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Synchronization Patterns in Network-of-Networks of Chaotic Systems via Cartesian Product*
4th IFAC Conference on Analysis and Control of Chaotic Systems 4th IFAC26-28, Conference on Analysis Control of Chaotic Systems August 2015. Tokyo, Japan and 4th IFAC Conference on and Control of Chaotic Systems 4th IFAC26-28, Conference on Analysis Analysis August 2015. Tokyo, Japan and Control of Chaotic Systems Available online at www.sciencedirect.com August August 26-28, 26-28, 2015. 2015. Tokyo, Tokyo, Japan Japan
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Synchronization Patterns in Synchronization Patterns in Synchronization Patterns in SynchronizationofPatterns in Network-of-Networks Chaotic Systems Network-of-Networks of Chaotic Systems Network-of-Networks of Chaotic Systems Network-of-Networks of Chaotic Systems via Cartesian Product via Cartesian Product via Cartesian Product via Cartesian Product
Kanaru Oooka ∗∗ Toshiki Oguchi ∗∗ Kanaru Oooka ∗∗ Toshiki Oguchi ∗∗ Kanaru Kanaru Oooka Oooka Toshiki Toshiki Oguchi Oguchi ∗ ∗ Dept. of Mechanical Engineering, Tokyo Metropolitan University ∗ Dept. of Mechanical Engineering, Tokyo Metropolitan University ∗ Dept. Mechanical Tokyo Metropolitan 1-1,of Hachioji-shi, Tokyo 192-0397 University Japan Dept. ofMinami-Osawa, Mechanical Engineering, Engineering, Tokyo Metropolitan University 1-1, Minami-Osawa, Hachioji-shi, Tokyo 192-0397 Japan 1-1, Minami-Osawa, Hachioji-shi, Tokyo 192-0397 Japan (e-mail: [email protected], [email protected]). 1-1, Minami-Osawa, Hachioji-shi, Tokyo 192-0397 Japan (e-mail: [email protected], [email protected]). (e-mail: [email protected], [email protected]). (e-mail: [email protected], [email protected]). Abstract: This paper considers the synchronization problem in network-of-networks of chaotic Abstract: This paper considers the synchronization problem in network-of-networks of chaotic Abstract: This paper the problem in of systems with time-delay. We have already proposed an estimation method for full/partial Abstract: This paper considers considers the synchronization synchronization problem in network-of-networks network-of-networks of chaotic chaotic systems with time-delay. We have already proposed an estimation method for full/partial systems with time-delay. time-delay. We have already proposed an estimation estimation method product. for full/partial full/partial synchronization conditionsWe for have networks of chaotic systems via the Cartesian In this systems with already proposed an method for synchronization conditions for networks of chaotic systems via the Cartesian product. In this synchronization conditions for networks of chaotic systems via the Cartesian product. In this paper, we attempt to estimate full/partial synchronization conditions for the Cartesian product synchronization conditions for networks of chaotic systems via the Cartesian product. In this paper, we attempt to estimate full/partial synchronization conditions for the Cartesian product paper, we attempt to estimate full/partial synchronization conditions for the Cartesian product networks with time-delay by using the scaling method for synchronization condition and the paper, we attempt to estimate full/partial synchronization conditions for the Cartesian product networks with time-delay by using the scaling method for synchronization condition and the networks with time-delay by using the scaling method for synchronization condition and the properties of the Cartesian product. The validity of the estimation method is tested through networks with time-delay by using the scaling method for synchronization condition and theaa properties of the Cartesian product. The validity of the estimation method is tested through properties of the Cartesian product. The validity of the estimation method is tested through aa numerical example. properties of the Cartesian product. The validity of the estimation method is tested through numerical example. numerical example. numerical example. © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Synchronization, Chaotic systems, Cartesian product, Networks, Graph Laplacian Keywords: Synchronization, Keywords: Synchronization, Synchronization, Chaotic Chaotic systems, systems, Cartesian Cartesian product, product, Networks, Networks, Graph Graph Laplacian Laplacian Keywords: Chaotic systems, Cartesian product, Networks, Graph Laplacian 1. INTRODUCTION scaling method to estimate synchronization conditions to 1. INTRODUCTION scaling method to estimate synchronization conditions to 1. INTRODUCTION scaling synchronization conditions to network-of-networks via the Cartesian product and showed 1. INTRODUCTION scaling method method to to estimate estimate synchronization conditions to network-of-networks via the Cartesian product and showed network-of-networks via the Cartesian product and showed the validity of the estimated conditions. Recently, there is much attention being given to synchro- network-of-networks via the Cartesian product and showed the validity of the estimated conditions. Recently, there attention being given synchrothe validity of this paper, weestimated estimate conditions. full/partial synchronization Recently, phenomena there is is much much attention being given to to synchronization in attention networks being of coupled systems in In the validity of the the estimated conditions. Recently, there is much given to synchroIn this paper, we estimate full/partial synchronization nization phenomena in networks of coupled systems in In this paper, we estimate full/partial synchronization conditions for chaotic systems with delay-couplings by nization phenomena in networks of coupled systems in applied mathematics, physics, information technology and this paper, we estimate full/partial synchronization nization phenomena in networks of coupled systemsand in In conditions for chaotic systems with delay-couplings by applied mathematics, physics, information technology conditions for chaotic systems with delay-couplings by applying the scaling method and there exist some patterns applied mathematics, physics, information technology and various fields. For practical systems, we need to consider conditionsthefor chaotic systems with exist delay-couplings by applied mathematics, physics, information technology and applying scaling method and there some patterns various fields. For practical systems, we need to consider applying the scaling method and there exist some patterns of partial synchronization that cannot be observed for novarious fields. For practical systems, we need to consider that there exist delays in signal transmission between applying the scaling method and there exist some patterns various fields. For practical systems, we need to consider of partial synchronization that cannot be observed for nothat exist delays in between of partial synchronization be for coupling. In addition,that we cannot show that partial synchrothat there there exist The delays in signal signal transmission transmission between coupled systems. synchronization condition for non- delay of partial synchronization that cannot be observed observed for nonothat there exist delays in signal transmission between delay coupling. In addition, we show that partial synchrocoupled systems. The synchronization condition for nondelay coupling. In addition, we show that partial synchronization patterns in the Cartesian product networks can coupled systems. The synchronization condition for nonlinear systems with delay-couplings can be derived in a delay coupling. In addition, we show that partial synchrocoupled systems. The synchronization condition for nonnization patterns in the Cartesian product networks can linear with delay-couplings can be derived nization patterns in product networks can detected by using the Cartesian eigenvectors of graph Laplacians linearofsystems systems with delay-couplings can be based derivedonin intheaaa be form the linear matrix inequalities can (LMIs) nization patterns in the the Cartesian product networks can linear systems with delay-couplings be derived in be detected by using the eigenvectors of graph Laplacians form of the linear matrix inequalities (LMIs) based on the be detected by using the eigenvectors of graph Laplacians of two original networks. form of the linear matrix inequalities (LMIs) based on the Lyapunov-Krasovskii theorem. However, it is well known be detected by using the eigenvectors of graph Laplacians form of the linear matrix inequalities (LMIs) based on the of two original networks. Lyapunov-Krasovskii theorem. However, it well Lyapunov-Krasovskii theorem. However, it is is because well known known that this approach causes conservative results the of of two two original original networks. networks. Lyapunov-Krasovskii theorem. However, it is well known that this approach causes conservative results because the that this approach causes conservative results because the Lyapunov-Krasovskii theorem gives a sufficient condition 2. SYNCHRONIZATION IN NETWORKS OF that this approach causes conservative results because the Lyapunov-Krasovskii theorem a sufficient condition 2. SYNCHRONIZATION IN NETWORKS OF Lyapunov-Krasovskii theorem gives gives sufficient condition for stability and the condition itself aahas inherent conser2. SYNCHRONIZATION IN CHAOTIC SYSTEMS Lyapunov-Krasovskii theorem gives sufficient condition 2. SYNCHRONIZATION IN NETWORKS NETWORKS OF OF for stability and the condition itself has inherent conserCHAOTIC SYSTEMS for stability and the condition itself has inherent conservativeness. Furthermore, to reduce the obtained inequality CHAOTIC SYSTEMS for stabilityFurthermore, and the condition itself has inherent conserCHAOTIC SYSTEMS vativeness. to reduce the obtained inequality vativeness.toFurthermore, Furthermore, to reduce reduce the obtained inequality Throughout this paper, we consider the following N idencondition the LMI condition, thethe obtained synchronizavativeness. to obtained inequality condition to LMI condition, the obtained synchronizaThroughout this paper, we consider the following N idencondition to the thetends LMI to condition, the obtained synchronization condition become the extremely restrictive. On tical Throughout this paper, condition to the LMI condition, obtained synchronizanonlinear systems: Throughout this paper, we we consider consider the the following following N N idenidention condition tends to become extremely restrictive. On tical