A partial synchronization theorem
Citation for published version (APA):Pogromsky, A. Y. (2008). A partial synchronization theorem. Chaos, 18(3), 037107-1/6. [037107]. https://doi.org/10.1063/1.2959145
DOI:
10.1063/1.2959145
Document status and date: Published: 01/01/2008
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A partial synchronization theorem
Alexander Yu. Pogromsky
Department of Mechanical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands
共Received 25 February 2008; accepted 26 June 2008; published online 22 September 2008兲 When synchronization sets in, coupled systems oscillate in a coherent way. It is possible to observe also some intermediate regimes characterized by incomplete synchrony which are referred to as partial synchronization. The paper focuses on analysis of partial synchronization in networks of linearly coupled oscillators. © 2008 American Institute of Physics.关DOI:10.1063/1.2959145兴
A complex network of coupled oscillators can exhibit a phenomenon called partial synchronization or clusteriza-tion. This phenomenon is characterized by coherent havior of some oscillators forming the clusters, while be-tween the clusters there might be no apparent agreement. The partial synchronization occurs in networks possess-ing symmetry in the couplpossess-ings. The paper studies the con-ditions leading to partial synchronization.
I. INTRODUCTION
Synchronization of dynamical systems is a topic that at-tracted attention during the last two–three decades. A large number of examples of synchronization in nature can be found in Refs.1–3, and references therein. A particular inter-est in this field is the so-called partial synchronization, or clustering, that is characterized by an agreement between several nodes of the networks, see, e.g., Refs.4–8.
If a given network of oscillators has a topology with some symmetries, this network can exhibit clustering phe-nomena that are characterized by existence and stability of invariant manifolds in the network. An approach initiated in
Ref. 4 and developed further by Belykh, Belykh, and
co-workers, see, e.g., Refs.5–7, is based on a graph-theoretical approach to find and characterize those manifolds with sub-sequent application of the direct Lyapunov method to prove its stability. Another approach which was initiated in Ref.9
is more algebraic by nature and is aimed at simultaneous study of existence and stability of those manifolds in a uni-fied framework.
The goal of this publication is to further develop this algebraic approach to avoid a commutation condition im-posed in Ref.9. For networks of complex topology this con-dition can be conservative as illustrated in this paper.
This paper follows the same lines as our previous work;9 however to make the presentation self-contained all neces-sary background material is presented here in a compact way. We refer to Ref. 9where the reader can find some explana-tions of the definiexplana-tions that will be used in this publication.
The paper is organized as follows: The problem state-ment is explained in Sec. II, where the network dynamics is outlined. In Sec. III the association between the symmetry and the linear invariant manifolds of the network is dis-cussed. Section IV begins with some background material
from control theory, after which we propose a proof of asymptotic stability of a compact subset of a specified linear invariant manifold. Section V contains an illustrative ex-ample.
Throughout the paper we use the following notations: Ik denotes the k⫻k identity matrix. The Euclidean norm in Rn is denoted simply as 兩·兩, 兩x兩2= xx, where defines
transpo-sition. The notation col共x1, . . . , xn兲 stands for the column vec-tor composed of the elements x1, . . . , xn. This notation will also be used in case where the components xiare vectors too. A function V : X→R+defined on a subset X ofRn, 0苸X is
positive definite if V共x兲⬎0 for all x苸X\兵0其 and V共0兲=0. It
is radially unbounded 共if X=Rn兲 or proper if V共x兲→⬁ as
兩x兩→⬁. If a quadratic form xPx with a symmetric matrix
P = Pis positive definite, then the matrix P is called posi-tive definite. For posiposi-tive definite matrices we use the
nota-tion P⬎0; moreover P⬎Q means that the matrix P−Q is
positive definite. For matrices A and B the notation A丢B 共the Kronecker product兲 stands for the matrix composed of submatrices AijB, i.e., A丢B =
冢
A11B A12B ¯ A1nB A21B A22B ¯ A2nB ] ] ] An1B An2B ¯ AnnB冣
where Aij, i, j = 1 . . . n, stands for the ijth entry of the n⫻n matrix A.
