
of an external equitable partition can be interpreted in terms
of invariant subspaces of the network adjacency matrix, a
notion that we exploit in our development. However, the

analysis in [22] is carried out with unweighted and undirected
networks and, as we show in this paper, the conditions in
[22] may not be necessary when dealing with directed and
weighted networks. Further, our approach relies on simple
notions from linear algebra, and leads to the development of
our control algorithm for the formation of desired patterns.
Paper contributions The contributions of this paper are
twofold. First, we consider a notion of exact cluster synchronization, where the phases of the oscillators within each
cluster remain equal to each other over time, and different
from the phases of the oscillators in different clusters. We
derive necessary and sufficient conditions for the formation
of a given synchronization pattern in directed and weighted
networks of Kuramoto oscillators. In particular we show
that cluster synchronization is possible if and only if (i)
the natural frequencies are equal within each cluster, and
(ii) for each cluster, the sum of the weights of the edges
from every separate group is the same for all nodes in
the cluster. Second, we leverage our characterization of
cluster synchronization to develop a control mechanism that
modifies the network weights so as to ensure the formation
of a desired synchronization pattern. Our control method
is optimal, in the sense that it determines the smallest
(measured by the Frobenius norm) network perturbation
for a given synchronization pattern, and it guarantees the
modification of only a desired subset of the edge weights.
Paper organization The rest of this paper is organized as
follows. Section II contains the problem setup and some
preliminary definitions. Section III contains our characterization of cluster synchronization, and Section IV contains
our structural control algorithm for the formation of a desired
synchronization pattern. Section V concludes the paper.
II. P ROBLEM SETUP AND PRELIMINARY NOTIONS
Consider a network of heterogenous Kuramoto oscillators
described by the digraph G = (V, E), where V = {1, . . . , n}
denotes the set of oscillators and E ⊆ V × V their interconnections. Let A = [aij ] be the weighted adjacency matrix of
G, where aij ∈ R if (i, j) ∈ E and aij = 0 otherwise. We
assume that G is strongly connected [23]. Let θi ∈ R denote
the phase of the i-th oscillator, whose dynamics reads as
θ̇i = ωi +

n
X

aij sin(θj − θi ),

j=1

where ωi is the natural frequency of the i-th oscillator. The
dynamics is a generalized version of the classic Kuramoto
model [24]. Depending on the interconnection graph G, the
adjacency matrix A, and the oscillators natural frequencies,
different oscillatory patterns are possible corresponding to
(partially) synchronized or chaotic states [25]. In this work
we are particularly interested in the case where the phases
of groups of oscillators evolve cohesively within each group,
yet independently from the phases of oscillators in different
groups. To formalize this discussion, let P = {P1 , . . . , Pm }
be a partition of V, that is, V = ∪m
i=1 Pi and Pi ∩ Pj = ∅ for
all i, j ∈ {1, . . . , m} with i 6= j. We restrict our attention
to the case m > 1. Throughout the paper we will assume

10

1

2

9
4

2

6
5

7

10

2

5

3

5

9

Fig. 1. A network of oscillators with partitions P1 = {1, 2, 3} and P2 =
{4, 5, 6}. The sum of the weights of all edges (i, j) is equal for each node
i of P1 (resp. P2 ), with j ∈ P2 (resp. j ∈ P1 ). In Section III we show that
this is a necessary condition for phase synchronization of the partition P.