nonlinear systems: tion condition tends to become extremely restrictive. On the other hand, Oguchi and Nijmeijer (2011) proposed tical nonlinear systems: tion condition tends to become extremely restrictive. On tical nonlinear systems: i i i i the other hand, Oguchi and Nijmeijer (2011) proposed x˙ i (t) = Axi (t) + f (xi (t)) + Bui (t) the alternative other hand, hand,approach Oguchi and and Nijmeijer a (2011) (2011) proposed an for deriving frequency-based the other Oguchi Nijmeijer proposed x = Ax f (xii (t)) + Bu ˙ i (t) Σii : (1) i (t) an for deriving aa frequency-based ii (t) (t) = = Cx Ax (t) + + x an alternative alternative approach approach for deriving frequency-based Σ (1) i : y˙˙ iii(t) synchronization conditionfor using the multi-variable circle Axii (t) + ff (x (x (t)) (t)) + + Bu Bui (t) (t) an alternative approach deriving a frequency-based i Σ :: x (1) y i (t) = Cxi (t) synchronization condition using the multi-variable circle Σ (1) i (t) yy i (t) = Cx synchronization condition using the and multi-variable circle criterion (Blimancondition (2000)).using Mimura Oguchi (2012) (t) = Cx (t) i n i i synchronization the multi-variable circle criterion (Bliman (2000)). Mimura and Oguchi (2012) ∈ R is the state, u ∈ R and y ∈ R are the where x criterionthat (Bliman (2000)). Mimura Mimura and Oguchi Oguchi (2012) showed the synchronization conditions of Lur’e sys- where xiii ∈ Rnnn is the state, uiii ∈ R and y iii ∈ R are the criterion (Bliman (2000)). and (2012) R is state, yy ∈ the where x showed that the synchronization conditions of Lur’e sysi =u ·∈ · ·R , Nand , respectively. B ∈the R output is the the for state, u1, and ∈ R R are areA, the where and x ∈ showed that the synchronization conditions of Lur’e sys- input tems with arbitrary network topologies can be estimated input and the output for ii = 1, ··∈·· ·· R ,, N ,, respectively. A, B showed that the synchronization conditions of Lur’e sysinput and the output for = 1, N respectively. A, B tems with arbitrary network topologies can be estimated and C are constant matrices. In addition, we assume that input and the output for i = 1, · · · , N , respectively. A, B tems with arbitrary network topologies can be estimated by scaling the synchronization condition ofbetwo identi- and Cnare constant matrices. In addition, we assume that n tems with arbitrary network topologies can estimated and C are constant matrices. In addition, we assume that by scaling the synchronization condition of two identif : R → R is a sufficiently smooth vector field. The nare constant n and C matrices. In addition, we assume that by scaling the synchronization condition of two identical systems with delay-couplings by eigenvalues of the ff :: R smooth vector field. The n → Rn is a sufficiently by scaling the synchronization condition of two identi→ R is sufficiently vector cal delay-couplings by eigenvalues of uii is givensmooth by f : R Rnfor →each Rn system is aa N sufficiently smooth vector field. field. The The cal systems systems with with delay-couplings by estimation eigenvaluesmethod of the the input corresponding graph Laplacian. This input for each system u i is given by cal systems with delay-couplings by eigenvalues of the i input for each system u is given by corresponding graph Laplacian. This estimation method N input for each system u is given by i j i corresponding graph Laplacian. This estimation method provides a less conservative synchronization condition. A N corresponding graph Laplacian. This estimation method ui (t) = − (2) N kij (y i (t − τ ) − y j (t − τ )) provides aa less synchronization condition. A u (2) i (t) = − kij (y i (t − τ ) − y j (t − τ )) providesestimation less conservative conservative synchronization condition. A similar method issynchronization independently condition. proposed by i (t) = − j=1 kij (y i (t − τ ) − y j (t − τ )) u (2) provides a less conservative A u k (y (t − τ ) − y (t − τ )) (2) (t) = − similar estimation method is independently proposed by ij j=1 similar estimation method is independently proposed by Steur et al. (2012) for coupled chaotic systems with similar estimation method is independently proposed by j=1 Steur et al. (2012) for coupled chaotic systems with j=1 time-delay, and kij represents a where τ is a constant Steur semi-passivity et al. al. (2012) (2012)property for coupled coupled chaotic systems systems with strict (Pogromsky (2008)).with For where τ is a constant time-delay, and kij represents a Steur et for chaotic represents a where ττ strength is aa constant constant time-delay, and kij strict semi-passivity property (Pogromsky (2008)). For that iftime-delay, there existsand a coupling betweena where is k strict semi-passivity property (Pogromsky (2008)). For coupling ij represents network systems with delay-couplings, an experimental coupling strength that if there exists aa coupling between strict semi-passivity property (Pogromsky (2008)). For coupling strength that if there exists coupling between network systems with delay-couplings, an experimental systems i and j, k = k = k, and otherwise k = 0. In ij ji ij coupling strength that if there exists a coupling between network systems with delay-couplings, an experimental result for systems the relationship between the minimum non-zero ii and j, k k = 0. In network with delay-couplings, an experimental ij = ji = systems k = that k = k, k, and and otherwise otherwise k kij result relationship between the non-zero systems this paper, we j, assume ij = 0. In i and and j, kij kji result for for the the relationship between theisminimum minimum ij = that ji = k, and otherwise kij = 0. In eigenvalue andrelationship synchronization region shown bynon-zero Neefs et systems this paper, we assume result for the between the minimum non-zero paper, we that eigenvalue and synchronization region is shown by et this we assume assume that eigenvalue and synchronization region is(2013) shown applied by Neefs Neefsthe et this al. (2010). and Furthermore, Oguchiregion et al.is i) paper, all couplings are bidirectional eigenvalue synchronization shown by Neefs et al. (2010). Furthermore, Oguchi et al. (2013) applied the i) all couplings are bidirectional al. (2010). (2010). Furthermore, Furthermore, Oguchi Oguchi et et al. al. (2013) applied the i) all couplings are bidirectional ii) each system is strongly semi-passive. al. (2013) applied the i) all couplings are bidirectional ⋆ This work was partially supported by the Japan Society for the ii) each system is strongly semi-passive. ii) each system is strongly semi-passive. ⋆ This work was partially supported by the Japan Society for the ii) each system is strongly semi-passive. ⋆ Here the network topology is represented by an undirected Promotion of Science (JSPS) Grant-in-Aid for Scientific Research work was partially supported by the Japan Society for the ⋆ This This workofwas partially supported by the for Japan SocietyResearch for the Here the network topology is represented by an undirected Promotion Science (JSPS) Grant-in-Aid Scientific Here the network topology is by graph G with N nodes, and it is characterized by the (No.26420424) Promotion of Science (JSPS) Grant-in-Aid for Scientific Research Here the network topology is represented represented by an an undirected undirected Promotion of Science (JSPS) Grant-in-Aid for Scientific Research graph G with N nodes, and it is characterized by (No.26420424) graph G with N nodes, and it is characterized by the the (No.26420424) graph G with N nodes, and it is characterized by the (No.26420424)
Copyright © 2015, 2015 IFAC 13 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2015 IFAC 13 Copyright © 2015 IFAC 13 Peer review under responsibility of International Federation of Automatic Copyright © 2015 IFAC 13 Control. 10.1016/j.ifacol.2015.11.003
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corresponding graph Laplacian L(G) defined by the degree matrix ∆(G) and the adjacency matrix A(G). The degree matrix is the diagonal matrix whose entries are the vertex-degrees of G on the diagonal, that is ∆(G) = diag(d(v1 ), · · · , d(vN )), where d(vi ) is the degree of the vertex vi . The adjacency matrix A(G) = [aij ] is the symmetric N × N matrix encoding of the adjacency relationship in the graph G and given by � 1 if (vi , vj ) ∈ E(G) aij = , 0 otherwise
where E(G) is the edge set of the graph G. Then the graph Laplacian L(G) is defined by L(G) = ∆(G) − A(G). If the graph is connected, the graph Laplacian L(G) has only one zero-eigenvalue. Then the input vector u(t) = [u1 (t) · · · uN (t)]T can be described by u(t) = −kL(G)y(t − τ ) = −k(L(G) ⊗ C)x(t − τ ) (3) where y(t) = [y 1 (t) · · · y N (t)]T is the output vector and ⊗ denotes the Kronecker product. Now, the dynamics of the total network system is described by � x(t) ˙ = (IN ⊗ A)x(t) + F (x(t)) −k(IN ⊗ B)(L(G) ⊗ C)x(t − τ ) (4) Σ: y(t) = (IN ⊗ C)x(t) 1
T
N
Fig. 1. Graphical representation of the scaling estimation method : S is the synchronization condition for K2 , and S2 , · · · , SN are the scaled synchronization conditions corresponding to the eigenvalues of L(G). Full synchronization is achieved in the region ovarlapped with all regions, and partial synchronization may be happened in the region overlapped with a part of regions. ψ(x1 (t), e12 (t)) .. Ψ(x1 (t), e(t)) = , . ψ(x1 (t), e1N (t))
T T
where x(t) = [(x (t)) · · · (x (t)) ] , IN is the identity matrix with N -dimension, and F (x(t)) = [f (x1 (t)) · · · f (xN (t))]T . To introduce an estimation method of the synchronization condition based on the scaling method (Mimura and Oguchi (2012)), we consider the complete graph KN with N nodes and define the synchronization error vector e(t) as 1 x (t) − x2 (t) e12 (t) .. e(t) := ... = = (M0 ⊗ In )x(t), (5) . 1 N e1N (t) x (t) − x (t)
where
where ψ(x1 (t), e1i (t)) = f (x1 (t)) − f (x1 (t) − e1i (t)) and 1N −1 = [1 · · · 1]T ∈ RN −1 . 2.1 Full synchronization in networks First, we consider the full synchronization problem. Full synchronization of systems to be considered here is defined as follows: Definition 1. (Oguchi and Nijmeijer (2011)) If there exists a positive real number r such that the trajectories xi (t) of the systems (1) with initial conditions φi , φj such that ||φi − φj ||C ≤ r satisfy ||xi (t) − xj (t)|| → 0 as t → ∞ for all i, j, then the coupled systems (1) and (2) are asymptotically synchronized. Here ||φ||C := max−τ ≤θ≤0 ||φ(θ)|| stands for the norm of a vector function φ, where || · || refers to the Euclidean vector norm.
