Exact exponential synchronization rate of high-dimensional Kuramoto models with identical oscillators and digraphs
Shanshan Peng, Jinxing Zhang, Jiandong Zhu, Jianquan Lu, Xiaodi Li
aa r X i v : . [ m a t h - ph ] F e b Exact exponential synchronization rate of high-dimensionalKuramoto models with identical oscillators and digraphs
Shanshan Peng a , b , Jinxing Zhang c , Jiandong Zhu a , Jianquan Lu b , Xiaodi Li c a School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210023, PRC b School of Mathematics, Southeast University, Nanjing, 211189, PRC c School of Mathematics and Statistics, Shandong Normal University, Jinan, 250358, PRC
Abstract
For the high-dimensional Kuramoto model with identical oscillators under a general digraph that has a directed spanning tree,although exponential synchronization was proved under some initial state constraints, the exact exponential synchronizationrate has not been revealed until now. In this paper, the exponential synchronization rate is precisely determined as the smallestnon-zero real part of Laplacian eigenvalues of the digraph. Our obtained result extends the existing results from the specialcase of strongly connected balanced digraphs to the condition of general digraphs owning directed spanning trees, which isthe weakest condition for synchronization from the aspect of network structure. Moreover, our adopted method is completelydifferent from and much more elementary than the previous differential geometry method.
Key words:
High-dimensional Kuramoto model; Exponential synchronization rate; Directed graph.
The well-known Kuramoto model proposed by YoshikiKuramoto in [8] is one of the most successful mathemat-ical models to describe collective behaviors of complexdynamical networks. Kuramoto model and its variousgeneralized forms have many applications in engineer-ing, neuroscience, physics and so on [18]. For Kuramotomodel, exponential synchronization is an typical collec-tive behavior and an interesting theoretical issue [6], [7],[20], [21]. The dynamics of the general Kuramoto modelis described by˙ θ i = ω i + k m X j =1 a ij sin( θ j − θ i ) , i = 1 , , . . . , m, (1) ⋆ This work is supported in part by National Natural ScienceFoundation (NNSF) of China under Grants 61673012 and11971240.
Email addresses: shanshan¯[email protected] (Shanshan Peng), [email protected] (JinxingZhang), [email protected] (Jiandong Zhu), [email protected] (Jianquan Lu), [email protected] (Xiaodi Li). where the m × m matrix ( a ij ) is the adjacency matrix ofthe interconnecting graph, θ i is the i -th oscillator’s phaseangle and ω i is the natural frequency. The n -dimensionalvector-valued Kuramoto model is described as follows:˙ r i = Ω i r i + k m X j =1 a ij ( r j − r Ti r j r Ti r i r i ) , i = 1 , , . . . , m, (2)where r i ∈ R n is the i -th oscillator’s state, Ω i is an n × n skew-symmetric matrix. When n = 2 and r i =[cos θ i , sin θ i ] T , by the n -dimensional Kuramoto model(2), it is easy to deduce the original Kuramoto model(1). If (2) has the identical oscillators, i.e. Ω i = Ω for i =1 , , . . . , m , one can assume, without loss of generality,that Ω i = 0 for i = 1 , , . . . , m [24]. Limited on the unitsphere S n − , dynamical network (2) with Ω i = 0 and k = 1 is reduced to˙ r i = m X j =1 a ij (cid:0) r j − ( r Ti r j ) r i (cid:1) , i = 1 , , . . . , m, (3)which is first proposed in [16] as a swarm model onspheres, and can be used to solve the max-cut problem.Due to Lohe’s pioneering literatures such as [11] and[12], a class of high-dimensional Kuramoto models is alsocalled Lohe model, which has potential applications to Preprint submitted