without loss of generality that, given P = {P
, . . . , Pm },
P1 i−1
the oscillators are labeled so that Pi = { j=1 |Pj | +
Pi
1, . . . , j=1 |Pj |}, where |Pj | denotes the cardinality of the
set Pj . While different notions of synchronization exist, we
will use the following definitions.
Definition 1: (Phase synchronization) For the network of
oscillators G = (V, E), the partition P = {P1 , . . . , Pm }
is phase synchronizable if, for some initial phases
θ1 (0), . . . , θn (0), it holds
θi (t) = θj (t),
for all times t ∈ R≥0 and i, j ∈ Pk , with k ∈ {1, . . . , m}.
Definition 2: (Frequency synchronization) For the network of oscillators G = (V, E), the partition P =
{P1 , . . . , Pm } is frequency synchronizable if, for some initial
phases θ1 (0), . . . , θn (0), it holds
θ̇i (t) = θ̇j (t),
for all times t ∈ R≥0 and i, j ∈ Pk , with k ∈ {1, . . . , m}.
Clearly, phase synchronization implies frequency synchronization, while the converse statement typically fails to hold.
Finally, we define the characteristic matrix associated with
a partition P of the network nodes, which will be used to
derive our synchronization conditions in Section III.
Definition 3: (Characteristic matrix) For the network of
oscillators G = (V, E) and the partition P = {P1 , . . . , Pm },
the characteristic matrix of P is VP ∈ Rn×m , where

VP = v1

v2

···

1

···
{z


vm ,

and

viT = 0
|

0

···
{z

Pi−1

j=1

|Pj |

0 1
} |

|Pi |

1 0 0 ···
} | P {z
n
j=i+1

|Pj |


0 .
}


We conclude this section with an illustrative example.
Example 1: (Setup and definitions) Consider the network
of Kuramoto oscillators in Fig. 1, with graph G = (V, E),
V = {1, 2, 3, 4, 5, 6} and partition P = {P1 , P2 }. The graph

cases where this assumption is satisfied are presented in
Fig. 2. A special case where Assumption (A1) is not satisfied
is discussed at the end of this section.
Theorem 3.1: (Cluster synchronization) For the network
of oscillators G = (V, E), the partition P = {P1 , . . . , Pm } is
phase synchronizable if and only if the following conditions
are simultaneously satisfied:
P
(i) the network weights satisfy k∈P` aik − ajk = 0 for
every i, j ∈ Pz and z, ` ∈ {1, . . . , m}, with z 6= `;
(a)

(ii) the natural frequencies satisfy ωi = ωj for every
k ∈ {1, . . . , m} and i, j ∈ Pk .
Proof: (If) Let θi = θj for all i, j ∈ Pk , k = 1, . . . , m.
Let i, j ∈ P` , and notice that
X X
aik sin(θk − θi ) − ajk sin(θk − θj )
θ̇i − θ̇j =
z6=` k∈Pz

=

X

sz`

z6=`

(b)
Fig. 2. For the network in Example 1 with natural frequencies ω =
[30, 30, 30, 10, 10, 10]T Fig. (a) shows the frequencies of the oscillators in
the clusters P1 = {1, 2, 3} and P2 = {4, 5, 6} as a function of time. Notice that Assumption A1 is satisfied over the entire time interval. In Fig. (b)
we let the frequencies be more homogeneous, ω = [19, 19, 19, 10, 10, 10]T .
Assumption A1 is satisfied within bounded time intervals, such as [t1 , t2 ].

G and the partition P are described by A and VP



1
0 0 0 0 0 10

1
0 0 0 5 0
5



1
0 0 0 0 10 0 



A=
 , VP = 0
9
0
0
0
0
0



0
0 9 0 0 0
0
0
0 7 2 2 0
0

as follows:

0
0

0
.
1

1
1


III. C ONDITIONS FOR CLUSTER SYNCHRONIZATION
In this section we derive necessary and sufficient conditions ensuring phase (hence frequency) synchronization of a
partition of oscillators. In particular, we show how synchronization of a partition depends both on the interconnection
structure and weights, as well as the oscillators natural
frequencies. We make the following technical assumption.
(A1) For the partition P = {P1 , . . . , Pm } there exists an
ordering of the clusters Pi and an interval of time
[t1 , t2 ], with t2 > t1 , such that for all times t ∈ [t1 , t2 ]:
max θ̇i > max θ̇i > · · · > max θ̇i .
i∈P1

i∈P2

i∈Pm

Assumption (A1) requires the phases of the oscillators
in different clusters to evolve with different frequencies, at
least in some interval of time. This assumption is in fact not
restrictive, as this is typically the case when the oscillators
in different clusters have different natural frequencies. Two