1 −1 0 (N −1)×N .. . M0 = ... ∈R . 1 0 −1
Applying the coordinate transformation 1 0 · · · 0 � 1 � .. . 1 −1 x (t) = . ⊗ In x(t), .. e(t) . . 0 . 1 0 −1 the dynamics (4) is rewritten as follows: � 1 � � 1 � x˙ (t) x (t) = (IN ⊗ A) + Ψ(x1 (t), e(t)) e(t) ˙ e(t) � � �� 1 0 1N −1 ⊗ BC x (t − τ ) −k e(t − τ ) 0 M0 LM0+ ⊗ BC
Let L(KN ) denote the graph Laplacian of the complete graph KN . Then it is diagonalizable and its eigenvalues are N with multiplicity N − 1 and a zero. Furthermore, since M0 LM0+ = diag(N, · · · , N ) holds, the dynamics of e(t) defined in (5) are blockdiagonalized and decomposed into N − 1 identical dynamics given by e˙ 1i = Ae1i (t) + ψ(x1 (t), e1i (t)) − kN BCe1i (t − τ ). (7) Now we consider the synchronization condition for two bidirectional coupled systems. The error dynamics is given by e˙ 12 = Ae12 (t) + ψ(x1 (t), e12 (t)) − 2kBCe12 (t − τ ). (8) If the origin of (8) is asymptotically stable, the coupled systems are asymptotically synchronized. Here we assume that S is a set of pairs (k, τ ) such that the systems achieve synchronization. Comparing (7) with (8), all coupled systems must be synchronized if ( kN 2 , τ ) ∈ S. Therefore, we can estimate the synchronization condition for N coupled
(6)
where M0+ denotes the pseudo-inverse matrix of M0 given by 0 0 . −1 . . + M0 = ∈ RN ×(N −1) , .. . 0 0 −1 14
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systems with complete graph network structure by scaling S with respect to the coupling strength k. Next, we consider the synchronization condition for more general networks G. In this case, the Laplacian L(G) has various eigenvalues, and partial synchronization may emerge. For undirected graph G, there always exists a nonsingular matrix P such that M0 LM0+ is diagonalized as P −1 M0 LM0+ P = diag(λ2 , · · · , λN ), where 0 = λ1 < λ2 ≤ · · · ≤ λN are eigenvalues of L(G). Applying the coordinate transformation e(t) = (P ⊗ In )¯ e(t), the error dynamics are transformed into e1i (t) + ψ ′ (x1 (t))¯ e1i (t) − λi kBC e¯1i (t − τ ) (9) e¯˙ 1i (t) = A¯
15
Fig. 2. Simplified sandglass network (left) and the complete graph with two nodes (right) the Cartesian product graph of G1 and G2 is represented as G = G1 ✷G2 , and V(G) = V1 × V2 . The graph Laplacian of a pair of graphs G1 and G2 and the Cartesian product graph G satisfy the following property.
where i = 2, · · · , N . Let Si be a set of pairs (k, τ ) that makes the origin of the error dynamics (9) asymptotically stable. Then the set Si (G) can be estimated by scaling S in the coupling strength axis with respect to the eigenvalue as follows: λi k , τ ∈ S} Si (G) = {(k, τ )| 2 ¯ Furthermore, if there exists S(G) = ∩i∈I Si (G) �= ∅(where I = {2, · · · , N }), the origin of the error dynamics are ¯ asymptotically stable for (k, τ ) ∈ S(G), that is, the network system achieves full synchronization satisfying x1 (t) = x2 (t) = · · · = xN (t).
L(G1 ✷G2 ) = L(G1 ) ⊗ IM + IN ⊗ L(G2 )
= L(G1 ) ⊕ L(G2 ) where N and M are the numbers of nodes of G1 and G2 , respectively, and ⊕ denotes the Kronecker sum. It is known that the eigenvalue and the corresponding eigenvector of the graph Laplacian of the Cartesian product graph satisfy following lemma. Lemma 3. (Mesbahi and Egerstedt (2010)) Let G1 and G2 be a pair of graphs on N and M nodes, respectively. Furthermore, assume that λ1 , λ2 , · · · , λN and µ1 , µ2 , · · · , µM are the eigenvalues of L(G1 ) and L(G2 ), respectively, corresponding to the eigenvectors v1 , v2 , · · · , vN and w1 , w2 , · · · , wM Then vi ⊗ wj , i = 1, 2, · · · , N, j = 1, 2, · · · , M is the eigenvector associated with the eigenvalue λi + µj of L(G1 ✷G2 )
2.2 Partial synchronization in networks Next we consider the partial synchronization problem. Partial synchronization in networks is defined as follows: Definition 2. (Mimura and Oguchi (2012)) If there exists a positive real number r such that the trajectories xi (t) and xj (t) of a part of systems i and j in networks with the initial conditions φi , φj such that ||φi − φj ||C ≤ r satisfy ||xi (t) − xj (t)|| → 0 as t → ∞ for some i, j, then the systems i and j are asymptotically synchronized and the network systems is called to be partially synchronized.
The second smallest eigenvalue of L(G1 ✷G2 ) is given by λ2 (G1 ✷G2 ) = min{λ2 (G1 ), λ2 (G2 )} As a result, the synchronization condition for the Cartesian product network can be estimated by using the scaling method with respect to the eigenvalues of the original two networks. We assume that the control inputs for each system in two original networks are given by uG1 (t) = −k1 L(G1 )y(t − τ ) for G1 uG2 (t) = −k2 L(G2 )y(t − τ ) for G2
Let the set I c{p} = I \ {p} and |I c{p} | = M . Then if there exists S¯c{p} (G) = ∩i∈I c{p} Si (G) �= ∅, the origin of M error dynamics e¯1i for i ∈ I c{p} is asymptotically stable for (k, τ ) ∈ S¯c{p} (G), and this implies the following simultaneous equations hold. (10) e¯1i (t) = (viT ⊗ In )x(t) = 0 for i ∈ I c{p} where vi is the corresponding eigenvector with the eigenvalue λi of L(G). Moreover, if there exists the solution of (10) in a form that xj (t) = xk (t) for some j, k ∈ I, the systems j and k achieve partial synchronization. If there is no time-delay, i.e. τ = 0, the partial synchronization pattern is unique. On the other hand, if τ > 0, another partial synchronization pattern that cannot be observed for τ = 0 may be emerged.
Then, the total dynamics of the system in the Cartesian product network G = G1 ✷G2 is rewritten as
x˙ = (IN M ⊗ A)x(t) + F (x) − ((k1 L(G1 ) ⊕ k2 L(G2 )) ⊗ BC)x(t − τ ) (11) where x = [(x(1,1) )T · · · (x(1,M) )T · · · (x(N,1) )T · · · (x(N,M) )T ]T . The synchronization condition for the Cartesian product network can be estimated by scaling the condition for K2 as k1 λi + k2 µj , τ ∈ S}, S(i,j) (G) = {(k1 , k2 , τ )| 2 and if there exists the set ¯ S(G) = ∩(i,j)∈I c{(1,1)} S(i,j) (G) �= ∅,
3. SYNCHRONIZATION CONDITION IN CARTESIAN PRODUCT NETWORK In this section, we consider synchronization in the networkof-networks via the Cartesian product. Here, we consider two undirected graphs G1 = (V1 , E1 ) and G2 = (V2 , E2 ), where Vi are node sets of the graph Gi . Then
N ×M
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4
λ2 λ3 λ4 λ5 λ6 λ7 λ8
3.5
Time delay τ
3 2.5 2 1.5 1
(a) PS1
(b) PS2
(c) PS3
(d) PS4
(e) PS5
(f) PS6
0.5 0 0
1
2 3 Coupling strength k
4
5
Fig. 3. Synchronization condition for G1 can be estimated by using scaled copies of scaling the synchronization condition for K2 with respect to the eigenvalue λi of L(G1 ). c{(1,1)}
where IN ×M = IN × IM \ {(1, 1)}, the system achieves ¯ Furfull synchronization for any pair (k1 , k2 , τ ) ∈ S(G). thermore, if there exists S¯c{(p,q)} (G) = ∩(i,j)∈I c{(p,q)} S(i,j) (G) �= ∅, c{(1,1)}
where I c{(p,q)} = IN ×M \ {(p, q)}, the system achieves partial synchronization for any pair (k1 , k2 , τ ) ∈ S¯c{(p,q)} (G) and partial synchronization patterns are estimated by solving the following simultaneous equations. ((vi ⊗ wj )T ⊗ Inn )x(t) = 0 for (i, j) ∈ I c{(p,q)} (12) As a result, we can estimate the partial synchronization conditions and synchronization patterns for the Cartesian network G = G1 ✷G2 from the eigenvalues and the corresponding eigenvectors of the graph Laplacian of the original two networks G1 and G2 .
Fig. 4. Partial synchronization patterns Then by applying the scaling method with respect to λi for i = 2, · · · , 8, we can estimate the synchronization condition as shown in Fig.3. The scaled region corresponding to λ4 is identical with the synchronization condition for K2 because λ4 = 2. For any pair (k, τ ) in the region overlapping all scaled regions, the network system achieves full synchronization. Furthermore, for any pair (k, τ ) in the region overlapping all of scaled regions except for counterpart of λ2 or λ8 , the network system can achieve partial synchronization. Fig.4 represents partial synchronization patterns for the Cartesian product network G and the same colored nodes mean the synchronized systems for each synchronization pattern.
