A method for computation and analysis of partial ...twr.cs.kuleuven.be/.../chapter_manifolds.pdf · and analysis of partial synchronization manifolds of networks with linear di usive

A method for computation and analysis ofpartial synchronization manifolds of delaycoupled systems

Libo Su, Wim Michiels, Erik Steur and Henk Nijmeijer.

Abstract In this chapter, we present a systematic method for computationand analysis of partial synchronization manifolds of networks with lineardiffusive couplings which is implemented as a MATLAB function. For thecomputation part, the method is developed based on a recently proposed al-gorithm for characterizing partial synchronization manifolds, with additionalapplicability to networks with different systems at the nodes. For the analysispart, it performs the decomposition of network dynamics into synchroniza-tion error dynamics and dynamics on manifolds in a systematic way, whichis important in the context of stability analysis of partial synchronizationmanifolds.

1 Introduction

During recent decades, attention has been attracted to synchronization of in-terconnected dynamical systems. This phenomenon has been widely observedby the scientific community. Examples can be found from nature to engineer-ing: fireflies flash synchronously, [1]; geese fly in flock during migration [2];vehicles move in platoon [3], etc. The most noticeable form of synchronous

Libo SuDepartment of Computer Science, KU Leuven, e-mail: [email protected]

Wim MichielsDepartment of Computer Science, KU Leuven, e-mail: [email protected]

Erik SteurDelft Center for Systems and Control, Delft University of Technology, e-mail:[email protected]

Henk NijmeijerDepartment of Mechanical Engineering, Eindhoven University of Technology, e-mail:[email protected]

1

2 Libo Su, Wim Michiels, Erik Steur and Henk Nijmeijer.

behavior is full synchronization, which refers to the situation where all sys-tems of a network behave identically [4]. Many researchers have proposedconditions for full synchronization of networks, see, e.g. , [5], [6], [7].However, networks may show a form of incomplete synchronization, or partialsynchronization, which refers to the phenomenon that only some but not allthe systems in the networks synchronize [4]. Partial synchronization has beenobserved in a variety of complex systems, for example collective behaviors ofneurons in parts of human brain, [8]. It has also attracted considerable at-tention recently, see, e.g. , [4], [9], [10], [11], [12], and the references therein.In practice, there may exist time delays between, and even inside the sys-tems in networks. Interestingly, these time delays may affect the existenceand stability of partial synchronization. Some studies have been conductedon partial synchronization of networked systems with time-delay couplings([13], [14], [15], [16], [17]). For example, in [13], a master stability function(see [6]) based approach is used to characterize cluster (partial) and groupsynchronization and also determine their stability of networks with time delaycouplings, where group synchronization is a specific type of partial synchro-nization where the nodes in different clusters host different local dynamics,see detailed definition in [13]. In [14], a decomposition method of networkdynamics around cluster states is proposed to derive delayed modal equa-tions about cluster states for stability analysis of partial synchronization innetworks with identical systems. In [15], an LMI-based condition for partialsynchronization is derived, and it is also shown that partial synchronizationis delay-independent for certain coupling strength larger than some thresholdprovided that the proposed condition is satisfied.In this work, we focus on partial synchronization in networks of systems in-teracting via linear time-delay couplings. To describe the patterns of partialsynchronization, the concept of partial synchronization manifolds is com-monly used, which are linear invariant subspaces of the state space of thesystems in networks, [16]. The existence of a partial synchronization man-ifold, by itself, does not imply that in experiments synchronous behaviorwithin each cluster of the corresponding partition occurs. The latter alsorequires the manifold to be stable. In [4], such manifolds are identified byexploiting the symmetry under permutation of the networks with identicalsystems coupled through diffusion. In [15], a so-called equitable partition isdefined to identify partial synchronization manifolds. More equivalent exis-tence conditions for partial synchronization manifolds for network of identicalsystems interacting via linear diffusive time-delay couplings have been pre-sented in [16]. Using one of the criteria, which utilizes the block structureof the reordered adjacency matrix of a network, an algorithm for computingpartial synchronization manifolds has been developed in [16]. In [18], an im-proved algorithm is developed which is more computationally efficient whenthe couplings are invasive, i.e., they do not vanish when full synchronizationemerges. Most work on partial synchronization considers networks consistingof identical systems. In some research, systems with different dynamics are

Method for partial synchronization manifolds 3

also considered, see, e.g. , [13], [19]. As an extension of [16] and [18], in thiswork, we first extend the method proposed in [16] for computing partial syn-chronization manifolds to allow for networks with (some but not all) differentsystems at the nodes. Furthermore, we complete this method by deriving thesynchronization error dynamics and the dynamics on the manifolds, whichserve as a foundation for stability analysis in future work.