II. PROBLEM STATEMENT
Consider k identical systems of the form
x˙j= f共xj兲 + Buj, yj= Cxj, 共1兲
where f is a smooth vector field, j = 1 , . . . , k, xj共t兲苸Rnis the state of the jth system, uj共t兲苸Rmand yj共t兲苸Rmare, respec-tively, the input and the output of the jth system, and B, C are constant matrices of appropriate dimension. We assume that matrix CB is similar to a positive definite matrix, and the k systems are interconnected through mutual linear output coupling,
uj= −␥j1共yj− y1兲 −␥j2共yj− y2兲 − ¯ −␥jk共yj− yk兲, 共2兲 where ␥ij are constants. With no loss of generality we as-sume in the sequel that CB is a positive definite matrix.
Define the k⫻k matrix ⌫ as
⌫ =
冢
兺
i=2 k ␥1i −␥12 ¯ −␥1k −␥21兺
i=1,i⫽2 k ␥2i ¯ −␥2k ] ] ] −␥k1 −␥k2 ¯兺
i=1 k−1 ␥ki冣
共3兲where all row sums are zero. With definition共3兲, the collec-tion of k systems 共1兲 with feedback 共2兲 can be rewritten in the more compact form
x˙ = F共x兲 + 共Ik丢B兲u, y = 共Ik丢C兲x, 共4兲 with the feedback given by
u = −共⌫丢Im兲y, 共5兲
where we denoted x = col共x1, . . . , xk兲, F共x兲
= col关f共x1兲, ... , f共xk兲兴苸Rkn, y = col共y1, . . . , yk兲, and u = col共u1, . . . , uk兲苸Rkm.
All main points have now been introduced in order to formulate a clear problem statement. Can we exploit symme-try in the network to identify its linear invariant manifolds, and benefit from a representation of the system as 共2,3兲, and/or 共4,5兲, typical for control purposes, in order to give conditions that guarantee stability of some selected partial 共or the full兲 synchronized states?
III. SYMMETRIES AND INVARIANT MANIFOLDS
If a given network possesses a certain symmetry, this
symmetry must be present in matrix ⌫. In particular, the
network may contain some repeating patterns, when consid-ering the arrangements of constants ␥ij, hence the permuta-tion of some elements will leave the network unchanged. The
matrix representation of a permutation of the set
兵1,2, ... ,k其 is a permutation matrix ⌸苸Rk⫻k. Permutation matrices are orthogonal, i.e., ⌸⌸=Ik, and they form a group with respect to the multiplication, so for any two per-mutation matrices⌸i,⌸jof the same size,⌸i⌸jis a permu-tation matrix too.
Rewrite the dynamics of共4,5兲 in the closed loop form
x˙ = F共x兲 + Gx, 共6兲
where G = −共Ik丢B兲共⌫丢Im兲共Ik丢C兲苸Rkn⫻kn, that can be simplified as G = −⌫丢BC. Let us recall here that given a dynamical system as Eq. 共6兲, the linear manifold AM=兵x 苸Rkn: Mx = 0其, with M 苸Rkn⫻kn
, is invariant if Mx˙ = 0 when-ever Mx = 0. That is, if at a certain time t0a trajectory is on the manifold, x共t0兲苸AM, then it will remain there for all time, x共t兲苸AMfor all t. The problem can be summarized in the following terms: given G and F共·兲 find a solution M to
MF关x共t0兲兴 + MGx共t0兲 = 0 共7兲
for all x共t0兲 for which Mx共t0兲=0. A natural way to solve Eq. 共7兲 is to exploit the symmetry of the network.
In representation 共6兲, we can establish conditions to identify those permutations that leave a given network in-variant. To this end we will establish conditions that guaran-tee that the set
ker共Ikn−⌸丢In兲 is invariant.
Let ⌺=⌸丢In for simplicity, and assume that at time t0x共t0兲 satisfies 共Ikn−⌺兲x共t0兲=0. Consider Eq. 共6兲, and
sup-pose that there is a solution X of the following system of linear equations:
共Ik−⌸兲⌫ = X共Ik−⌸兲. 共8兲
Since ⌸ is a permutation matrix, it also follows that ⌺F共x兲 = F共⌺x兲. If we multiply both sides of Eq. 共6兲 by Ikn−⌺, we obtain, at time t0,
共Ikn−⌺兲x˙共t0兲 = F关x共t0兲兴 − F关⌺x共t0兲兴
+共Ik丢B兲共X丢Im兲共Ik丢C兲共Ikn−⌺兲x共t0兲 = 0
because we assumed 共Ikn−⌺兲x共t0兲=0. Therefore,
共Ikn−⌺兲x共t兲=0 for all t, and we can reformulate this result as: Lemma 1: Given a permutation matrix⌸ such that Eq.