to Automatica 9 February 2021 uantum systems. More general Kuramoto models de-fined on some matrix manifolds can be seen in [1], [15],[19].For the case of complete graphs, many theoretical resultson the synchronization of high-dimensional Kuramotomodel have been achieved such as [5] , [16] and [10].In our early paper [24], the synchronization limited onan open half-sphere is proved for the case of generalundirected connected graphs. If the states of particles arenot limited on an open half-sphere, some almost globalsynchronization results can also be obtained for the caseof undirected connected graphs [14], [25].For general directed graphs, the synchronization of thehigh-dimensional Kuramoto model is proved in [9] by us-ing a differential geometry method. However, in our pa-per [23], a simpler approach based on LaSalle invarianceprinciple is adopted to demonstrate the synchronizationunder the general directed graph condition. Just like theoriginal Kuramoto model, high-dimensional Kuramotomodel also has the dynamical property of exponentialsynchronization (see [4], [5], [9] and [22]). But so far,the exact exponential synchronization rate is only ob-tained for a very special kind of digraphs, i.e. stronglyconnected balanced digraphs. In our recent paper [22],the exponential synchronization is achieved for a generaldigraph admitting a spanning tree, but the exact expo-nential synchronization rate has not yet been obtained.In this paper, for the high-dimensional Kuramoto modelunder a general digraph containing a spanning tree, it isproved the exponential synchronization rate is also ex-actly Re( λ ) just like the case of strongly connected bal-anced digraphs considered in [9], where λ is the eigen-value of the Laplacian L with the smallest non-zero realpart.The rest of this paper is organized as follows. Section2 gives some preliminaries and the problem statement.Section 3 includes our main results. Section 4 is a sum-mary. For high-dimensional Kuramoto model (3), exponentialsynchronization is defined as follows:
Definition 2.1
It is said that the exponential synchro-nization for (3) is achieved if there exists a µ > and aconstant c > such that || r i ( t ) − r j ( t ) || ≤ c e − µt ∀ i, j = 1 , , . . . , m. (4) Moreover, we say that the exponential synchronizationrate is at least µ when (4) holds. The minimum µ sat-isfying (4) is just called the exponential synchronizationrate. The first result on the exponential synchronization of(3) under digraphs is obtained in [9], which is based ondifferential geometry method. The main contribution ofTheorem 1 and Corollary 1 of [9] can be rewritten asfollows:
Proposition 2.1
Consider the high-dimensional Ku-ramoto model (3) . If the interconnecting graph is stronglyconnected and balanced, then the local exponential syn-chronization is achieved and the exact exponential syn-chronization rate is the smallest non-zero real part of theLaplacian eigenvalues of the interconnecting digraph.
The target of this paper is to get the exact exponen-tial synchronization rate under the more general digraphcondition and the framework of matrix Riccati differen-tial equation proposed in our early paper [22].Let e ij = 1 − r Ti r j = 12 || r i − r j || . (5)It is easily seen that e ij = e ji , e ii = 0 and 0 ≤ e ij ≤ i, j = 1 , , . . . , m . Let E ( t ) = ( e ij ( t )) ∈ R m × m . By[22], the dynamics of E ( t ) is described by matrix Riccatidifferential equation˙ E = − LE − EL T − α ( E ) T − α T ( E )+Λ( E ) E + E Λ( E ) , (6)where L is the Laplacian matrix of the digraph, α ( E ) = ( α ( E ) , α ( E ) , . . . , α m ( E )) T ∈ R m , (7)with each α i ( E ) = m