The structure of this chapter is as follows. Section 2 shows how to char-acterize and compute partial synchronization manifolds, summarizing the re-sults from [16] and [18]. Section 3 shows the synchronization errors dynamicsand dynamics on the partial synchronization manifolds. Section 4 presentsnumerical examples. Section 5 provides the conclusions.

2 Characterization and computation of partialsynchronization manifolds

In this section, we introduce some relevant concepts on partial synchroniza-tion manifolds from [16], and also extend the existence condition for partialsynchronization manifolds and software in [16] and [18] to allow for networkswith different systems at the nodes. As an extension of [16] and [18], we followthe settings and convections from them throughout this section.

2.1 Characterization

We consider a network consisting of N systems whose coupling is representedby a directed graph G = (V, E , A).

• V is a finite set of nodes with cardinality |V| = N ;• E ⊂ V × V is the ordered set of edges, where the edge (i, j) points from

node i to node j;• A =

(ai,j)∈ RN×N is the weighted adjacency matrix, where ai,j > 0

represents the weight of edge (i, j) when (i, j) ∈ E , and ai,j = 0 when(i, j) /∈ E .

Note that the graphs considered here are simple and strongly connected.A graph is simple if it has no self-loops or multiple edges, where self-loopsare edges joining a node to itself and multiple edges are two or more edgesconnecting an ordered pair of nodes [20]. A graph is strongly connected if, forany two nodes i, j in it, there exist a directed path from i to j and a directedpath from j to i, [21].Every node hosts a dynamical system of the following form

4 Libo Su, Wim Michiels, Erik Steur and Henk Nijmeijer.xi(t) = fi(xi(t)) +Biui(t)yi(t) = Cixi(t)

(1)

where i ∈ V, state xi(t) ∈ Rn, function fi : Rn → Rn, input(s) ui(t) ∈ Rm,output(s) yi(t) ∈ Rm, input matrices Bi ∈ Rn×m and output matrices Ci ∈Rm×n, i = 1, . . . , N .The systems (1) interact via either one of the following two types of diffusivecouplings:

ui(t) = k∑j∈Ni

ai,j [yj(t− τ)− yi(t)] (2)

orui(t) = k

∑j∈Ni

ai,j [yj(t− τ)− yi(t− τ)], (3)

where, Ni is the neighboring set of node i, i.e., N := j ∈ V | (i, j) ∈ E,and τ is the time-delay. Coupling (2) is called invasive coupling since thecoupling does not vanish when all the nodes synchronize; while coupling (3)is called non-invasive coupling since the coupling vanishes when all the nodessynchronize, [22].For the coupled systems (1), (2) or (1), (3), a solution is a partial synchro-nization solution if there exist i, j ∈ V with i 6= j such that

xi(t) = xj(t), ∀t ≥ t0, (4)

whenever xi(t) = xj(t) for t ∈ [t0 − τ, t0]. Here xi(·) is the solution of nodei in the network.In order to describe the clustering of nodes in a graph G, the concept ofpartition is used. A partition P is a set of nonempty, disjoint subsets of Vsatisfying the condition that V is the union of these subsets. These subsetsare refereed as parts of the partition. In this way, a part represents a cluster ofthe nodes. The number of parts is denoted by κ. A partition is parameterizedby a permutation matrix Π. While, the permutation matrix Π is associatedwith an equivalent relation ∼, which is such that if i ∼ j if the ij-th entry ofΠ is equal to 1. Note that κ = dim ker(IN −Π).Denoting the state of the coupled network by xt ∈ C([−τ, 0],RNn),

xt(θ) :=

xt,1(θ)xt,2(θ)

...xt,N (θ)

=

x1(t+ θ)x2(t+ θ)

...xN (t+ θ)

, θ ∈ [−τ, 0],

i.e., the function segment x(t + θ), θ ∈ [−τ, 0], conditions of the form (4)can then be expressed as

xt(θ) = (Π ⊗ In)xt(θ), ∀θ ∈ [−τ, 0], ∀t ≥ t0, (5)


or xt ∈M(Π), ∀t ≥ t0 whenever xt0 ∈M(Π), where

M(Π) := φ ∈ C([−τ, 0],RNn) |φ(θ) = col(φ1(θ), φ2(θ), . . . , φN (θ)),

φi(θ) ∈ Rn, i = 1, . . . , N, φ(θ) ∈ ker(INn −Π ⊗ In)∀θ ∈ [−τ, 0]

is the set of partially synchronous states induced by the permutation matrixΠ.