共8兲 has a solution X, the set
ker共Ikn−⌸丢In兲 共9兲
is a linear invariant manifold for system共6兲.
IV. STABILITY ANALYSIS A. Semipassivity
Consider systems of the form
x˙ = f共x兲 + Bu, y = Cx, 共10兲
where x苸Rn is the state, u苸Rmis the input, y苸Rmis the output, the matrices B, C are of corresponding dimensions, f is smooth enough to ensure existence and uniqueness of so-lutions for admissible u共t兲. Suppose it is possible to find a scalar nonnegative function V defined on Rn, whose deriva-tive satisfies, along the solutions of Eq.共10兲, the inequality
V˙ 共x,u兲 艋 yTu − H共x兲, 共11兲
where function H :Rn→R is non-negative outside some ball,
∃⬎ 0, ∀ 兩x兩 艌⇒ H共x兲 艌 共兩x兩兲 共12兲
for some continuous non-negative function defined for 兩x兩艌. In this case system共10兲 is called semipassive. This notion was introduced in Ref.11, while in Ref.12an equiva-lent notion was called quasipassivity. If function H is posi-tive outside some ball, i.e., Eq.共12兲holds for some continu-ous positive function , then system共10兲is said to be strictly semipassive. In brief, a semipassive system behaves like a passive system for sufficiently large兩x兩.
It is important to observe that the dissipation inequality
共11兲 can be rewritten in an equivalent way as follows:
V
xf共x兲 艋 − H共x兲,
V
xB = x
TCT.
Suppose that system共10兲is strictly semipassive and the stor-age function V satisfying the dissipation inequality 共11兲 is radially unbounded, that is V共x兲→⬁ when 兩x兩 →⬁, then any feedback u =共y兲 satisfying the inequality
yT共y兲 艋 0 共13兲
makes the closed loop system ultimately bounded. This state-ment can be proven just by considering the storage function V as a Lyapunov function candidate.
Therefore, the following result is valid:
Lemma 2: If system(1)is strictly semipassive with radi-ally unbounded storage functions and the symmetrized ma-trix ⌫+⌫ is positive semidefinite, then all solutions of the coupled systems (1) and (2) exist for all t艌0 and are ulti-mately bounded.
The technical proof of this statement and more general related results can be found in Ref.11.
B. Convergent systems
Consider a dynamical system of the form
z˙ = q关z,w共t兲兴, 共14兲
with z苸Rl, driven by the external signal w共t兲 taking values from some compact set. This system is said to be convergent13 共see also Ref. 14兲 if for any bounded signal
w共t兲 defined on the whole time interval 共−⬁, +⬁兲 there is a unique bounded, globally asymptotically stable solution z¯共t兲 defined on the same interval共−⬁, +⬁兲, from which it follows that
lim
t→⬁兩z共t兲 − z¯共t兲兩 = 0, 共15兲
for all initial conditions. In systems of this type the limit mode is solely determined by the external excitation w共t兲, not by the initial conditions of z. From the existence of a unique mode z¯共t兲, it obviously descends that two identical copies of convergent system z1 and z2, Eq. 共14兲 must
syn-chronize, that is, if Eq.共15兲holds, lim
t→⬁兩z1共t兲 − z2共t兲兩 = 0
holds as well. Convergence is then closely related to syn-chronization, hence it is important to find conditions ensur-ing it. Recently, an importance of the concept of convergent systems was recognized in control community with a poten-tial application to observer design. In Ref.15, a bit stronger notion was called incremental global asymptotic stability 共␦GAS兲; therein the necessary and sufficient conditions for
␦GAS were formulated in terms of the existence of
Lyapunov functions. We present here a slight modification of a sufficient condition obtained by Demidovich:13if there is a positive definite symmetric l⫻l matrix P such that all eigen-values i共Q兲 of the symmetric matrix
Q共z,w兲 =1 2
冋
P冉
q z共z,w兲冊
+冉
q z共z,w兲冊
T P册
共16兲are negative and separated from zero, i.e., there is␦⬎0 such that
i共Q兲 艋 −␦⬍ 0, 共17兲
with i = 1 , . . . , l for all z, w苸Rl, then system共14兲is conver-gent, and there exists a quadratic function W共兲=P
sat-isfying
W共z1− z2兲
关q共z1,w兲 − q共z2,w兲兴 艋 −␣兩z1− z2兩2, 共18兲
for some␣⬎0. This condition is a slight modification of the Demidovich theorem on convergent systems in the case P = Il.