X

aik − ajk = 0,

k∈Pz

where we have used conditions (i) and (ii), and where szl =
sin(θz −θ` ) depends on the clusters z and ` but not on i, j, k.
Thus, when conditions (i) and (ii) are satisfied, θ ∈ Im(VP )
implies θ̇ ∈ Im(VP ), the image of VP . Im(VP ) is invariant
and the network is phase synchronizable (θ(0) ∈ Im(VP )).
(Only if) We first show that condition (i) is necessary for
phase synchronization. Assume that the network is phase
synchronized. Let i, j ∈ P` . At all times is must hold that
X X
0 = θ̈i − θ̈j =
aik cos(θk − θi )(θ̇k − θ̇i )
z6=` k∈Pz

−

X X

ajk cos(θk − θj )(θ̇k − θ̇j )
(1)

z6=` k∈Pz

=

X

cz` vz`

z6=`

X

aik − ajk ,

k∈Pz

|

{z

}

dz

where cz` = cos(θz − θ` ) and vz` = θ̇z − θ̇` depend on the
clusters z and `, but not on i, j, k. From (A1), possibly after
reordering the clusters, in some nontrivial interval we have
max θ̇i > max θ̇i > · · · > max θ̇i .
i∈P1

i∈P2

i∈Pm

Thus, (1) implies that, either dz = 0 for all z (thus
implying condition (i)), or the functions cz` vz` must be
linearly dependent at all times in the interval. Assume by
contradiction that the functions czl vzl are linearly dependent
at all times in the above interval. Then it must hold that
X dn
dz n cz` vz` = 0,
dt
z6=`

n

d
for every nonnegative integer n, where dt
n denotes n-times
differentiation. In other words, not only the functions cz` vz`
must be linearly dependent, but also all their derivatives, at
some times in the above interval. Let d1 6= 0 (if d1 = 0,
simply select the first nonzero coefficient), and i, j 6∈ P1 .
Because of assumption (A1), there exists an integer n such
dn
dn
that d1 dt
n c1` v1`  dz dtn cz` vz` , for all z 6= 1. Thus, the

functions cz` vz` cannot be linearly dependent. We conclude
that statement (i) is necessary for phase synchronization.
We now prove that, when the network is phase synchronized, statement (i) implies statement (ii). This shows that
statement (ii) is necessary for phase synchronization. Let the
network be phase synchronized, and let i, j ∈ P` . We have
X
X
0 = θ̇i − θ̇j = ωi − ωj +
sz`
aik − ajk ,
z6=`

k∈Pz

|

{z

=0

min
∆

s.t.

}

where sz` = sin(θz − θ` ) does not depend on the indices
i, j, k (see above), and where we have used that statement
(i) is necessary for phase synchronization. To conclude,
ωi = ωj , and statement (ii) is also necessary for phase
synchronization.
Remark 1: (Necessity of assumption A1) Consider a network of oscillators with adjacency matrix


0 a12 0
0
a21 0 a23 0 

A=
 0 a32 0 a34  ,
0
0 a43 0
and natural frequencies ωi = ω̄ for all i ∈ {1, . . . , 4}.
Notice that condition (i) in Theorem 3.1 is not satisfied.
Let θ1 (0) = θ2 (0) and θ3 (0) = θ4 (0) = θ1 (0) + π, and
notice that θ̇i = ω̄ at all times and for all i ∈ {1, . . . , 4}
(Assumption A1 is not satisfied). In other words, the partition
P = {P1 , P2 }, with P1 = {1, 2} and P2 = {3, 4} is phase
synchronized, independently of the interconnection weights
among the oscillators. Thus, condition (i) in Theorem may
not be necessary when Assumption A1 is not satisfied. 
Let A B denote the Hadamard product between A and
B [26], and Im(VP )⊥ the orthogonal subspace to Im(VP ).
Corollary 3.2: (Matrix condition for synchronization)
Condition (i) in Theorem 3.1 is equivalent to V̄PT ĀVP = 0,
where V̄P ∈ Rn×(n−m) satisfies Im(V̄P ) = Im(VP )⊥ , and
Ā = A − A

typically not satisfied for arbitrary partitions and interconnection weights. In this section we develop a control mechanism
to modify the oscillators’ interconnection weights so as to
guarantee synchronization of a given partition. Specifically,
we study the following minimization problem:

VP VPT .