Definition 1. [16] The set M(Π) with permutation matrix Π for which1 < κ < N is a partial synchronization manifold for the coupled systems (1),(2), or (1), (3), if and only if it is positively invariant under the dynamics(1), (2), or (1), (3), respectively.

Given a partition P, the nodes can be relabelled by clusters, i.e., , the firstκ1 nodes belong to cluster 1, the second κ2 nodes belong to cluster 2, and soon,

1, . . . , κ1, κ1 + 1, . . . , κ1 + κ2, . . .,where κi, i = 1, . . . , κ are the sizes of parts of P. Mathematically, this canbe done by using another permutation matrix R, which is refereed as thereordering matrix. A N × N permutation matrix R is a reordered matrix ifit satisfies

R>ΠR =

ΠC(κ1) 0

ΠC(κ2). . .

0 ΠC(κκ)

,

κ∑`=1

k` = N, (6)

i.e., R>ΠR is a block diagonal matrix with κ blocks ΠC(κ`), each of whichis a κ` × κ`-dimensional cyclic permutation matrix.Using R, the reordered adjacency matrix can be constructed as follows.

R>AR =

A11 A12 · · · A1κ

A21 A22 · · · A2κ

.... . .

. . ....

Aκ1 Aκ2 · · · Aκκ

, Aij ∈ Rki×kj . (7)

Now, we can introduce the existence condition for partial synchronizationmanifolds. To allow for systems to have different dynamics (fi(·) 6= fj(·),Bi 6= Bj and Ci 6= Cj , for some i, j ∈ V), we extend Theorem 3 and 4 in [16]as follows.

Theorem 1. Given an adjacency matrix A and a permutation matrix Π ofthe same dimension. Assume system (1) is left-invertible (the system input-output map is injective), then the following statements are equivalent:

1)M(Π) is a partial synchronization manifold for (1) and (2), respectively(1) and (3);


2) all blocks, respectively all off-diagonal blocks, of the reordered adjacencymatrix (7), partitioned in blocks of size ki × kj have constant row-sums,and F = (Π ⊗ In)F , B = (Π ⊗ In)B, C = (Π ⊗ In)C, with

F =

f1(·)f2(·)

...fN (·)

,B =

B1

B2

...BN

, C =

C>1C>2

...C>N

.Here, the conditions F = (Π⊗In)F , B = (Π⊗In)B and C = (Π⊗In)C indi-cate all the nodes in the same cluster host systems with the same dynamics.

2.2 Computation

Here, we briefly introduce the algorithm for computing the partial synchro-nization manifolds for networks with identical/non-identical dynamical sys-tems at the nodes. It is an extension of the algorithm in [16] and [18] whichonly work for networks with identical systems.The algorithm consists of two steps:

• generate possible partitions;• check the viability of a partition (i.e., check whether a partition is related

to a partial synchronization manifold).

For the first step, a vector AR ∈ NN indicating (in)equalities of row-sums ofthe adjacency matrix is exploited in [18] to limit the number of partitions tobe generated for networks with invasive couplings, since for any two nodesto be in the same cluster of a viable partition, their corresponding rows inthe adjacency matrix should have the same row sum. Here, we use a similarvector AF ∈ NN = (Fi), i = 1, 2, . . . , N representing the types of dynamicalsystems in each node. Fi = Fj if and only if fi(·) = fj(·), Bi = Bj andCi = Cj for any i, j ∈ 1, 2, . . . , N.For networks with invasive couplings (2), we can use the Algorithm 1 togenerate the partitions. Here, each partition is presented by a N -digit coded1 · · · di · · · dN . For example, for a network of 6 nodes 1, 2, 3, 4, 5, 6, the code001201 represents the partition 1, 2, 5, 3, 6, 4.For networks with non-invasive couplings (3), the procedure is similar withthe exception that there is no AR, i.e., when adding the sequential system i,Di = dj |Fi = Fj , j = 1, . . . , i− 1 ∪ max(d1, d2, . . . , di−1) + 1.With the partitions generated, the viability of these partitions can be checkedusing the row-sum test from Theorem 1.