C. On global asymptotic stability of the partial synchronization manifolds
A permutation matrix ⌸ satisfying Eq.共8兲 for some X
defines a linear invariant manifold of system 共6兲, given by Eq.共9兲. This expression stands for a set of linear equations of the form
xi− xj= 0 共19兲
for some i and j that can be read off from the nonzero
ele-ments of the ⌸ matrix under consideration. Therefore, we
can identify a particular manifold associated with a particular matrix⌸ by the correspondent set I⌸of pairs i, j for which Eq. 共19兲holds.
In this section we are going to investigate asymptotic stability of partial synchronization as asymptotic stability of sets. In order to find a Lyapunov function which proves sta-bility of the partial synchronization manifold, one can seek a Lyapunov function candidate as a sum of two functions, the first one dependent on the input-output relations of systems
共1兲 and the second one dependent on the way the systems
interact via coupling. The best way to carry this out is to find a globally defined coordinate change that allows us to exploit minimum phaseness.
Let us first differentiate yj, y˙j= Cf共xj兲 + CBuj.
Then, choosing some n − m coordinates zj, complementary to yj, it is possible to rewrite system共1兲 in the form
z˙j= q共zj兲,yj, y˙j= a共zj,yj兲 + CBuj, 共20兲 where zj苸Rn−m, and q and a are some vector functions. It is
important to emphasize that the coordinate change
xj哫col共zj, yj兲 can be linear, if CB is nonsingular, and that, owing to the linear input-output relations, this transformation is globally defined. This transformation is explicitly com-puted in, for example, Ref.16. As the reader may expect, for more complicated input-output relations, this coordinate transformation may not be globally defined. Conditions on the existence of this normal form can be found in Refs. 10
and17, for example. In the equation for zjin Eq.共20兲, yjacts as a forcing input, hence we can apply properties of
conver-gent systems, if matrix Q共z,w兲 defined for q in Eq.共20兲has negative eigenvalues, separated from zero.
The purpose of this section is to prove the following theorem:
Theorem 1: Suppose that
(i) Each free system (1) is strictly semipassive with re-spect to the input uj and output yj with a radially unbounded storage function.
(ii) There exists a positive definite matrix P such that Eq.
(17)holds with some␦⬎0 for the matrix Q defined as in Eq.(16)for q as in Eq. (20).
(iii) The symmetrized matrix ⌫+⌫ is positive semidefi-nite.
(iv) There is a k⫻k matrix solution X of the following linear equation
共Ik−⌸兲⌫ = X共Ik−⌸兲. (v) CB is positive definite.
Let
⬘
be the minimal eigenvalue of 12共X+X兲 under the restriction that the eigenvectors of 12共X+X兲 are taken from the set range共Ik−⌸兲.Then all solutions of network (4, 5) are ultimately bounded and there exists a positive¯ such that if
⬘
⬎¯ the set ker共Ikn−⌸丢In兲 contains a globally asymptotically stable compact subset.We sketch the proof of Theorem 1. To make the presen-tation more transparent we omitted some standard technical details which can be found in similar proofs, of related re-sults, presented in Refs.11and18. Our approach is inspired by the results on feedback-passive systems as presented in Ref.19. In the proof we are mostly focused on the approach to find the Lyapunov function guaranteeing stability of the partial synchronization mode. As we previously introduced the notation y = col共y1, . . . , yk兲, let us denote with z苸Rkmthe vector col共z1, . . . , zk兲. Since the derivative of z-variables in Eq. 共20兲does not depend on the coupling, while the deriva-tive of y-variables does, we can search for a Lyapunov func-tion in the form
V共z,y兲 = V1共z兲 + V2共y兲.
Let us start with function V1. According to assumption
共iii兲 there is a positive definite radially unbounded function W共兲 defined on Rn−m which satisfies the partial differential inequality 共18兲 for all zi, zj苸Rn−m, w苸Rm. Then we con-struct the function V1 as
V1共z兲 =
兺
共i,j兲苸I⌸
W共zi− zj兲.