(2)

Proof: Let Ā = [āij ] and A = [aij ]. Notice that āij =
aij when i and j belong to different clusters, and āij = 0
when i and j belong to the same cluster. Thus,
(P
/ Pj ,
k∈Pj aik , if i ∈
[ĀVP ]ij =
0,
if i ∈ Pj .
v̄iT x

Select V̄P so that V̄P = [v̄1 · · · v̄n−m ] and
= xr − xs ,
with r, s ∈ P` , for a vector x of compatible dimension. Then,
(P
/ Pj ,
k∈Pj ark − ask , r, s ∈
T
[V̄P ĀVP ]ij =
0,
r, s ∈ Pj ,
where r, s are the nonzero indices of v̄i .
IV. C ONTROL OF CLUSTER SYNCHRONIZATION
In the previous section we derive conditions on the network of oscillators to guarantee phase and frequency synchronization. These conditions are rather stringent, and are

k∆k2F


V̄PT Ā + ∆ VP = 0

(3b)

∆∈H

(3c)

(3a)

where k∆kF denotes the Frobenius norm of the matrix ∆, Ā
is as in (2), and H encodes a desired sparsity pattern of the
perturbation matrix ∆. For example, H may represent the set
of matrices compatible with the graph G = (V, E), that is,
H = {M : M ∈ R|V|×|V| and mij = 0 if (i, j) 6∈ E}. The
constraint (3b) reflects the invariance condition in Corollary
3.2 and, together with condition (ii) in Theorem 3.1, ensures
synchronization of the partition P. Thus, the minimization
problem (3) determines the smallest perturbation of the
interconnection weights that guarantees synchronization of
a partition P and satisfies desired sparsity constraints. It
should be observed that, given the solution ∆∗ to (3), the
modified adjacency matrix is A + ∆∗ even if the constraint
(3b) is expressed in terms of Ā. This follows from the
fact that connections among nodes of the same cluster do
not affect the synchronization properties of the partition
P = {P1 , . . . , Pm } (see Corollary 3.2).
To solve the minimization problem (3), we define the
following minimization problem by including the sparsity
constraints (3c) into the cost function:
min
∆

s.t.

k∆

T

Hk2F

(4a)


V̄P Ā + ∆ VP = 0

(4b)

where denotes elementwise division, and H satisfies hij =
1 if there exists a matrix M ∈ H such that mij 6= 0, and
hij = 0 otherwise. Clearly, the minimization problems (3)
and (4) are equivalent, in the sense that ∆∗ is a (feasible)
solution to (3) if and only if it has finite cost in (4).
Theorem 4.1: (Synchronization via structured perturbation) Let T = [VP V̄P ], and let


Ã11 Ã12
= T −1 ĀT.
Ã21 Ã22
The minimization problem (3) has a solution if and only if
there exists a matrix Λ satisfying
X = (V̄P ΛVPT )

H, and Ã21 = V̄PT XVP .

Moreover, if it exists, a solution ∆∗ to (3) is
 ∗

˜
˜∗
∆
∆
∗
−1
11
12
∆ = T ˜∗
˜∗ T ,
∆21 ∆
22
˜ ∗ = −V T XVP , ∆
˜ ∗ = −V T X V̄P , ∆
˜ ∗ = −Ã21 ,
where ∆
11
12
21
P
P
∗
T
˜ = −V̄ X V̄P .
and ∆
22
P
Proof: We adopt the method of Lagrange multipliers
to derive the optimality conditions for the problem (4). The

Lagrangian is
L(∆, Λ) =

n X
n
X

2 −1
δij
hij +

i=1 j=1

m
X

T
λT
i V̄P (Ā + ∆)vi ,

i=1

where Λ = [λ1 , . . . , λm ] ∈ R(n−m)×m is a matrix collecting
vectors of Lagrange multipliers, and vi ∈ Rn is the i-th
column of VP . By equating the partial derivatives of L to
zero we obtain the following optimality conditions:
∂L
= 0 ⇒ V̄P (Ā + ∆)vi = 0,
∂λi
m
X
∂L
T
= 0 ⇒ 2δij h−1
+
λT
k v̄i vjk = 0,
ij
∂δij