4. EXAMPLES We consider synchronization of the Hindmarsh-Rose neuron systems (Hindmarsh and Rose (1984)) given by i x˙ (t) = −(xi1 (t))3 + 3(xi2 (t))2 + xi2 (t) − xi3 (t) 1 +a + ui (t) (13) i i 2 i x˙ (t) = 1 − 5(x1 (t)) − x2 (t) 2i i i x˙ 3 (t) = b(4(x1 (t) + c) − x3 (t) where a = 3.25, b = 0.005, c = 1.618. We consider the synchronization problem in the Cartesian product network G = G1 ✷K2 , where G1 denotes a simplified sandglass network and K2 denotes the complete graph with two nodes as shown in Fig.2. The inputs for systems in each network with delaycouplings is given by � uG1 (t) = −k1 L(G1 )x1 (t − τ ) for G1 (14) uK2 (t) = −k2 L(K2 )x1 (t − τ ) for K2
4.1 Case τ = 0 First, we consider the synchronization problem in the Cartesian product network G for τ = 0. From equation (10), the full/partial synchronization conditions for the network G1 are estimated as follows:
Now, we focus on the synchronization problem for G1 . The eigenvalues of the graph Laplacian of G1 are given by √ √ λ1 = 0, λ2 = 3 − √3, λ3 = 4 − 6, λ4 = 2, √ λ8 = 4 + 6. λ5 = 4, λ6 = 3 + 3, λ7 = 6,
• a partial synchronization pattern such that x1 = x3 �= x4 = x5 �= x6 = x8 �= x2 �= x7 2k 2k appears for λ3 ≤ k < λ2 • full synchronization appears for
16
2k λ2
≤k
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17
Fig. 5. Estimated synchronization bifurcation diagram for G without delay error
1 0 −10 error
1
1
x(4, I) − x(5, I)
1
x(6, I) − x(8, I)
0
200 400 600−10 200 400 600−10 200 400 600 x(1, I) − x(6, I) x(2, I) − x(7, I) x(3, I) − x(8, I)) 1 1 0
Fig. 7. Full synchronization region of (k1 , k2 , τ ) for the Cartesian product network G
0
200 400 600−10 200 400 600−10 200 400 600 x(1, I) − x(1, II) x(4, I) − x(4, II) x(6, I) − x(6, II) 1 1
0 −10
1 0
0 −10
error
x(1, I) − x(3, I)
0 200
400 600−10 time[s]
0 200
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200
400 600 time[s]
Fig. 6. Synchronization errors for (k1 , k2 , τ ) = (0.7, 1.0, 0) where k ≈ 0.45 is the minimum coupling strength that the complete graph K2 achieves synchronization for τ = 0. The estimated synchronization bifurcation diagram of (k1 , k2 ) for the Cartesian product network G for τ = 0 is illustrated in Fig.5. In this case, all full/partial synchronization conditions are expressed as square regions in the bifurcation diagram. In addition, this diagram shows that four partial synchronization patterns from PS1 to PS4 shown in Fig.4(a)-(d), respectively, emerge for delayfree coupling. Fig.6 represents the synchronization errors in numerical simulations for (k1 , k2 , τ ) = (0.7, 1.0, 0) ∈ S¯c{(2,1)} (G). For these parameters, we can observe the partial synchronization pattern PS3 as shown in Fig.4(c).
Fig. 8. Partial synchronization regions of (k1 , k2 , τ ) for the Cartesian product network G timated full/partial synchronization conditions (k1 , k2 , τ ) for the system (13) in the Cartesian product network G is shown in Figs.7 and 8. Since the estimated regions of the full/partial synchronization conditions become smaller as increasing τ as shown in Figs.7 and 8, the region of full synchronization is expressed as a triangle and the regions of partial synchronization have some oblique boundary lines in the estimated synchronization bifurcation diagram as shown in Fig.9. As shown in Fig.9, for τ > 0, the Cartesian product network G also has partial synchronization patterns PS5 and PS6 shown in Fig.4(e) and (f), respectively, which cannot be observed for τ = 0. Furthermore there exists not only lower bound but also upper one of the coupling strengths for the synchronization condition. c{(1,1)} Let the set I c{(8,2)} = I8×2 \ {(8, 2)}. Then c{(8,2)} ¯ S (G) = ∩(i,j)∈I c{(8,2)} S(i,j) (G) �= ∅
4.2 Case τ > 0 Next we focus on the system (13) coupled with the delayed control input (14), i.e. τ > 0. If τ = 0.2, the full/partial synchronization conditions for network G1 are estimated from (10) as follows: • a partial synchronization pattern such that x1 = x3 �= x4 = x5 �= x6 = x8 �= x2 �= x7 2k 2k appears for λ3 ≤ k < λ2 2k
¯
• full synchronization appears for λ2 ≤ k ≤ λ2k8 • a partial synchronization pattern such that x1 = x6 �= x2 = x7 �= x3 = x8 �= x4 �= x5 ¯ ¯ appears for λ2k8 < k ≤ λ2k7
and the system achieves partial synchronization for any pair (k1 , k2 , τ ) ∈ S¯c{(8,2)} (G). Furthermore, the partial synchronization pattern is estimated from the solutions of the simultaneous equations (12) as follows:
where k ≈ 0.65 and k¯ ≈ 6.25 are the minimum and maximum coupling strengths that the complete graph K2 achieves synchronization for τ = 0.2, respectively. The es17
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slightly different from the estimation based on the scaling method. 5. CONCLUSION In this paper, we considered synchronization patterns emerged in the Cartesian product networks of chaotic systems with/without time-delay. We showed that the synchronization condition can be estimated by applying scaling method based on the eigenvalue of the graph Laplacian. If there exists time-delay, different pattern of partial synchronization may be emerged that cannot be observed for delay-free coupling. Furthermore, synchronization patterns for the Cartesian product network can be estimated using eigenvectors of the graph Laplacian of graphs composing the Cartesian product network. The validity of the estimated condition and synchronization patterns for the Cartesian product network was tested through a numerical simulation. Throughout this paper, we dealt with bidirectionally coupled systems. This restriction on networks guarantees that the synchronization error dynamics is diagonalizable, and the synchronization problem can be reduced to the stability of the synchronization error dynamics of two coupled systems. The estimation method of synchronization conditions based on the scaling is completely reliant on this property.
Fig. 9. The estimated synchronization bifurcation diagram for G = G1 (k1 )✷K2 (k2 ) with τ = 0.2. error
1 0
−1 0
error
1
error
1
error
1
1
x(3, I) − x(8, I)
0
REFERENCES
−1 −1 200 400 600 0 200 400 600 0 200 400 600 x(1, I) − x(1, II) x(2, I) − x(2, II) x(3, I) − x(3, II) 1 1 0
0
−1 −1 200 400 600 0 200 400 600 0 200 400 600 x(1, I) − x(8, II) x(2, I) − x(7, II) x(3, I) − x(6, II) 1 1 0
0
−1 −1 200 400 600 0 200 400 600 0 200 400 600 x(4, I) − x(5, I) x(4, I) − x(5, II) x(5, I) − x(4, II) 1 1
0
−1 0
1 0
0
0
−1 0
x(2, I) − x(7, I)
−1 −1 200 400 600 0 200 400 600 0 200 400 600 x(1, II) − x(6, II) x(2, II) − x(7, II) x(3, II) − x(8, II) 1 1
0
−1 0
1 0
0
−1 0
error
x(1, I) − x(3, I)
0 200
−1 400 600 0 time[s]
0 200
−1 400 600 0 time[s]
200
400 600 time[s]
Fig. 10. The synchronization errors for (k1 , k2 , τ ) = (1.5, 1.3, 0.2) (1,I) x = x(6,I) = x(3,II) = x(8,II) (2,I) x = x(7,I) = x(2,II) = x(7,II) (3,I) x = x(8,I) = x(1,II) = x(6,II) (4,I) x = x(5,II) �= x(5,I) = x(4,II)
Fig.10 represents the synchronization errors for (k1 , k2 , τ ) = (1.5, 1.3, 0.2) ∈ S¯c{(8,2)} (G), we can observe the partial synchronization pattern PS6 as shown in Fig.4(f). Remark 4. In full synchronization, all dynamics of xi (t) are same as that of an uncoupled system. On the other hand, in partial synchronization, the dynamics is not same as an uncoupled system because even if partial synchronization occurs, the coupling terms never vanish. Therefore, the actual regions for partial synchronization may be 18
Bliman, P.A. (2000). Extension of Popov absolute stability criterion to non-autonomous systems with delays. International Journal of Control, 73(15), 1349-1361. Mimura, T., Oguchi, T. (2012). Partial Synchronization of Lur’e Type Nonlinear Systems with Delay Couplings. In Proceedings of the 3rd IFAC. Steur, E., Oguchi, T., Leeuwen, C.V., Nijmeijer, H. (2012). Partial synchronization in diffusively time-delay coupled oscillator networks. Chaos, 22, 043144. Pogromsky, A.Y. (2008). A partial synchronization theorem. Chaos, 18, 037107. Oguchi, T., Oomen, W., Murguia, C., Nijmeijer, H. (2013). Partial Synchronization in Cartesian Product Networks of Coupled Nonlinear Systems without/with Delays. In Proceedings of JJAC (in Japanese). Oguchi, T., Nijmeijer, H. (2011) A synchronization condition for coupled nonlinear systems with time-delay : A frequency domain approach Int. J. Bifurcation and Chaos, Vol.21, No.9. Neefs, P.J., Steur, E. and Nijmeijer, H. (2010) Network complexity and synchronous behavior-an experimental approach. International Journal of Neural Systems, 20(3), 233-247. Mesbahi, M., Egerstedt, M. (2010). Graph Theoretic Methods in Multiagent Networks, 20(3). Princeton University Press. Hindmarsh, J.L., Rose, R.M. (1984). A Model for Neuronal Burstting Using Three Coupled Differential Equations. Proceedings of the Royal Society of London, B 221, 87102.