Algorithm 1: Generate partitions (invasive coupling case), [18]

Input: AR = [R1 · · · RN ]>, AF = [F1 · · · FN ]>

Output: set of partition codes d1 · · · dN for systems 1, . . . , Nlet D1 = 0 be the set of partition codes for system 1 (d1=0);for i=2,. . . ,N do

set the set of partition codes for systems 1, . . . , i empty;foreach element in the set of partition codes for systems 1, . . . , i− 1 do

retrieve d1, . . . , di−1;compute the set Di =dj | Rj = Ri&Fi = Fj , j = 1, . . . , i− 1︸︷︷︸System i allowed in the same clsuters of other systems

due to same row sums and same dynamics

∪max(d1, . . . , di−1) + 1︸︷︷︸System i in a new clsuter

;

append every element of Di to the current code d1 · · · di−1 and add allcorresponding codes d1 · · · di in the set of partition codes for systems1, . . . , i;

end

end

3 Dynamics decomposition of partially synchronizednetwork

In this section, we present a systematic approach to split the dynamics of thenetwork into two parts: 1) dynamics of synchronization error; 2) dynamicsof solutions lying on the partial synchronization manifolds. The stability ofthe synchronization error reveals the stability of the partial synchronizationmanifolds.For simplicity, we assume the systems are ordered in a preliminary phaseaccording to a viable partition P associated with Π in κ clusters. The statevariables are also re-indexed such that xi,j denote the j-th state in the i-thcluster, that is,

x1,1, x1,2, . . . , x1,κ1cluster 1,

x2,1, x2,2, . . . , x2,κ2cluster 2,

......

xκ,1, xκ,2, . . . , xκ,κκ cluster κ,

where κi as the number of nodes in cluster i and κ1 + · · ·κκ = N .In order to split the dynamics on the manifold from the error dynamics, wedefine another new set of coordinates


z = Hx =

x1,1x2,1

...xκ,1E1

E2

...Eκ

, with Ei =

ei,2...ei,κi

=

xi,2 − xi,1...xi,κi − xi,1

, i = 1, . . . , κ. (8)

Here, H ∈ RnN×nN is an invertible matrix describing the transformationof the coordinates, and Ei denote the synchronization errors within clusteri. We refer the systems corresponding to x1,1, x2,1, . . . , xκ,1 as the referencesystems of each cluster.Since Π is a viable partition, the row sum of all (off-diagonal) blocks in(7) are constant for networks with invasive (non-invasive) couplings. LetRi,j be the value of the row sums of the (i, j)-th block of matrix A, withi, j ∈ 1, . . . , κ and, in case of non-invasive coupling, i 6= j. Additionally,for i = j, we define Ri,j = 0 in case of non-invasive coupling. Also, the dy-namical systems at the nodes in each cluster are the same. In this case, letus denote the dynamics of the nodes in cluster i by gi, Bi, Ci, i = 1, 2, . . . , κ.That is, g1 = f1 = f2 = · · · = fκ1 , g2 = fκ1+1 = fκ1+1 = · · · = fκ1+κ2 ,B1 = B1 = B2 = · · · = Bκ1

, C1 = C1 = C2 = · · · = Cκ1, and so on.

3.1 Synchronization error dynamics

3.1.1 Network with invasive coupling

Let

Ri =

κ∑j=1

Ri,j , i = 1, . . . , κ

denote the row sums of the adjacency matrix, corresponding to each cluster.Then the error dynamics can be expressed as


e1,2(t)...

e1,κ1(t)...

eκ,2(t)...

eκ,κκ(t)

=

g1(x1,2(t))− g1(x1,1(t))...

g1(x1,κ1(t))− g1(x1,1(t))...

gκ(xκ,2(t))− gκ(xκ,1(t))...

gκ(xκ,κκ(t))− gκ(xκ,1(t))

−

kB1C1R1e1,2(t)...

kB1C1R1e1,κ1(t)...

kBκCκRκeκ,2(t)...

kBκCκRκeκ,κκ(t)

+ kB(Ared ⊗ Im)C

e1,2(t− τ)...

e1,κ1(t− τ)...

eκ,2(t− τ)...

eκ,κκ(t− τ)

, (9)

withB = diag(Iκ1−1 ⊗ B1, . . . , Iκκ−1 ⊗ Bκ),

C = diag(Iκ1−1 ⊗ C1, . . . , Iκκ−1 ⊗ Cκ),

andAred = T1AT

T1 − T2ATT1 , (10)

where T1, T2 ∈ R(N−κ)×N are defined as

T1 =

0 1 · · · 0...

. . ....

1 00 · · · 0 1

. . .

0 1 · · · 0...

. . ....

1 00 · · · 0 1

, T2 =

1 0 · · · 01 0 · · · 0...