(5a)

∆∗ = −V̄P V̄PT ĀVP VPT .
Proof: Because hij = 1 for all i and j, the optimality
condition (6b) becomes

(5b)

∆ + V̄P ΛVPT = 0.

k=1

where v̄i is the i-th row of V̄P and vjk is the entry (j, k) of
the matrix VP . Finally, (5a) and (5b) can be rewritten as
V̄PT (Ā + ∆)VP = 0,

(6a)

H + V̄P ΛVPT = 0,

(6b)

∆

where the factor 2 of (5b) has been included into the
Lagrange multipliers. Applying the change of coordinates
˜ −1 , with Id the
T = [VP V̄P ], Ā = T ÃT −1 and ∆ = T ∆T
identity matrix of dimension d, equation (6a) becomes
˜ −1 VP =
V̄PT T (Ã + ∆)T

˜ 11

 Ã11 + ∆
0 In−m
˜ 21
Ã21 + ∆

˜ 12
Ã12 + ∆
˜ 22
Ã22 + ∆

˜ ∗ = −Ã21 . Equation (6b) is equivalent to
which leads to ∆
21
T
∆ + (V̄P ΛVP ) H = 0, which can be decomposed as

 
˜ 11 ∆
˜ 12 V T

 ∆
P +(V̄ ΛV T )
VP V̄P ˜
H = 0.
P
P
˜
V̄PT
| {z } ∆
21 ∆22
{z
}
|
T
T −1

Consequently,
(V̄P ΛVPT )

H = 0.

We now pre- and post-multiply both sides of the above
equality by V̄PT and VP , respectively, and obtain
Λ = V̄PT ĀVP , ∆∗ = −V̄P V̄PT ĀVP VPT ,
where we have used (6a), VPT VP = I, and V̄PT V̄P = I.
We now present an example where we modify the network
weights to ensure synchronization of a desired partition.
Example 2: (Enforcing synchronization of a partition)
Consider the network in Fig. 3(a). The dashed edges and the
solid edges represent constrained and uncostrained edges,
respectively. The corresponding matrices Ā and H read as


 
Im
= 0,
0

˜ 11 VPT − V̄P Ã12 VPT + VP ∆
˜ 12 V̄PT + V̄P ∆
˜ 22 V̄PT )+
(VP ∆

Theorem 4.1 characterizes the smallest (measured by the
Frobenius norm) structured network perturbation that ensures
synchronization of a given partition. Without constraints, the
optimal perturbation has a straightforward expression.
Corollary 4.2: (Unconstrained minimization problem)
Let H = {M : mij 6= 0 for all i and j}. The minimization
problem (3) is always feasible, and its solution is

(7)

Let X = (V̄P ΛVPT ) H. Recall that V̄PT V̄P = In−m ,
= Im , and V̄PT VP = 0. By pre-multiplicating equation
(7) by V̄PT and post-multiplicating it by VP , we obtain
VPT VP

−Ã21 + V̄PT XVP = 0,
which is a system of linear equations, that can be solved with
respect to the unknown Λ. Following the same reasoning of
above, we can obtain the following other three equations that
˜ 11 , ∆
˜ 12 , and ∆
˜ 22 :
entirely determine the solution ∆
˜ 11 + V T XVP = 0, ∆
˜ 12 + V T X V̄P = 0, and
∆
P
P
˜ 22 + V̄PT X V̄P = 0.
∆
Finally, the optimal matrix ∆∗ , solution to the problem (4),
is given in original coordinates as
 ∗