P l =1 a il e il , and Λ( E ) = diag( α ( E )) isthe diagonal matrix with the diagonal elements com-posed of α ( E ), α ( E ), . . . , α m ( E ).For the linear space R m × m , denote by S m the subspacecomposed of all the symmetric real matrices with all thediagonal entries being 0. Then the synchronization er-ror equation (6) is the dynamics restricted on S m . Let S m and K m be the m -order symmetric matrix subspaceand the m -order skew-symmetric matrix subspace, re-spectively. Then R m = S m ⊕ K m , S m = S m ⊕ D m , (8)where D m is the m -order diagonal matrix subspace and ⊕ denotes the direct sum. In our early paper [22], Lyapunov functions are designedby using the left eigenvector of the Laplacian matrix ofthe digraph and consequently the exponential synchro-nization is proved. However, we cannot get the exact ex-ponential synchronization rate by the approach of Lya-punov functions.2n this section, we turn to the Lyapunov’s first method,i.e. the approximate exponential linearization method toget the exact synchronization rate. It is straightforwardto check that the approximate linearized system of (6)can be rewritten as˙ E = − (cid:0) LE + EL T − ˆ L vec( E ) T m − m (vec( E )) T ˆ L T (cid:1) , (9)where vec( · ) : R m × m → R m denotes the column-stacking operator,ˆ L = L L . . . L m , L i = Row i ( L ) , i = 1 , , . . . , m. Considering (9), we define a linear transformation on R m × m by T ( X ) = LX + XL T − ˆ L vec( X ) T m − m (vec( X )) T ˆ L T . (10)So the exponential decay rate of E ( t ) is just the smallestnon-zero real part of eigenvalues of T ( X ) restricted on S m . It is not easy to directly compute the eigenvalues of T ( X ) restricted on S m . We first investigate propertiesof T ( X ) on R m . Proposition 3.1
Consider the linear transformation T ( X ) defined by (10) and the Lyapunov mapping S ( X ) = LX + XL T . For the subspaces S m , D m and K m , the following statements hold: (i) if X ∈ S m , then T ( X ) ∈ S m ; (ii) if X ∈ K m , then the projection of T ( X ) onto K m isjust S ( X ) . PROOF. (i) If X = X T , it is easy to check that (cid:0) T ( X ) (cid:1) T = T ( X ). Moreover, a straightforward compu-tation shows that δ T i T ( X ) δ i = 2 (cid:0) δ T i LXδ i − δ T i ˆ L vec( X ) T m δ i (cid:1) = 2 (cid:0) L i X i − δ T i ˆ L vec( X ) (cid:1) = 0 , where δ i is the m -dimensional vector whose i -th entry isone and the others are zero. So T ( X ) ∈ S m .(ii) If X = − X T , then( S ( X )) T = X T L T + LX T = −S ( X ) . Thus S ( X ) ∈ K m . From (10), it follows that T ( X ) = S ( X ) − ( ˆ L vec( X ) T m + m (vec( X )) T ˆ L T ) , where S ( X ) ∈ K m and ˆ L vec( X ) T m + m (vec( X )) T ˆ L T ∈ S m .So the projection of T ( X ) onto K m is just S ( X ). (cid:4) Denote by B , B and B the base of S m , D m and K m , re-spectively. By conclusion (i) of Proposition 3.1, we have T ( S m ) ⊂ S m and T ( D m ) ⊂ S m . So T [ B , B , B ] = [ B , B , B ] T T T T T . (11)By the properties of the Lyapunov mapping S ( · ) [2], [3],we get S [ B , B , B ] = [ B , B , B ] S S S S S . (12)From conclusion (ii) of Proposition 3.1, it follows that T = S . Fortunately, the eigenvalues of S , i.e. thespectrum of the Lyapunov mapping S ( · ) restricted on K m , have been revealed. Lemma 3.1 [3] Assume that the eigenvalues of L be λ , λ , λ , . . . , λ m . Then the eigenvalues of the Lyapunovmapping S ( · ) restricted on K m are { λ i + λ j | ≤ i Let A, B, C ∈ R n × n satisfy AB = BC and B = 0 . Then the eigenvalue set of A + B is the same asthat of A . PROOF. Let rank B = r . When r = 0 the assertion ofthe lemma is obviously right. When r > 0, there is an n -by- r matrix M