Along the solutions of the closed loop system, the derivative of V1共z兲 satisfies V˙1共z,y兲 =
兺
共i,j兲苸I⌸ W共zi− zj兲 关q共zi,yi兲 − q共zj,yj兲兴 艋 −␣兺
共i,j兲苸I⌸ 兩zi− zj兩2 +兺
共i,j兲苸I⌸ W共zi− zj兲 关q共zj,yi兲 − q共zj,yj兲兴.The next step is to find the second part of the Lyapunov function, i.e., function V2. It is clear that if x苸ker共Ikn−⌸
丢In兲, then necessarily y苸ker共Ikm−⌸丢Im兲. So, on this in-variant manifold, the quantity 共y兲=共Ikm−⌸丢Im兲y is identi-cally zero. We can therefore construct function V2 as
V2共y兲 = 1 2兩共y兲兩 2 =1 2y T共I km−⌸丢Im兲T共Ikm−⌸丢Im兲y =1 2共i,j兲苸I
兺
⌸ 兩yi− yj兩2,which is positive definite with respect to, and zero on the set ker共Ikm−⌸丢Im兲. Differentiating V2gives
V˙2共z,y兲 =
兺
j=1 k V2共yj兲 yj a共zj,yj兲 − U共y兲 where using assumption iv,U共y兲 =1
2yT共Ikm−⌸丢Im兲T关共X + XT兲丢CB兴共Ikm−⌸丢Im兲y. It follows that
U共y兲 艌
⬘
yT共Ikm−⌸丢Im兲T共Ikm−⌸丢Im兲y,where  is the minimal eigenvalue of matrix CB and
⬘
is the minimal eigenvalue of 12共X+X兲 under the restriction that the eigenvectors of 12共X+X兲 are taken from the set range共Ik−⌸兲.We proceed now to evaluate the derivative of V. From the previous intermediate results it follows that
V˙ 共z,y兲 艋 −␣
兺
共i,j兲苸I⌸ 兩zi− zj兩2− 2⬘
V2共y兲 +兺
j=1 k V2共yj兲 yj a共zj,yj兲 +兺
共i,j兲苸I⌸ W共zi− zj兲 关q共zj,yi兲 − q共zj,yj兲兴. Note that for any compact set ⍀ there exist some positive numbers C1, C2, C3 such that the following estimates arevalid on⍀:
冏
兺
j=1 k V2共yj兲 yj a共zj,yj兲冏
=冏
兺
共i,j兲苸I⌸共yi− yj兲T关a共zi,yi兲 − a共zj,yj兲兴
冏
艋
冏
兺
共i,j兲苸I⌸共yi − yj兲T关a共zi,yi兲 − a共zi,yj兲兴冏
+冏
兺
共i,j兲苸I⌸共yi − yj兲T关a共zi,yj兲 − a共zj,yj兲兴冏
艋 C1V2共y兲 + C2兺
共i,j兲苸I⌸ 兩zi− zj兩兩yi− yj兩 and冏
共i,j兲苸I兺
⌸ W共zi− zj兲 关q共zj,yi兲 − q共zj,yj兲兴冏
艋 C3兺
共i,j兲苸I⌸ 兩zi− zj兩 · 兩yi− yj兩.Now we are going to use strict semipassivity of the systems forming the diffusive network. Recall that strict semipassiv-ity implies ultimate boundedness of all the solutions, that is, all the solutions approach in a finite time some compact set ⍀ which can be chosen independently on ⌫. On this compact set the derivative of V is a quadratic form with respect to 兩zi− zj兩 and 兩yi− yj兩. It is clear then that, if the value of
⬘
is large enough共that is, ⬘
is greater than a positive computable threshold ¯兲, due to Eq. 共18兲, the derivative of V共z,y兲 is nonpositive on this set. After some algebra, an explicit for-mula for ¯ is derived as¯ =1 
冋
C1 2 + 共C2+ C3兲2 4␣册
. 共21兲This argument proves that the set ker共Ikn−⌸丢In兲 contains a globally asymptotically stable compact subset for
⬘
⬎¯. 䊏V. EXAMPLE
In this section we consider an example of a network of diffusively coupled systems depicted in Fig. 1. Matrix ⌫ in this case is defined as
⌫ =
冤
␥1 k1 0 k1 k2 k4 0 0 k1 ␥1 k1 0 k4 k2 0 0 0 k1 ␥1 k1 0 0 k2 k4 k1 0 k1 ␥1 0 0 k4 k2 k2 k4 0 0 ␥2 k3 0 k3 k4 k2 0 0 k3 ␥2 k3 0 0 0 k2 k4 0 k3 ␥2 k3 0 0 k4 k2 k3 0 k3 ␥2冥
with ␥1= − 2k1− k2− k4, ␥2= − k2− 2k3− k4.Consider the following permutation matrix:
⌸ =
冤
0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0冥
.The set ker共I8−⌸兲 is given by x1= x2= x3= x4, x5= x6= x7= x8.