......

1 0 · · · 0. . .

1 0 · · · 01 0 · · · 0...

......

1 0 · · · 0

.

The linearized error dynamics are described by

10 Libo Su, Wim Michiels, Erik Steur and Henk Nijmeijer. δE1(t)...

δEκ(t)

=

Iκ1−1 ⊗ (∂g1∂x (x1,1(t))− kR1B1C1) 0. . .

0 Iκκ−1 ⊗ (∂gκ∂x (xκ,1(t))− kRκBκCκ)

δE1(t)

...δEκ(t)

+ kB(Ared ⊗ Im)C

δE1(t− τ)...

δEκ(t− τ)

. (11)

The first term in (10) corresponds to matrix A where the rows and columns,corresponding to the reference systems (described by x1,1, . . . , xκ,1) are takenout. The second term has the following form, which corresponds to matrixA where the columns corresponding to the reference systems are taken out,and the rows corresponding to the reference systems are repeated for (κi−1)times in the rows corresponding to cluster i.

T2ATT1 =

a1,2 · · · a1,κ1

......

a1,2 · · · a1,κ1

a1,κ1+2 · · · a1,κ1+κ2

......

a1,κ1+2 · · · a1,κ1+κ2

· · ·

......

aN−κκ+1,2 · · · aN−κκ+1,κ1

......

aN−κκ+1,2 · · · aN−κκ+1,κ1

aN−κκ+1,κ1+2 · · · aN−κκ+1,κ1+κ2

......

aN−κκ+1,κ1+2 · · · aN−κκ+1,κ1+κ2

· · ·

.

Note that if Bi = B and Ci = C for all i = 1, . . . , N , the term kB(Ared⊗Im)Creduces to kAred ⊗BC.The following result concerns the spectrum of Ared.

Theorem 2. Let σΠ(A) be the part of the spectrum of A, corresponding to theinvariant subspace ker(I −Π), i.e., all eigenvaules of A whose correspondingeigenspaces belong to ker(I −Π). Then we have

σ(Ared) = σ(A) \ σΠ(A). (12)

Proof. When the set M(Π) is a partial synchronization manifold, ker(I−Π)is a right invariant subspace of A, see [16] for details. That is,

ker(I −Π) = κ∑i=1

civi|ci ∈ R,


where vi are the (generalized) eigenvectors of A, corresponding to σΠ(A).Using H defined in (8), we can construct matrix A as follows

A = HAH−1 =

[A11 A12

0 Ared

], (13)

where A11 ∈ Rκ×κ, A12 ∈ Rκ×(N−κ) and 0 ∈ R(N−κ)×κ.Based on the structure of A, we have

σ(A) = σ(A) = σ(A11)⊕ σ(Ared).

As vi ∈ ker(I − Π), i.e., vi = Πvi, the vectors wi = Hvi will have thefollowing form

wi =

[wi0

](14)

where wi ∈ Rκ and 0 ∈ RN−κ, i = 1, 2, . . . , κ.If for some i ∈ 1, 2, . . . , κ, vi is an eigenvector of A, wi is an eigenvector ofA, which indicates

A

[wi0

]=

[A11 A12

0 Ared

] [wi0

]= λi

[wi0

].

Therefore,A11wi = λiwi.

A similar argument holds for the generalized eigenvectors of A. It follows that

σ(A11) = σΠ(A).

Hence, we can conclude (12), completing the proof.

3.1.2 Network with non-invasive coupling

When defining a series of vectors Ri ∈ Rκi−1 with their elements being therow sums of the rows in A corresponding to xi,2, . . . , xi,κi for i = 1, . . . , κ,the error dynamics of networks with non-invasive couplings can be expressedas


e1,2(t)...

e1,κ1(t)

...eκ,2(t)

...eκ,κκ(t)

=

g1(x1,2(t))− g1(x1,1(t))...

g1(x1,κ1(t))− g1(x1,1(t))

...gκ(xκ,2(t))− gκ(xκ,1(t))

...gκ(xκ,κκ(t))− gκ(xκ,1(t))

− kB(Lred ⊗ Im)C

e1,2(t− τ)...

e1,κ1(t− τ)...

eκ,2(t− τ)...

eκ,κκ(t− τ)

,

(15)with

Lred = diag(R1, . . . , Rκ

)−Ared. (16)

The linearized error dynamics is given by δE1(t)...

δEκ(t)

=

Iκ1−1 ⊗ ∂g1∂x (x1,1(t)) 0

. . .