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Synchronization Patterns in Synchronization Patterns in Synchronization Patterns in SynchronizationofPatterns in Network-of-Networks Chaotic Systems Network-of-Networks of Chaotic Systems Network-of-Networks of Chaotic Systems Network-of-Networks of Chaotic Systems via Cartesian Product via Cartesian Product via Cartesian Product via Cartesian Product
Kanaru Oooka ∗∗ Toshiki Oguchi ∗∗ Kanaru Oooka ∗∗ Toshiki Oguchi ∗∗ Kanaru Kanaru Oooka Oooka Toshiki Toshiki Oguchi Oguchi ∗ ∗ Dept. of Mechanical Engineering, Tokyo Metropolitan University ∗ Dept. of Mechanical Engineering, Tokyo Metropolitan University ∗ Dept. Mechanical Tokyo Metropolitan 1-1,of Hachioji-shi, Tokyo 192-0397 University Japan Dept. ofMinami-Osawa, Mechanical Engineering, Engineering, Tokyo Metropolitan University 1-1, Minami-Osawa, Hachioji-shi, Tokyo 192-0397 Japan 1-1, Minami-Osawa, Hachioji-shi, Tokyo 192-0397 Japan (e-mail: [email protected], [email protected]). 1-1, Minami-Osawa, Hachioji-shi, Tokyo 192-0397 Japan (e-mail: [email protected], [email protected]). (e-mail: [email protected], [email protected]). (e-mail: [email protected], [email protected]). Abstract: This paper considers the synchronization problem in network-of-networks of chaotic Abstract: This paper considers the synchronization problem in network-of-networks of chaotic Abstract: This paper the problem in of systems with time-delay. We have already proposed an estimation method for full/partial Abstract: This paper considers considers the synchronization synchronization problem in network-of-networks network-of-networks of chaotic chaotic systems with time-delay. We have already proposed an estimation method for full/partial systems with time-delay. time-delay. We have already proposed an estimation estimation method product. for full/partial full/partial synchronization conditionsWe for have networks of chaotic systems via the Cartesian In this systems with already proposed an method for synchronization conditions for networks of chaotic systems via the Cartesian product. In this synchronization conditions for networks of chaotic systems via the Cartesian product. In this paper, we attempt to estimate full/partial synchronization conditions for the Cartesian product synchronization conditions for networks of chaotic systems via the Cartesian product. In this paper, we attempt to estimate full/partial synchronization conditions for the Cartesian product paper, we attempt to estimate full/partial synchronization conditions for the Cartesian product networks with time-delay by using the scaling method for synchronization condition and the paper, we attempt to estimate full/partial synchronization conditions for the Cartesian product networks with time-delay by using the scaling method for synchronization condition and the networks with time-delay by using the scaling method for synchronization condition and the properties of the Cartesian product. The validity of the estimation method is tested through networks with time-delay by using the scaling method for synchronization condition and theaa properties of the Cartesian product. The validity of the estimation method is tested through properties of the Cartesian product. The validity of the estimation method is tested through aa numerical example. properties of the Cartesian product. The validity of the estimation method is tested through numerical example. numerical example. numerical example. © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Synchronization, Chaotic systems, Cartesian product, Networks, Graph Laplacian Keywords: Synchronization, Keywords: Synchronization, Synchronization, Chaotic Chaotic systems, systems, Cartesian Cartesian product, product, Networks, Networks, Graph Graph Laplacian Laplacian Keywords: Chaotic systems, Cartesian product, Networks, Graph Laplacian 1. INTRODUCTION scaling method to estimate synchronization conditions to 1. INTRODUCTION scaling method to estimate synchronization conditions to 1. INTRODUCTION scaling synchronization conditions to network-of-networks via the Cartesian product and showed 1. INTRODUCTION scaling method method to to estimate estimate synchronization conditions to network-of-networks via the Cartesian product and showed network-of-networks via the Cartesian product and showed the validity of the estimated conditions. Recently, there is much attention being given to synchro- network-of-networks via the Cartesian product and showed the validity of the estimated conditions. Recently, there attention being given synchrothe validity of this paper, weestimated estimate conditions. full/partial synchronization Recently, phenomena there is is much much attention being given to to synchronization in attention networks being of coupled systems in In the validity of the the estimated conditions. Recently, there is much given to synchroIn this paper, we estimate full/partial synchronization nization phenomena in networks of coupled systems in In this paper, we estimate full/partial synchronization conditions for chaotic systems with delay-couplings by nization phenomena in networks of coupled systems in applied mathematics, physics, information technology and this paper, we estimate full/partial synchronization nization phenomena in networks of coupled systemsand in In conditions for chaotic systems with delay-couplings by applied mathematics, physics, information technology conditions for chaotic systems with delay-couplings by applying the scaling method and there exist some patterns applied mathematics, physics, information technology and various fields. For practical systems, we need to consider conditionsthefor chaotic systems with exist delay-couplings by applied mathematics, physics, information technology and applying scaling method and there some patterns various fields. For practical systems, we need to consider applying the scaling method and there exist some patterns of partial synchronization that cannot be observed for novarious fields. For practical systems, we need to consider that there exist delays in signal transmission between applying the scaling method and there exist some patterns various fields. For practical systems, we need to consider of partial synchronization that cannot be observed for nothat exist delays in between of partial synchronization be for coupling. In addition,that we cannot show that partial synchrothat there there exist The delays in signal signal transmission transmission between coupled systems. synchronization condition for non- delay of partial synchronization that cannot be observed observed for nonothat there exist delays in signal transmission between delay coupling. In addition, we show that partial synchrocoupled systems. The synchronization condition for nondelay coupling. In addition, we show that partial synchronization patterns in the Cartesian product networks can coupled systems. The synchronization condition for nonlinear systems with delay-couplings can be derived in a delay coupling. In addition, we show that partial synchrocoupled systems. The synchronization condition for nonnization patterns in the Cartesian product networks can linear with delay-couplings can be derived nization patterns in product networks can detected by using the Cartesian eigenvectors of graph Laplacians linearofsystems systems with delay-couplings can be based derivedonin intheaaa be form the linear matrix inequalities can (LMIs) nization patterns in the the Cartesian product networks can linear systems with delay-couplings be derived in be detected by using the eigenvectors of graph Laplacians form of the linear matrix inequalities (LMIs) based on the be detected by using the eigenvectors of graph Laplacians of two original networks. form of the linear matrix inequalities (LMIs) based on the Lyapunov-Krasovskii theorem. However, it is well known be detected by using the eigenvectors of graph Laplacians form of the linear matrix inequalities (LMIs) based on the of two original networks. Lyapunov-Krasovskii theorem. However, it well Lyapunov-Krasovskii theorem. However, it is is because well known known that this approach causes conservative results the of of two two original original networks. networks. Lyapunov-Krasovskii theorem. However, it is well known that this approach causes conservative results because the that this approach causes conservative results because the Lyapunov-Krasovskii theorem gives a sufficient condition 2. SYNCHRONIZATION IN NETWORKS OF that this approach causes conservative results because the Lyapunov-Krasovskii theorem a sufficient condition 2. SYNCHRONIZATION IN NETWORKS OF Lyapunov-Krasovskii theorem gives gives sufficient condition for stability and the condition itself aahas inherent conser2. SYNCHRONIZATION IN CHAOTIC SYSTEMS Lyapunov-Krasovskii theorem gives sufficient condition 2. SYNCHRONIZATION IN NETWORKS NETWORKS OF OF for stability and the condition itself has inherent conserCHAOTIC SYSTEMS for stability and the condition itself has inherent conservativeness. Furthermore, to reduce the obtained inequality CHAOTIC SYSTEMS for stabilityFurthermore, and the condition itself has inherent conserCHAOTIC SYSTEMS vativeness. to reduce the obtained inequality vativeness.toFurthermore, Furthermore, to reduce reduce the obtained inequality Throughout this paper, we consider the following N idencondition the LMI condition, thethe obtained synchronizavativeness. to obtained inequality condition to LMI condition, the obtained synchronizaThroughout this paper, we consider the following N idencondition to the thetends LMI to condition, the obtained synchronization condition become the extremely restrictive. On tical Throughout this paper, condition to the LMI condition, obtained synchronizanonlinear systems: Throughout this paper, we we consider consider the the following following N N idenidention condition tends to become extremely restrictive. On tical nonlinear systems: tion condition tends to become extremely restrictive. On the other hand, Oguchi and Nijmeijer (2011) proposed tical nonlinear systems: tion condition tends to become extremely restrictive. On tical nonlinear systems: i i i i the other hand, Oguchi and Nijmeijer (2011) proposed x˙ i (t) = Axi (t) + f (xi (t)) + Bui (t) the alternative other hand, hand,approach Oguchi and and Nijmeijer a (2011) (2011) proposed an for deriving frequency-based the other Oguchi Nijmeijer proposed x = Ax f (xii (t)) + Bu ˙ i (t) Σii : (1) i (t) an for deriving aa frequency-based ii (t) (t) = = Cx Ax (t) + + x an alternative alternative approach approach for deriving frequency-based Σ (1) i : y˙˙ iii(t) synchronization conditionfor using the multi-variable circle Axii (t) + ff (x (x (t)) (t)) + + Bu Bui (t) (t) an alternative approach deriving a frequency-based i Σ :: x (1) y i (t) = Cxi (t) synchronization condition using the multi-variable circle Σ (1) i (t) yy i (t) = Cx synchronization condition using the and multi-variable circle criterion (Blimancondition (2000)).using Mimura Oguchi (2012) (t) = Cx (t) i n i i synchronization the multi-variable circle criterion (Bliman (2000)). Mimura and Oguchi (2012) ∈ R is the state, u ∈ R and y ∈ R are the where x criterionthat (Bliman (2000)). Mimura Mimura and Oguchi Oguchi (2012) showed the synchronization conditions of Lur’e sys- where xiii ∈ Rnnn is the state, uiii ∈ R and y iii ∈ R are the criterion (Bliman (2000)). and (2012) R is state, yy ∈ the where x showed that the synchronization conditions of Lur’e sysi =u ·∈ · ·R , Nand , respectively. B ∈the R output is the the for state, u1, and ∈ R R are areA, the where and x ∈ showed that the synchronization conditions of Lur’e sys- input tems with arbitrary network topologies can be estimated input and the output for ii = 1, ··∈·· ·· R ,, N ,, respectively. A, B showed that the synchronization conditions of Lur’e sysinput and the output for = 1, N respectively. A, B tems with arbitrary network topologies can be estimated and C are constant matrices. In addition, we assume that input and the output for i = 1, · · · , N , respectively. A, B tems with arbitrary network topologies can be estimated by scaling the synchronization condition ofbetwo identi- and Cnare constant matrices. In addition, we assume that n tems with arbitrary network topologies can estimated and C are constant matrices. In addition, we assume that by scaling the synchronization condition of two identif : R → R is a sufficiently smooth vector field. The nare constant n and C matrices. In addition, we assume that by scaling the synchronization condition of two identical systems with delay-couplings by eigenvalues of the ff :: R smooth vector field. The n → Rn is a sufficiently by scaling the synchronization condition of two identi→ R is sufficiently vector cal delay-couplings by eigenvalues of uii is givensmooth by f : R Rnfor →each Rn system is aa N sufficiently smooth vector field. field. The The cal systems systems with with delay-couplings by estimation eigenvaluesmethod of the the input corresponding graph Laplacian. This input for each system u i is given by cal systems with delay-couplings by eigenvalues of the i input for each system u is given by corresponding graph Laplacian. This estimation method N input for each system u is given by i j i corresponding graph Laplacian. This estimation method provides a less conservative synchronization condition. A N corresponding graph Laplacian. This estimation method ui (t) = − (2) N kij (y i (t − τ ) − y j (t − τ )) provides aa less synchronization condition. A u (2) i (t) = − kij (y i (t − τ ) − y j (t − τ )) providesestimation less conservative conservative synchronization condition. A similar method issynchronization independently condition. proposed by i (t) = − j=1 kij (y i (t − τ ) − y j (t − τ )) u (2) provides a less conservative A u k (y (t − τ ) − y (t − τ )) (2) (t) = − similar estimation method is independently proposed by ij j=1 similar estimation method is independently proposed by Steur et al. (2012) for coupled chaotic systems with similar estimation method is independently proposed by j=1 Steur et al. (2012) for coupled chaotic systems with j=1 time-delay, and kij represents a where τ is a constant Steur semi-passivity et al. al. (2012) (2012)property for coupled coupled chaotic systems systems with strict (Pogromsky (2008)).with For where τ is a constant time-delay, and kij represents a Steur et for chaotic represents a where ττ strength is aa constant constant time-delay, and kij strict semi-passivity property (Pogromsky (2008)). For that iftime-delay, there existsand a coupling betweena where is k strict semi-passivity property (Pogromsky (2008)). For coupling ij represents network systems with delay-couplings, an experimental coupling strength that if there exists aa coupling between strict semi-passivity property (Pogromsky (2008)). For coupling strength that if there exists coupling between network systems with delay-couplings, an experimental systems i and j, k = k = k, and otherwise k = 0. In ij ji ij coupling strength that if there exists a coupling between network systems with delay-couplings, an experimental result for systems the relationship between the minimum non-zero ii and j, k k = 0. In network with delay-couplings, an experimental ij = ji = systems k = that k = k, k, and and otherwise otherwise k kij result relationship between the non-zero systems this paper, we j, assume ij = 0. In i and and j, kij kji result for for the the relationship between theisminimum minimum ij = that ji = k, and otherwise kij = 0. In eigenvalue andrelationship synchronization region shown bynon-zero Neefs et systems this paper, we assume result for the between the minimum non-zero paper, we that eigenvalue and synchronization region is shown by et this we assume assume that eigenvalue and synchronization region is(2013) shown applied by Neefs Neefsthe et this al. (2010). and Furthermore, Oguchiregion et al.is i) paper, all couplings are bidirectional eigenvalue synchronization shown by Neefs et al. (2010). Furthermore, Oguchi et al. (2013) applied the i) all couplings are bidirectional al. (2010). (2010). Furthermore, Furthermore, Oguchi Oguchi et et al. al. (2013) applied the i) all couplings are bidirectional ii) each system is strongly semi-passive. al. (2013) applied the i) all couplings are bidirectional ⋆ This work was partially supported by the Japan Society for the ii) each system is strongly semi-passive. ii) each system is strongly semi-passive. ⋆ This work was partially supported by the Japan Society for the ii) each system is strongly semi-passive. ⋆ Here the network topology is represented by an undirected Promotion of Science (JSPS) Grant-in-Aid for Scientific Research work was partially supported by the Japan Society for the ⋆ This This workofwas partially supported by the for Japan SocietyResearch for the Here the network topology is represented by an undirected Promotion Science (JSPS) Grant-in-Aid Scientific Here the network topology is by graph G with N nodes, and it is characterized by the (No.26420424) Promotion of Science (JSPS) Grant-in-Aid for Scientific Research Here the network topology is represented represented by an an undirected undirected Promotion of Science (JSPS) Grant-in-Aid for Scientific Research graph G with N nodes, and it is characterized by (No.26420424) graph G with N nodes, and it is characterized by the the (No.26420424) graph G with N nodes, and it is characterized by the (No.26420424)
Copyright © 2015, 2015 IFAC 13 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2015 IFAC 13 Copyright © 2015 IFAC 13 Peer review under responsibility of International Federation of Automatic Copyright © 2015 IFAC 13 Control. 10.1016/j.ifacol.2015.11.003
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corresponding graph Laplacian L(G) defined by the degree matrix ∆(G) and the adjacency matrix A(G). The degree matrix is the diagonal matrix whose entries are the vertex-degrees of G on the diagonal, that is ∆(G) = diag(d(v1 ), · · · , d(vN )), where d(vi ) is the degree of the vertex vi . The adjacency matrix A(G) = [aij ] is the symmetric N × N matrix encoding of the adjacency relationship in the graph G and given by � 1 if (vi , vj ) ∈ E(G) aij = , 0 otherwise
where E(G) is the edge set of the graph G. Then the graph Laplacian L(G) is defined by L(G) = ∆(G) − A(G). If the graph is connected, the graph Laplacian L(G) has only one zero-eigenvalue. Then the input vector u(t) = [u1 (t) · · · uN (t)]T can be described by u(t) = −kL(G)y(t − τ ) = −k(L(G) ⊗ C)x(t − τ ) (3) where y(t) = [y 1 (t) · · · y N (t)]T is the output vector and ⊗ denotes the Kronecker product. Now, the dynamics of the total network system is described by � x(t) ˙ = (IN ⊗ A)x(t) + F (x(t)) −k(IN ⊗ B)(L(G) ⊗ C)x(t − τ ) (4) Σ: y(t) = (IN ⊗ C)x(t) 1
T
N
Fig. 1. Graphical representation of the scaling estimation method : S is the synchronization condition for K2 , and S2 , · · · , SN are the scaled synchronization conditions corresponding to the eigenvalues of L(G). Full synchronization is achieved in the region ovarlapped with all regions, and partial synchronization may be happened in the region overlapped with a part of regions. ψ(x1 (t), e12 (t)) .. Ψ(x1 (t), e(t)) = , . ψ(x1 (t), e1N (t))
T T
where x(t) = [(x (t)) · · · (x (t)) ] , IN is the identity matrix with N -dimension, and F (x(t)) = [f (x1 (t)) · · · f (xN (t))]T . To introduce an estimation method of the synchronization condition based on the scaling method (Mimura and Oguchi (2012)), we consider the complete graph KN with N nodes and define the synchronization error vector e(t) as 1 x (t) − x2 (t) e12 (t) .. e(t) := ... = = (M0 ⊗ In )x(t), (5) . 1 N e1N (t) x (t) − x (t)
where
where ψ(x1 (t), e1i (t)) = f (x1 (t)) − f (x1 (t) − e1i (t)) and 1N −1 = [1 · · · 1]T ∈ RN −1 . 2.1 Full synchronization in networks First, we consider the full synchronization problem. Full synchronization of systems to be considered here is defined as follows: Definition 1. (Oguchi and Nijmeijer (2011)) If there exists a positive real number r such that the trajectories xi (t) of the systems (1) with initial conditions φi , φj such that ||φi − φj ||C ≤ r satisfy ||xi (t) − xj (t)|| → 0 as t → ∞ for all i, j, then the coupled systems (1) and (2) are asymptotically synchronized. Here ||φ||C := max−τ ≤θ≤0 ||φ(θ)|| stands for the norm of a vector function φ, where || · || refers to the Euclidean vector norm.