˜
˜∗
∆
∆
∗
−1
11
12
∆ =T
˜∗ T .
−Ã21 ∆
22

0
0

0
Ā = 
9
0
0

0
0
0
0
9
7

0
0
0
0
0
2

0
5
0
0
0
0

0
0
10
0
0
0



12
0

1
5


0
1
,H = 

0
0
1
0
0
1

1
0
1
1
1
0

1
1
0
1
1
0

0
0
0
0
1
1

0
1
1
1
0
1


0
0

1
.
1


1
0

Notice that H allows only a subset of interconnections to be
modified, specifically, those corresponding to its unit entries.
It can be shown that, because condition (i) in Theorem
3.1 is not satisfied (equivalently V̄PT ĀVP 6= 0), the network
is not phase synchronizable (see Fig. 3(b) and 3(c) for an
evolution of the oscillators’ phases and frequencies). From
Theorem 4.1 we obtain the optimal perturbation that ensures
synchronization, which leads to the network in Fig. 3(d).
Notice that the network in Fig. 3(d) satisfies condition (i)
in Theorem 3.1. In fact, when the natural frequencies are
equal within each cluster (condition (ii) in Theorem 3.1),
the clusters evolve cohesively; see Fig. 3(e) and 3(f).

V. C ONCLUSION
In this work we study cluster synchronization in networks
of Kuramoto oscillators. We derive necessary and sufficient
conditions on the network interconnection weights and on the
oscillators’ natural frequencies to guarantee that the phases
of groups of oscillators evolve cohesively with one another,
yet independently from the phases of oscillators belonging to different groups. Additionally, we develop a control
mechanism to modify the edges of a network to ensure the
formation of desired clusters. Our control method is optimal,
as it determines the smallest perturbation (measured by the
Frobenius norm) for a desired synchronization pattern that is
compatible with a pre-specified set of structural constraints.

1

12
2

9
4

2

6
5

7
2

5

3
10

9

5

(a)

1

(c)

(e)

(f)

2

12
2

9

(b)

6

3
1

5

7

11
9

4

5

2

5
2

(d)

Fig. 3. Fig. (a) shows the network in Example 2, where the dashed (resp. solid) edges correspond to the zero (resp. unit) entries of H. The partition
P = {P1 , P2 }, with P1 = {1, 2, 3} and P2 = {4, 5, 6}, is not synchronizable because, for instance, the sum of the weights of the incoming edges to
nodes 1 and 2 is different (see Theorem 3.1). Fig. (b) and (c) show the phases and frequencies of the oscillators as a function of time. Fig. (d) shows the
modified network obtained from Theorem 4.1, which satisfies condition (i) in Theorem 3.1 and leads to a synchronizable partition P. When the natural
frequencies are selected to satisfy condition (ii) in Theorem 3.1, the oscillators’ phases and frequencies are synchronized as illustrated in Fig. (e) and (f).

R EFERENCES
[1] F. L. Lewis, H. Zhang, K. Hengster-Movric, and A. Das. Introduction
to synchronization in nature and physics and cooperative control for
multi-agent systems on graphs. In Cooperative Control of Multi-Agent
Systems, pages 1–21. Springer, 2014.
[2] S. H. Strogatz. From Kuramoto to Crawford: exploring the onset
of synchronization in populations of coupled oscillators. Physica D:
Nonlinear Phenomena, 143(14):1–20, 2000.
[3] F. A. S. Ferrari, R. L. Viana, S. R. Lopes, and R. Stoop. Phase
synchronization of coupled bursting neurons and the generalized
Kuramoto model. Neural Networks, pages 107–118, 2015.
[4] F. Dörfler and F. Bullo. Synchronization in complex networks of phase
oscillators: A survey. Automatica, 50(6):1539–1564, 2014.
[5] N. Takashi and A. E. Motter. Comparative analysis of existing
models for power-grid synchronization. New Journal of Physics,
17(1):015012, 2015.
[6] T. Danino, O. Mondragon-Palomino, L. Tsimring, and J. Hasty. A
synchronized quorum of genetic clocks. Nature, 463(7279):326–330,
Jan 2010.
[7] J. Kim, D. Shin, S. H. Jung, P. Heslop-Harrison, and K. Cho. A design
principle underlying the synchronization of oscillations in cellular
systems. Journal of Cell Science, 123(4):537–543, 2010.
[8] D. A. Paley, N. E. Leonard, R. Sepulchre, D. Grunbaum, and J. K.
Parrish. Oscillator models and collective motion. IEEE Control
Systems Magazine, 27(4):89–105, 2007.
[9] A. Schnitzler and J. Gross. Normal and pathological oscillatory
communication in the brain. Nature Reviews Neuroscience, 6(4):285–
296, Apr 2005.
[10] H. Constance, B. Hagai, and P. Brown. Pathological synchronization
in parkinson’s disease: networks, models and treatments. Trends in
Neurosciences, 30(7):357–364, 2007.
[11] M. Banaie, Y. Sarbaz, M. Pooyan, S. Gharibzadeh, and F. Towhidkhah.
Modeling huntingtons disease considering the theory of central pattern
generators. In Advances in Computational Intelligence, pages 11–19.
Springer, 2009.
[12] K. Lehnertz, S. Bialonski, M. T. Horstmann, D. Krug, A. Rothkegel,
M. Staniek, and T. Wagner. Synchronization phenomena in human epileptic brain networks. Journal of Neuroscience Methods,
183(1):42–48, 2009.
[13] Y. Kuramoto. Self-entrainment of a population of coupled non-linear
oscillators. In International symposium on mathematical problems in
theoretical physics, pages 420–422, Berlin, Heidelberg, 1975.