and a r -by- n matrix N such that B = M N, rank M = rank N = r. (14)Since M has a full column rank, one can construct anonsingular matrix T = [ M, T ]. Let T − = [ P T , Q T ] T .Then " I r I n − r = T − T = " P M P T QM QT , (15)which implies that QM = 0 , QB = QM N = 0 . (16)Since N has a full row rank and BM N = B = 0, wehave that BM = 0. Thus T − BT = " PQ B [ M T ] = " P BT . (17)From AM N = AB = BC and (16), it follows that QAM = QBCN T ( N N T ) − = 0 . (18)Thus T − AT = " PQ A [ M T ] = " P AM P AT QAT . (19)By (17) and (19), the proof is complete. (cid:4) Proposition 3.2 Assume that the Laplacian matrix L satisfies rank ( L ) = m − and the eigenvalues of L are λ = 0 , λ , λ , . . . , λ m . Then the characteristic polyno-mial of T ( · ) is s m m Y i =2 ( s − λ i ) m Y j =2 ( s − λ j ) Y ≤ q
Since rank( L ) = m − L m = 0, thereexists a nonsingular matrix P = [ m , P ] which results the Jordan canonical form of L as follows: J = P − LP = " J = λ b . . . . . . λ m − b m − λ m , (21)where every λ i is nonzero and each b i is one or zero. Itis easy to check that( P − ⊗ P − )( L ⊗ I m + I m ⊗ L − m ⊗ ˆ L )( P ⊗ P )= J ⊗ I m + I m ⊗ J − δ ⊗ P − ˆ L ( P ⊗ P )= J ⊗ I m + I m ⊗ J − δ ⊗ P − ˆ L [ m ⊗ P, P ⊗ P ]= J ⊗ I m + I m ⊗ J − δ ⊗ [ P − LP, P − ˆ L ( P ⊗ P )]= J ⊗ I m + I m ⊗ J − δ ⊗ [ J, P − ˆ L ( P ⊗ P )]= " m × m ˜ J ⊗ I m + " J I m − ⊗ J − " J P − ˆ L ( P ⊗ P )0 0 = " m × m − P − ˆ L ( P ⊗ P )˜ J ⊗ I m + I m − ⊗ J . (22)From (22), it follows that the characteristic polynomialof L ⊗ I m + I m ⊗ L − m ⊗ ˆ L is just (20).In the following, we use Lemma 3.3 to complete theproof. Let A = L ⊗ I m + I m ⊗ L − m ⊗ ˆ L,B = ˆ L ⊗ m = m L m L . . . m L m . From L m = 0, it follows that B = 0, ( I m ⊗ L ) B = 0and ( m ⊗ ˆ L ) B = 0 . Now, we construct a matrix C suchthat ( L ⊗ I m ) B = BC , which is equivalent to l ij m L j = m L i C ij , ∀ ≤ i, j ≤ m. (23)When L i = 0, we have l ij = 0, which means that C ij can be any m -by- m matrix. When L i = 0, it is easyto check that C ij = l ij L T i ( L i L T i ) − L j satisfies (23). So,there exists a matrix C such that AB = BC . Therefore,by Lemma 3.3, A − B and A have the same characteristicpolynomial. The proof is complete. (cid:4) By Proposition 3.2, we have obtained all the eigenvaluesof the linear transformation T ( · ). Now we can determinethe exact exponential synchronization rate.4 heorem 3.1 Consider the high-dimensional Ku-ramoto model (3) limited on the unit sphere S n − . If theinterconnecting digraph with the adjacency matrix ( a ij ) has a spanning tree, then the exponential synchroniza-tion rate is exactly the smallest non-zero real part of theeigenvalues of Laplacian matrix L . PROOF. Since the interconnecting digraph with theadjacency matrix ( a ij ) has a spanning tree, the Lapla-cian matrix L satisfies rank L = m − T ( · ) asshown in (20), where λ = 0, λ , . . . , λ m are the eigen-values of L . By Lemma 3.2 and λ = 0, we get the thecharacteristic polynomial of T in (11) as follows: m Y i =2 ( s − λ i ) Y ≤ q 0. Considering | e ij ( t ) | = e ij ( t ) = || r i ( t ) − r j ( t ) || ,we conclude that the exponential synchronization rateis Re( λ ). (cid:4) Remark 3.1 The graph condition that the digraph hasa directed spanning tree is the weakest condition forsynchronization. Actually, if the digraph has no a di-rected spanning tree, there exist at least two independentstrongly connected components. Since there is no any in-formation interaction between two independent stronglyconnected components, it is impossible to achieve syn-chronization for all the possible initial states. 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