1 −1 0 (N −1)×N .. . M0 = ... ∈R . 1 0 −1
Applying the coordinate transformation 1 0 · · · 0 � 1 � .. . 1 −1 x (t) = . ⊗ In x(t), .. e(t) . . 0 . 1 0 −1 the dynamics (4) is rewritten as follows: � 1 � � 1 � x˙ (t) x (t) = (IN ⊗ A) + Ψ(x1 (t), e(t)) e(t) ˙ e(t) � � �� 1 0 1N −1 ⊗ BC x (t − τ ) −k e(t − τ ) 0 M0 LM0+ ⊗ BC
Let L(KN ) denote the graph Laplacian of the complete graph KN . Then it is diagonalizable and its eigenvalues are N with multiplicity N − 1 and a zero. Furthermore, since M0 LM0+ = diag(N, · · · , N ) holds, the dynamics of e(t) defined in (5) are blockdiagonalized and decomposed into N − 1 identical dynamics given by e˙ 1i = Ae1i (t) + ψ(x1 (t), e1i (t)) − kN BCe1i (t − τ ). (7) Now we consider the synchronization condition for two bidirectional coupled systems. The error dynamics is given by e˙ 12 = Ae12 (t) + ψ(x1 (t), e12 (t)) − 2kBCe12 (t − τ ). (8) If the origin of (8) is asymptotically stable, the coupled systems are asymptotically synchronized. Here we assume that S is a set of pairs (k, τ ) such that the systems achieve synchronization. Comparing (7) with (8), all coupled systems must be synchronized if ( kN 2 , τ ) ∈ S. Therefore, we can estimate the synchronization condition for N coupled
(6)
where M0+ denotes the pseudo-inverse matrix of M0 given by 0 0 . −1 . . + M0 = ∈ RN ×(N −1) , .. . 0 0 −1 14
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systems with complete graph network structure by scaling S with respect to the coupling strength k. Next, we consider the synchronization condition for more general networks G. In this case, the Laplacian L(G) has various eigenvalues, and partial synchronization may emerge. For undirected graph G, there always exists a nonsingular matrix P such that M0 LM0+ is diagonalized as P −1 M0 LM0+ P = diag(λ2 , · · · , λN ), where 0 = λ1 < λ2 ≤ · · · ≤ λN are eigenvalues of L(G). Applying the coordinate transformation e(t) = (P ⊗ In )¯ e(t), the error dynamics are transformed into e1i (t) + ψ ′ (x1 (t))¯ e1i (t) − λi kBC e¯1i (t − τ ) (9) e¯˙ 1i (t) = A¯
15
Fig. 2. Simplified sandglass network (left) and the complete graph with two nodes (right) the Cartesian product graph of G1 and G2 is represented as G = G1 ✷G2 , and V(G) = V1 × V2 . The graph Laplacian of a pair of graphs G1 and G2 and the Cartesian product graph G satisfy the following property.
where i = 2, · · · , N . Let Si be a set of pairs (k, τ ) that makes the origin of the error dynamics (9) asymptotically stable. Then the set Si (G) can be estimated by scaling S in the coupling strength axis with respect to the eigenvalue as follows: λi k , τ ∈ S} Si (G) = {(k, τ )| 2 ¯ Furthermore, if there exists S(G) = ∩i∈I Si (G) �= ∅(where I = {2, · · · , N }), the origin of the error dynamics are ¯ asymptotically stable for (k, τ ) ∈ S(G), that is, the network system achieves full synchronization satisfying x1 (t) = x2 (t) = · · · = xN (t).
L(G1 ✷G2 ) = L(G1 ) ⊗ IM + IN ⊗ L(G2 )
= L(G1 ) ⊕ L(G2 ) where N and M are the numbers of nodes of G1 and G2 , respectively, and ⊕ denotes the Kronecker sum. It is known that the eigenvalue and the corresponding eigenvector of the graph Laplacian of the Cartesian product graph satisfy following lemma. Lemma 3. (Mesbahi and Egerstedt (2010)) Let G1 and G2 be a pair of graphs on N and M nodes, respectively. Furthermore, assume that λ1 , λ2 , · · · , λN and µ1 , µ2 , · · · , µM are the eigenvalues of L(G1 ) and L(G2 ), respectively, corresponding to the eigenvectors v1 , v2 , · · · , vN and w1 , w2 , · · · , wM Then vi ⊗ wj , i = 1, 2, · · · , N, j = 1, 2, · · · , M is the eigenvector associated with the eigenvalue λi + µj of L(G1 ✷G2 )
2.2 Partial synchronization in networks Next we consider the partial synchronization problem. Partial synchronization in networks is defined as follows: Definition 2. (Mimura and Oguchi (2012)) If there exists a positive real number r such that the trajectories xi (t) and xj (t) of a part of systems i and j in networks with the initial conditions φi , φj such that ||φi − φj ||C ≤ r satisfy ||xi (t) − xj (t)|| → 0 as t → ∞ for some i, j, then the systems i and j are asymptotically synchronized and the network systems is called to be partially synchronized.
The second smallest eigenvalue of L(G1 ✷G2 ) is given by λ2 (G1 ✷G2 ) = min{λ2 (G1 ), λ2 (G2 )} As a result, the synchronization condition for the Cartesian product network can be estimated by using the scaling method with respect to the eigenvalues of the original two networks. We assume that the control inputs for each system in two original networks are given by uG1 (t) = −k1 L(G1 )y(t − τ ) for G1 uG2 (t) = −k2 L(G2 )y(t − τ ) for G2
Let the set I c{p} = I \ {p} and |I c{p} | = M . Then if there exists S¯c{p} (G) = ∩i∈I c{p} Si (G) �= ∅, the origin of M error dynamics e¯1i for i ∈ I c{p} is asymptotically stable for (k, τ ) ∈ S¯c{p} (G), and this implies the following simultaneous equations hold. (10) e¯1i (t) = (viT ⊗ In )x(t) = 0 for i ∈ I c{p} where vi is the corresponding eigenvector with the eigenvalue λi of L(G). Moreover, if there exists the solution of (10) in a form that xj (t) = xk (t) for some j, k ∈ I, the systems j and k achieve partial synchronization. If there is no time-delay, i.e. τ = 0, the partial synchronization pattern is unique. On the other hand, if τ > 0, another partial synchronization pattern that cannot be observed for τ = 0 may be emerged.
Then, the total dynamics of the system in the Cartesian product network G = G1 ✷G2 is rewritten as
x˙ = (IN M ⊗ A)x(t) + F (x) − ((k1 L(G1 ) ⊕ k2 L(G2 )) ⊗ BC)x(t − τ ) (11) where x = [(x(1,1) )T · · · (x(1,M) )T · · · (x(N,1) )T · · · (x(N,M) )T ]T . The synchronization condition for the Cartesian product network can be estimated by scaling the condition for K2 as k1 λi + k2 µj , τ ∈ S}, S(i,j) (G) = {(k1 , k2 , τ )| 2 and if there exists the set ¯ S(G) = ∩(i,j)∈I c{(1,1)} S(i,j) (G) �= ∅,
3. SYNCHRONIZATION CONDITION IN CARTESIAN PRODUCT NETWORK In this section, we consider synchronization in the networkof-networks via the Cartesian product. Here, we consider two undirected graphs G1 = (V1 , E1 ) and G2 = (V2 , E2 ), where Vi are node sets of the graph Gi . Then
N ×M
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4
λ2 λ3 λ4 λ5 λ6 λ7 λ8
3.5
Time delay τ
3 2.5 2 1.5 1
(a) PS1
(b) PS2
(c) PS3
(d) PS4
(e) PS5
(f) PS6
0.5 0 0
1
2 3 Coupling strength k
4
5
Fig. 3. Synchronization condition for G1 can be estimated by using scaled copies of scaling the synchronization condition for K2 with respect to the eigenvalue λi of L(G1 ). c{(1,1)}
where IN ×M = IN × IM \ {(1, 1)}, the system achieves ¯ Furfull synchronization for any pair (k1 , k2 , τ ) ∈ S(G). thermore, if there exists S¯c{(p,q)} (G) = ∩(i,j)∈I c{(p,q)} S(i,j) (G) �= ∅, c{(1,1)}
where I c{(p,q)} = IN ×M \ {(p, q)}, the system achieves partial synchronization for any pair (k1 , k2 , τ ) ∈ S¯c{(p,q)} (G) and partial synchronization patterns are estimated by solving the following simultaneous equations. ((vi ⊗ wj )T ⊗ Inn )x(t) = 0 for (i, j) ∈ I c{(p,q)} (12) As a result, we can estimate the partial synchronization conditions and synchronization patterns for the Cartesian network G = G1 ✷G2 from the eigenvalues and the corresponding eigenvectors of the graph Laplacian of the original two networks G1 and G2 .
Fig. 4. Partial synchronization patterns Then by applying the scaling method with respect to λi for i = 2, · · · , 8, we can estimate the synchronization condition as shown in Fig.3. The scaled region corresponding to λ4 is identical with the synchronization condition for K2 because λ4 = 2. For any pair (k, τ ) in the region overlapping all scaled regions, the network system achieves full synchronization. Furthermore, for any pair (k, τ ) in the region overlapping all of scaled regions except for counterpart of λ2 or λ8 , the network system can achieve partial synchronization. Fig.4 represents partial synchronization patterns for the Cartesian product network G and the same colored nodes mean the synchronized systems for each synchronization pattern.