0 Iκκ−1 ⊗ ∂gκ∂x (xκ,1(t))

δE1(t)

...δEκ(t)

− kB(Lred ⊗ Im)C

δE1(t− τ)...

δEκ(t− τ)

. (17)

Again, if Bi = B and Ci = C for all i = 1, . . . , N , the term kB(Lred ⊗ Im)Creduces to kLred ⊗BC.The following result concerns the spectrum of Lred.

Theorem 3. Let σΠ(L) be the part of the spectrum of L, corresponding to theinvariant subspace ker(I −Π), i.e., all eigenvaules of L whose correspondingeigenspaces belong to ker(I −Π). Then we have

σ(Lred) = σ(L) \ σΠ(L). (18)

Proof. Replace A, A11, A12 and Ared with L, L11, and Lred in the proof ofTheorem 2.

3.2 Dynamics on the synchronization manifolds

3.2.1 Network with invasive coupling

For a network with invasive coupling, the dynamics on the synchronizationmanifolds can be described by a network of κ coupled systems:


xi,1(t) = gi(xi,1(t)) + kBi

κ∑j=1

Ri,j

(Cjxj,1(t− τ)− Cixi,1(t)

), (19)

where i = 1, . . . , κ.This reduced network can be represented by graph G = (V, E , Asyn), where

|V| = κ and Asyn = [Ri,j ]κi,j=1 ∈ Rκ×κ.

Note that the “reduced” adjacency matrix [Ri,j ]κi,j=1 has in general diagonal

elements different from zero, implying the presence of self-loops in (19).

3.2.2 Network with non-invasive coupling

For a network with non-invasive coupling, the dynamics on the synchroniza-tion manifolds can be described by a network of κ coupled systems:

xi,1(t) = fi(xi,1(t)) + kBi

κ∑j=1, j 6=i

Ri,j

(Cjxj,1(t− τ)− Cixi,1(t− τ)

), (20)

where i = 1, . . . , κ.This reduced network can be represented by graph G = (V, E , Asyn), where

|V| = κ and Asyn = [Ri,j ]κi,j=1 ∈ Rκ×κ. Note that Rij = 0 if i = j.

It is important to note that this coupling type may affect the dynamics onthe synchronization manifold. Hence, strictly speaking, the terminology “non-invasive” only applies to fully synchronous solutions.

Note: For a stable partial synchronization manifold, to observe the cor-responding partial synchrony in practice, it also requires that xi,1(t) 6=xj,1(t),∀t ≥ t0, for every i 6= j ∈ 1, . . . , κ, which means the reduced net-work has no stable partial synchronization manifold.

3.3 Software

The algorithm proposed in Section 2 and the results we presented in thissection are implemented in the software such that it is able to:

• compute partial synchronization manifolds given a network (adjacency ma-trix A and coupling type, dynamical system types of each nodes)

• compute the matrices describing the synchronization error dynamics (Ared

or Lred) and the dynamics on the manifolds (Asyn) given a viable partition.

The second part can be achieved by using (10) or (16), and Asyn = [Rij ] easily.

The input of the software includes the adjacency matrix A of a network. If A


contains non-integer elements, a pre-defined tolerance is required for perform-ing the row sum test. If A only contains integer elements, the calculation inthe software can be performed in exact arithmetic. If the adjacency matrix Ais decomposable in the way defined in [16], (i.e., A = ω1A1+. . .+ωrAr, whereωi, i = 1, . . . , r are rationally independent and Ai, i = 1, . . . , r are adjacencymatrices with non-negative integer entries), and if the input A1, A2, . . . , Aris provided, the calculations can also be done in exact arithmetic.The software is implemented as a MATLAB function and can be downloadedfrom the following link:http://twr.cs.kuleuven.be/research/software/delay-control/manifolds/.

4 Example

We present a didactic example of a network with 4 interconnected systems toshow the analysis process presented in the preceding sections. The systemsare of the following form.

xi(t) = A0xi(t) +Bui(t),

yi(t) = Cxi(t),(21)

with

A0 =

[−ε 1−Ω2 −ε

], B =

[01

], C =

[1 0], Ω, ε ∈ R≥0 and i = 1, 2, 3, 4.