[14] J. Gómez-Gardeñes, Y. Moreno, and A. Arenas. Synchronizability
determined by coupling strengths and topology on complex networks.
Physical Review E, 75:066106, Jun 2007.
[15] Z. Zhang, A. Sarlette, and Z. Ling. Synchronization of Kuramoto oscillators with non-identical natural frequencies: a quantum dynamical
decoupling approach. In IEEE Conf. on Decision and Control, pages
4585–4590, December 2014.
[16] L. M. Pecora, F. Sorrentino, A. M. Hagerstrom, T. E. Murphy, and
R. Roy. Cluster synchronization and isolated desynchronization in
complex networks with symmetries. Nature communications, 5, 2014.
[17] F. Sorrentino, L. M. Pecora, A. M. Hagerstrom, T. E. Murphy,
and R. Roy. Complete characterization of the stability of cluster
synchronization in complex dynamical networks. Science Advances,
2(4), 2016.
[18] W. Lu, B. Liu, and T. Chen. Cluster synchronization in networks of
coupled nonidentical dynamical systems. Chaos: An Interdisciplinary
Journal of Nonlinear Science, 20(1):013120, 2010.
[19] T. Dahms, J. Lehnert, and E. Schöll. Cluster and group synchronization
in delay-coupled networks. Physical Review E, 86:016202, Jul 2012.
[20] C. Favaretto, D. S. Bassett, A. Cenedese, and F. Pasqualetti. Bode
meets kuramoto: Synchronized clusters in oscillatory networks. In
IEEE American Control Conference, pages 2799–2804, Seattle, WA,
USA, May 2017.
[21] C. Favaretto, A. Cenedese, and F. Pasqualetti. Cluster synchronization
in networks of kuramoto oscillators. In IFAC World Congress, 2017.
To Appear.
[22] M. T. Schaub, N. O’Clery, Y. N. Billeh, J. Delvenne, R. Lambiotte,
and M. Barahona. Graph partitions and cluster synchronization in
networks of oscillators. Chaos, 26(9):094821, 2016.
[23] C. D. Godsil and G. F. Royle. Algebraic Graph Theory, volume 207
of Graduate Texts in Mathematics. Springer, 2001.
[24] Y. Kuramoto. Self-entrainment of a population of coupled non-linear
oscillators. In H. Araki, editor, Int. Symposium on Mathematical
Problems in Theoretical Physics, volume 39 of Lecture Notes in
Physics, pages 420–422. Springer, 1975.
[25] M. Mirchev, L. Basnarkov, F. Corinto, and L. Kocarev. Cooperative
phenomena in networks of oscillators with non-identical interactions and dynamics. IEEE Transactions on Circuits and Systems I,
61(3):811–819, 2014.
[26] R. A. Horn and C. R. Johnson. Matrix Analysis. Cambridge University
Press, 1985.