4. EXAMPLES We consider synchronization of the Hindmarsh-Rose neuron systems (Hindmarsh and Rose (1984)) given by i x˙ (t) = −(xi1 (t))3 + 3(xi2 (t))2 + xi2 (t) − xi3 (t) 1 +a + ui (t) (13) i i 2 i x˙ (t) = 1 − 5(x1 (t)) − x2 (t) 2i i i x˙ 3 (t) = b(4(x1 (t) + c) − x3 (t) where a = 3.25, b = 0.005, c = 1.618. We consider the synchronization problem in the Cartesian product network G = G1 ✷K2 , where G1 denotes a simplified sandglass network and K2 denotes the complete graph with two nodes as shown in Fig.2. The inputs for systems in each network with delaycouplings is given by � uG1 (t) = −k1 L(G1 )x1 (t − τ ) for G1 (14) uK2 (t) = −k2 L(K2 )x1 (t − τ ) for K2
4.1 Case τ = 0 First, we consider the synchronization problem in the Cartesian product network G for τ = 0. From equation (10), the full/partial synchronization conditions for the network G1 are estimated as follows:
Now, we focus on the synchronization problem for G1 . The eigenvalues of the graph Laplacian of G1 are given by √ √ λ1 = 0, λ2 = 3 − √3, λ3 = 4 − 6, λ4 = 2, √ λ8 = 4 + 6. λ5 = 4, λ6 = 3 + 3, λ7 = 6,
• a partial synchronization pattern such that x1 = x3 �= x4 = x5 �= x6 = x8 �= x2 �= x7 2k 2k appears for λ3 ≤ k < λ2 • full synchronization appears for
16
2k λ2
≤k
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17
Fig. 5. Estimated synchronization bifurcation diagram for G without delay error
1 0 −10 error
1
1
x(4, I) − x(5, I)
1
x(6, I) − x(8, I)
0
200 400 600−10 200 400 600−10 200 400 600 x(1, I) − x(6, I) x(2, I) − x(7, I) x(3, I) − x(8, I)) 1 1 0
Fig. 7. Full synchronization region of (k1 , k2 , τ ) for the Cartesian product network G
0
200 400 600−10 200 400 600−10 200 400 600 x(1, I) − x(1, II) x(4, I) − x(4, II) x(6, I) − x(6, II) 1 1
0 −10
1 0
0 −10
error
x(1, I) − x(3, I)
0 200
400 600−10 time[s]
0 200
400 600−10 time[s]
200
400 600 time[s]
Fig. 6. Synchronization errors for (k1 , k2 , τ ) = (0.7, 1.0, 0) where k ≈ 0.45 is the minimum coupling strength that the complete graph K2 achieves synchronization for τ = 0. The estimated synchronization bifurcation diagram of (k1 , k2 ) for the Cartesian product network G for τ = 0 is illustrated in Fig.5. In this case, all full/partial synchronization conditions are expressed as square regions in the bifurcation diagram. In addition, this diagram shows that four partial synchronization patterns from PS1 to PS4 shown in Fig.4(a)-(d), respectively, emerge for delayfree coupling. Fig.6 represents the synchronization errors in numerical simulations for (k1 , k2 , τ ) = (0.7, 1.0, 0) ∈ S¯c{(2,1)} (G). For these parameters, we can observe the partial synchronization pattern PS3 as shown in Fig.4(c).
Fig. 8. Partial synchronization regions of (k1 , k2 , τ ) for the Cartesian product network G timated full/partial synchronization conditions (k1 , k2 , τ ) for the system (13) in the Cartesian product network G is shown in Figs.7 and 8. Since the estimated regions of the full/partial synchronization conditions become smaller as increasing τ as shown in Figs.7 and 8, the region of full synchronization is expressed as a triangle and the regions of partial synchronization have some oblique boundary lines in the estimated synchronization bifurcation diagram as shown in Fig.9. As shown in Fig.9, for τ > 0, the Cartesian product network G also has partial synchronization patterns PS5 and PS6 shown in Fig.4(e) and (f), respectively, which cannot be observed for τ = 0. Furthermore there exists not only lower bound but also upper one of the coupling strengths for the synchronization condition. c{(1,1)} Let the set I c{(8,2)} = I8×2 \ {(8, 2)}. Then c{(8,2)} ¯ S (G) = ∩(i,j)∈I c{(8,2)} S(i,j) (G) �= ∅
4.2 Case τ > 0 Next we focus on the system (13) coupled with the delayed control input (14), i.e. τ > 0. If τ = 0.2, the full/partial synchronization conditions for network G1 are estimated from (10) as follows: • a partial synchronization pattern such that x1 = x3 �= x4 = x5 �= x6 = x8 �= x2 �= x7 2k 2k appears for λ3 ≤ k < λ2 2k
¯
• full synchronization appears for λ2 ≤ k ≤ λ2k8 • a partial synchronization pattern such that x1 = x6 �= x2 = x7 �= x3 = x8 �= x4 �= x5 ¯ ¯ appears for λ2k8 < k ≤ λ2k7
and the system achieves partial synchronization for any pair (k1 , k2 , τ ) ∈ S¯c{(8,2)} (G). Furthermore, the partial synchronization pattern is estimated from the solutions of the simultaneous equations (12) as follows:
where k ≈ 0.65 and k¯ ≈ 6.25 are the minimum and maximum coupling strengths that the complete graph K2 achieves synchronization for τ = 0.2, respectively. The es17
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slightly different from the estimation based on the scaling method. 5. CONCLUSION In this paper, we considered synchronization patterns emerged in the Cartesian product networks of chaotic systems with/without time-delay. We showed that the synchronization condition can be estimated by applying scaling method based on the eigenvalue of the graph Laplacian. If there exists time-delay, different pattern of partial synchronization may be emerged that cannot be observed for delay-free coupling. Furthermore, synchronization patterns for the Cartesian product network can be estimated using eigenvectors of the graph Laplacian of graphs composing the Cartesian product network. The validity of the estimated condition and synchronization patterns for the Cartesian product network was tested through a numerical simulation. Throughout this paper, we dealt with bidirectionally coupled systems. This restriction on networks guarantees that the synchronization error dynamics is diagonalizable, and the synchronization problem can be reduced to the stability of the synchronization error dynamics of two coupled systems. The estimation method of synchronization conditions based on the scaling is completely reliant on this property.
Fig. 9. The estimated synchronization bifurcation diagram for G = G1 (k1 )✷K2 (k2 ) with τ = 0.2. error
1 0
−1 0
error
1
error
1
error
1
1
x(3, I) − x(8, I)
0
REFERENCES
−1 −1 200 400 600 0 200 400 600 0 200 400 600 x(1, I) − x(1, II) x(2, I) − x(2, II) x(3, I) − x(3, II) 1 1 0
0
−1 −1 200 400 600 0 200 400 600 0 200 400 600 x(1, I) − x(8, II) x(2, I) − x(7, II) x(3, I) − x(6, II) 1 1 0
0
−1 −1 200 400 600 0 200 400 600 0 200 400 600 x(4, I) − x(5, I) x(4, I) − x(5, II) x(5, I) − x(4, II) 1 1
0
−1 0
1 0
0
0
−1 0
x(2, I) − x(7, I)
−1 −1 200 400 600 0 200 400 600 0 200 400 600 x(1, II) − x(6, II) x(2, II) − x(7, II) x(3, II) − x(8, II) 1 1
0
−1 0
1 0
0
−1 0
error
x(1, I) − x(3, I)
0 200
−1 400 600 0 time[s]
0 200
−1 400 600 0 time[s]
200
400 600 time[s]
Fig. 10. The synchronization errors for (k1 , k2 , τ ) = (1.5, 1.3, 0.2) (1,I) x = x(6,I) = x(3,II) = x(8,II) (2,I) x = x(7,I) = x(2,II) = x(7,II) (3,I) x = x(8,I) = x(1,II) = x(6,II) (4,I) x = x(5,II) �= x(5,I) = x(4,II)
Fig.10 represents the synchronization errors for (k1 , k2 , τ ) = (1.5, 1.3, 0.2) ∈ S¯c{(8,2)} (G), we can observe the partial synchronization pattern PS6 as shown in Fig.4(f). Remark 4. In full synchronization, all dynamics of xi (t) are same as that of an uncoupled system. On the other hand, in partial synchronization, the dynamics is not same as an uncoupled system because even if partial synchronization occurs, the coupling terms never vanish. Therefore, the actual regions for partial synchronization may be 18
Bliman, P.A. (2000). Extension of Popov absolute stability criterion to non-autonomous systems with delays. International Journal of Control, 73(15), 1349-1361. Mimura, T., Oguchi, T. (2012). Partial Synchronization of Lur’e Type Nonlinear Systems with Delay Couplings. In Proceedings of the 3rd IFAC. Steur, E., Oguchi, T., Leeuwen, C.V., Nijmeijer, H. (2012). Partial synchronization in diffusively time-delay coupled oscillator networks. Chaos, 22, 043144. Pogromsky, A.Y. (2008). A partial synchronization theorem. Chaos, 18, 037107. Oguchi, T., Oomen, W., Murguia, C., Nijmeijer, H. (2013). Partial Synchronization in Cartesian Product Networks of Coupled Nonlinear Systems without/with Delays. In Proceedings of JJAC (in Japanese). Oguchi, T., Nijmeijer, H. (2011) A synchronization condition for coupled nonlinear systems with time-delay : A frequency domain approach Int. J. Bifurcation and Chaos, Vol.21, No.9. Neefs, P.J., Steur, E. and Nijmeijer, H. (2010) Network complexity and synchronous behavior-an experimental approach. International Journal of Neural Systems, 20(3), 233-247. Mesbahi, M., Egerstedt, M. (2010). Graph Theoretic Methods in Multiagent Networks, 20(3). Princeton University Press. Hindmarsh, J.L., Rose, R.M. (1984). A Model for Neuronal Burstting Using Three Coupled Differential Equations. Proceedings of the Royal Society of London, B 221, 87102.