Here ui describes the couplings between the systems:

ui = k∑j∈Ni

aij [yj(t− τ)− yi(t− τ)], (22)

where aij are the elements of the adjacency matrix A defined as below

A =

0 0.1 2 1

0.1 0 1 24 2 0 0.33 3 0.3 0

, (23)

and where the coupling strength k and the time delay τ are considered asparameters.Since the network only contains 4 nodes which host the same linear system,it is possible to characterize and analyze the partial manifolds of the networkanalytically.Combining the systems of all the nodes, we have

Method for partial synchronization manifolds 15x1(t)x2(t)x3(t)x4(t)

= I ⊗A0

x1(t)x2(t)x3(t)x4(t)

− kL⊗BCx1(t− τ)x2(t− τ)x3(t− τ)x4(t− τ)

. (24)

Here is L is the weighted Laplacian matrix, defined by

L := D −A =

3.1 −0.1 −2 −1−0.1 −3.1 −1 −2−4 −2 −6.3 −0.3−3 −3 −0.3 6.3

, (25)

where

D =

∑Nj=1 a1j 0

. . .

0∑Nj=1 aNj

=

3.1 0 0 00 3.2 0 00 0 6.3 00 0 0 6.3

.The eigenvalues and eigenvectors of L are

σ(L) =

0

2.939.006.87

, V =

0.50 0.56 −0.32 0.150.50 −0.79 −0.32 −0.220.50 0.18 0.63 −0.630.50 −0.19 0.63 0.73

. (26)

First, we explain how to compute the stability regions of (21)-(22) in the(k, τ) parameter space. The characteristic function of (24) is given by

γ(λ; k, τ) = det(I ⊗ (λI −A0)− L⊗ kBCe−λτ ). (27)

As in [23], this function can be fractorized as

γ(λ; k, τ) =

4∏i=1

γi(λ; k, τ), (28)

withγi(λ; k, τ) = det(λI −A0 − kBCλi(L)e−λτ ), i = 1, 2, 3, 4. (29)

Denoting k = λi(L)k, each corresponding characteristic equation is of thefollowing form

(λ+ ε)2 +Ω2 + ke−λτ = 0. (30)

To analyze the stability of (24) at the equilibrium, substitute λ = jω, ω > 0,yields:

−ω2 + ε2 + 2jεω +Ω2 = −ke−jωτ . (31)


Equating the module and phase of the left side with the right side of (31),we have

(−ω2 + ε2 +Ω2)2 + (2εω)2 = k2, (32)

arctan(2εω

−ω2 + ε2 +Ω2) + 2πl = −ωτ, l ∈ Z. (33)

From these two equations, we can express k and τ as function of ω, i.e.,k := k(ω) and τ := τ(ω). Therefore, by sweeping ω, we can depict thecrossing curves of the time delay system numerically in the parameter space(τ, k). Fig. 1 shows the stability crossing curves of the system with Ω = 1and ε = 0.02. These values are used in the remainder of the example. Thestability region is marked with the symbol “S”. Note that for ω = 0, we havek = −(ε2 +Ω2).Next, we analyze the partial manifolds of this system using the procedure andsoftware we presented in the preceding section. Using the software developedin this work, we can easily identify a viable partition P = 1, 2, 3, 4 andderive the separated dynamics.The synchronization error dynamics are described by[

E1(t)

E2(t)

]=

[A0 00 A0

] [E1(t)E2(t)

]− kLred ⊗BC

[E1(t− τ)E2(t− τ)

], (34)

with

Lred =

[3.2 −1−1 6.6

], σ(Lred) =

[2.936.87

]. (35)

We can see that σ(Lred) ⊂ σ(L).By comparing (24) and (34), we observe they are of the same form (butdifferent dimensions). Therefore, the factorized characteristic function is

2∏i=1

γi(λ; k, τ) =

2∏i=1

det(λI −A0 − kBCλi(Lred)e−λτ ), (36)

which is part of (29).The dynamics on the manifold are given by[

x1x3

]=

[A0 00 A0

] [x1(t)x3(t)

]+

[kBCR12(x3(t− τ)− x1(t− τ))kBCR21(x1(t− τ)− x3(t− τ))

], (37)

i.e., [x1x3

]=

[A0 00 A0

] [x1(t)x3(t)

]+ kLsyn ⊗BC

[x1(t− τ)x3(t− τ)

], (38)

where

Lsyn =

[−R12 R12

R21 −R21

]=

[−3 36 −6

], σ(Lsyn) =

[09

]. (39)


We can see that σ(Lred) = σ(L) \ σ(Lsyn), which verifies Theorem 3.Again, the factorized characteristic function of (38) shown below is part of(29):

2∏i=1

γi(λ; k, τ) =

2∏i=1

det(λI −A0 + εI − kBCλi(Lsyn)e−λτ ). (40)

A comparision between (26) on one hand, and (35), (39) on the other hand,yields additional information that the eigenvalues λ1(L) = 0 and λ3(L) = 9are related to the dynamics on the manifolds, and that the eigenvaluesλ2(L) = 2.93 and λ4(L) = 6.87 are related to the synchronization errorsdynamics. Note that the decomposition can already be observed from thepatterns of the eigenvectors in (26), see also Theorem 2 in [24]. However,when the network become complicated (e.g. increased size), analysis by handmay become extremely difficult. In this case, the procedure we proposed willshow its advantage, as it is automatic and more systematic.