As one can easily verify matrices ⌸ and ⌫ do not commute
and therefore the theorem presented in Ref. 9cannot be ap-plied to study stability of this invariant set. At the same time there is a solution X to Eq.共8兲for the given⌫ and ⌸ 共it was verified numerically for different k1, k2, k3, k4using singular value decomposition兲 and thus Theorem 1 presented in this paper can be utilized to investigate stability of this invariant manifold.
To apply the partial synchronization theorem the other conditions of the theorem should be verified. They depend on the input-output properties of the individual dynamics form-ing the network. Suppose the individual dynamics are given by the Lorenz system
x˙j,1=共xj,2− xj,1兲 + uj, x˙j,2= rxj,1− xj,2− xj,1xj,3, x˙j,3= − bxj,3+ xj,1xj,2,
with, r, b⬎0. The Lorenz system with input ujand output yj= xj,1is strictly semipassive. To prove this statement, con-sider the following storage function candidate:
V共xj,1,xj,2,xj,3兲 =
1
2关共xj,1兲2+共xj,2兲2+共xj,3−− r兲2兴. Calculating the derivative of this function along the solutions of the system yields
V˙ 共xj,1,xj,2,xj,3,u兲 = xj,1u − H共xj,1,xj,2,xj,3兲, where H共xj,1,xj,2,xj,3兲 =共xj,1兲2+共xj,2兲2+ b
冉
xj,3− + r 2冊
2 − b共+ r兲 2 4 .It is easy to see that the condition H艋0 determines an ellip-soid in R3. This fact proves strict semipassivity of system
共22兲 with input uj and output yj= xj,1. Hence, in a diffusive
network of any number of Lorenz systems with outputs yj
= xj,1 and inputs uj, all solutions are ultimately bounded. If we think of output yjas a driving input for the remain-ing part共q-subsystem in the theorem conditions兲 of the Lo-renz system, we have
x˙j,2= − xj,2+ ryj− yjxj,3, x˙j,3= − bxj,3+ yjxj,2,
which is convergent. Applying Demidovich’s result we see that, using P = I2, matrix Q共xj, yj兲 in Eq.共16兲is given by
FIG. 1. 共Color online兲 A network of eight coupled identical systems with symmetric coupling.
Q共xj,yj兲 = diag共− 1 − b兲.
Now one can apply the theorem to conclude that if the parameters of the coupling matrix are appropriately chosen, the network of coupled Lorenz systems possesses partially synchronous modes.
There are other systems that satisfy conditions imposed on the input-output properties of individual dynamics of the network. This is the case if one takes, for example the Lo-renz or Hindmarsh–Rose system, see Refs.9and20for de-tails.
VI. CONCLUSION
In this paper we have demonstrated an approach, based on the second Lyapunov method, to study partial synchroni-zation regimes in a network of linearly coupled identical dynamical systems. We presented a theorem that allows us to cope with more general networks than those presented in our previous work.9 For relatively simple ringlike networks the corresponding symmetry group is isomorphic to the powers of unity and the analysis of such systems can be performed by the result of Ref.9. For more complex networks, commu-tativity of the corresponding symmetry group can be an is-sue, as illustrated by the example in this paper. To classify all permutations that satisfy Eq.共8兲for a given topology one can
use a special property of matrix ⌫+⌫ 共all row sums are
zeros兲 and the seminal Birkhoff–von Neumann theorem on doubly stochastic matrices. This result allows us to represent a doubly stochastic matrix as a convex combination of per-mutation matrices. Since the partial synchronization mode is defined by another permutation matrix, the theorem due to Birkhoff and von Neumann can be a very useful tool in the analysis of partial synchronization in complex net-works. That can constitute an interesting topic for future re-search.
ACKNOWLEDGMENTS
This work was partially supported by the Dutch-Russian mathematical program “Dynamics and Control of Hybrid
Mechanical Systems” 共NWO Grant No. 047.017.018兲.
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