0 1 2 3 4 5 6 7 8 9−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

S

S

S

τ

k=kλi(L)

Stability crossing curves in the (τ, k) parameter space

Case 1Case 2

Fig. 1 Stability Regions of (21)-(22) with Ω = 1 and ε = 0.02.

Now, let us check the stability crossing curve of the time delay system de-scribed by (31). From Fig. 1, we can see that the parameters k and τ impactthe stability of the partial synchronization the partial synchronization man-ifolds. To illustrate the impact, we show two following cases

• τ = 5.0 and k = 0.05;


• τ = 5.0 and k = 0.08.

For Case 1, we have [k1 k2 k3 k4] = [0 0.15 0.45 0.34]. The parameter pairs(τ, ki) are shown in Fig. 1. All the points are inside the stability region.Therefore, all the dynamics are stable.For Case 2, we have [k1 k2 k3 k4] = [0 0.23 0.75 0.55]. The parameter pairs(τ, ki) are shown in Fig. 1. As we can see from it, the pair (τ, k3) is outsideof the stability region while the other three pairs are inside the stabilityregion. From the analysis above, we know k3 is related to the dynamics onthe manifold. Therefore, the partial synchronization manifold is stable, whilethe dynamics on the manifold is unstable. Fig. 4 shows the simulation resultsof the network. Note that the differences between y1 and y2, respectively,between y3 and y4 are invisible after some time.Finally, we also add a nonlinear term in each system.

xi(t) = A0xi(t) +Bui(t) + h(xi(t)),

yi(t) = Cxi(t),(41)

where

h(xi(t)) =

[−vi(t)(v2i (t) + w2

i (t))−vi(t)(v2i (t) + w2

i (t))

], (42)

and

xi(t) =

[vi(t)wi(t)

], vi, wi ∈ R.

The effect induced by the nonlinear term is that after the stability of thesynchronized equilibrium is lost by increasing k, the partially synchronoussolutions converge to a limit cycle. The latter is shown in Fig. 3.It is worth mentioning that matrix (25) reveals that the coupling of the nodeswithin clusters is weak, and that the coupling of the nodes between clustersis strong. In this sense, it might be surprising that, when k is increased,the dynamics on the manifold associated with P = 1, 2, 3, 4 becomeunstable first, instead of breaking the synchrony within the clusters. However,this can be explained by the presence of the time delay and the sensitivity ofhigh gain feedback with respect to it.

5 Conclusion

In this chapter, we present a method to compute and systematically ana-lyze the partial synchronization manifolds of delay coupled systems, whichis realized as a MATLAB software. First, we extend the method from [16]and [18] to allow the systems in the networks to be different. Secondly, wesplit the dynamics of the network into two parts: 1) synchronization errordynamics; 2) dynamics on manifolds. The stability of the synchronization


0 20 40 60 80 100 120 140−5

−4

−3

−2

−1

0

1

2

3

4

5

time t

solutiony

Network of 4 Nodes

y1

y2

y3

y4

Fig. 2 Simulation results of the network with τ = 5.00 and k = 0.08

error dynamics is a direct indicator of the stability of partial synchroniza-tion manifolds. We also specify the relations between the spectrum of theseparated dynamics and that of the original networks. To demonstrate theprocedure we propose for analysis of partial synchronization manifolds, wepresent a didactic example. Using the procedure along with the software, thedynamics of the network can be systematically separated, which can be usedfor stability analysis of the manifolds.


0 20 40 60 80 100 120 140 160 180 200−0.6

−0.4

−0.2

0

0.2

0.4

0.6

time t

solutiony

Network of 4 Nodes

y1

y2

y3

y4

Fig. 3 Simulation of the network (τ = 5.00, k = 0.08) with the nonlinear term

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Acknowledgements This research was supported by the project UCoCoS, fundedby the European Unions Horizon 2020 research and innovation program under theMarie Skodowska-Curie Grant Agreement No 675080.

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