Document Version Publisher s PDF, also known as Version of Record (includes final page, issue and volume numbers)

Transcript

1 Partial Synchronization in Cartesian Product Networks of Coupled Nonlinear Systems Without/with Delays Oguchi, T.; Nijmeijer, H.; Murguia Rendon, C.G.; Oomen, W.A.W.A. Published in: 56th Japan Joint Automatic Control Published: 01/01/2013 Document Version Publisher s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication: A submitted manuscript is the author's version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. The final author version and the galley proof are versions of the publication after peer review. The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication Citation for published version (APA): Oguchi, T., Nijmeijer, H., Murguia Rendon, C. G., & Oomen, W. A. W. A. (2013). Partial Synchronization in Cartesian Product Networks of Coupled Nonlinear Systems Without/with Delays. In 56th Japan Joint Automatic Control (pp ). Tokyo: T. Oguchi. General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. Users may download and print one copy of any publication from the public portal for the purpose of private study or research. You may not further distribute the material or use it for any profit-making activity or commercial gain You may freely distribute the URL identifying the publication in the public portal? Take down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Download date: 27. Jun. 2018

2 631 () ( ) Willard Oomen (Eindhoven Univ. of Tech.) Carlos Murguia (Eindhoven Univ. of Tech.) Henk Nijmeijer (Eindhoven Univ. of Tech.) Partial Synchronization in Cartesian Product Networks of Coupled Nonlinear Systems without/with Delays T. Oguchi (Tokyo Metropolitan Univ.), W. Oomen (Eindhoven Univ. of Tech.) C. Murguia (Eindhoven Univ. of Tech.) and H. Nijmeijer (Eindhoven Univ. of Tech.) Abstract This paper considers the synchronization problem in the Cartesian product networks of chaotic systems without/with delays in the couplings. By scaling the synchronization condition for two identical systems with bidirectional delayed coupling with respect to the coupling strength, we can estimate conditions for full/partial synchronization in networks of coupled systems. In this report, we show that full/partial synchronization patterns in the Cartesian product networks of two network systems can be estimated by the same approach. The validity of the estimated synchronization conditions is tested by numerical simulations and experiments with electric circuits. Key Words: Chaos synchronization, Networks, Time-delays, The Cartesian product 1 2 N 1) { ẋ i (t) = Ax i (t)+f(x i )+Bu i (t) Σ i : (1) y i (t) = Cx i (t) x i R n, u i,y i R f : R n R n CB > 0 u i N 3) u i (t) = k ij (y i (t l) y j (t l)) (2) j=1 Lyapunov-Krasovskii k ij 0 (i,j) k ij = k, k ij = 0 l 0 l = 0 u(t) = [u 1 (t) u N (t)] T Lur e u(t) = k(l(g) C)x(t l) 2, 3) 4, 5) 6) ẋ(t) =(I N A)x(t)+F(x) k(i N B)(L(G) C)x(t l) (3) F(x) F(x) = [ f(x 1 ) T f(x N ) T] T, L(G) G D(G) A(G) L(G) = D(G) A(G) [No.13-23] 第 56 回自動制御連合講演会 ( 新潟市 )

3 2.1 Σ i,σ j ϕ i ϕ j C r ϕ 1,ϕ j t ] [ ] [ẋ1 (t) x x i (t) x j (t) 0 r=(i ė(t) N A 0 ) 1 (t) +Ψ(x e(t) 1,e) Σ i Σ j [ ][ ] ϕ C := max l θ 0 ϕ(θ) 0 a12 BC a k 1N BC x1 (t τ) 0 ML(K N )M + BC e(t τ) Σ i,σ j x 1 (t) x 2 (t) e(t) :=. = (M I n )x(t) x 1 (t) x N (t) l = M = Wu-Chua 7). m 1... R (N 1) N G 1 m G G 1 α 1. α 1 λ 2 (G 1 ) = α 2 λ 2 (G 2 ) α 2 G 2 M + = 1.. λ 2 (G i )G i... 0 RN (N 1) 0 1 8) Ψ = [ f(x 1 ) T ψ T (x 1,e 12 ) ψ T (x 1,e 1N ) ] T 2, 3, 4, 5) a 1j A(1,j) N = 2 L(K 2 ) ML(K N )M + = diag(n,,n) [ ] 1 1 L(K 2 ) = e(t) 1 1 N 1 [ ] x1 (t) ( [ ] 1 0 ) ė 1j = Ae 1j (t)+ψ(x 1,e 1j ) kbce 1j (t l) = I e 12 (t) 1 1 n x(t) k N k = ] [ ] [ẋ1 (t) x1 (t) λ 2 (K N )k = Nk ( 2,l) S =(I ė 12 (t) 2 A) +Ψ(x e 12 (t) 1,e 12 ) N [ ][ ] N = 2 0 BC x1 (t l) k (4) S N 0 2BC e 12 (t l) [ ] [ ] f(x Ψ(x 1,e 12 ) = 1 ) f(x1 ) 0N( N 1) S := f(x 1 ) f(x 1 e 12 ) ψ(x 1,e 12 ) 1, ė 12 = Ae 12 (t) + ψ(x 1,e 12 ) 2kBCe 12 (t l) (k,l) S ML(G)M + P 1 ML(G)M + P = N diag(λ 2,,λ N ) P λ i L(G) 0 = λ 1 < λ 2 λ N ψ(x 1,e)e = e = (P I n )ē [ ] x1 (t) = 1 1. e(t) I n x(t) N := (M 0 I n )x(t), (5) Σ i : ē i (t) = Aē i (t)+ψ (x 1 )ē i (t) λ i+1 kbcē i (t l) (6)

4 l l (a) S k S N 1 S 2 Fig. 1: Scaling method :(a) Synchronization condition for N = 2, and (b) the estimated synchronization condition for N coupled systems. (b) S 1 S k L(G 1 G 2 ) = L(G 1 ) I m +I n L(G 2 ) = L(G 1 ) L(G 2 ) (8) G 1 G 2 n, m i = 1,...,N 1.Σ i 1 10) G 1 G 2 n m (k,l) S i S i λ 1,λ 2,...,λ n S i = {(k,l) ( λ µ 1,µ 2,...,µ m L(G 1 ) L(G 2 ) i+1k,l) S} 2 u 1,u 2,...,u n v 1,v 2,...,v m N = 2 2/λ L(G 1 G 2 ) λ i + µ j i+1 S = i I S i ( I I = {1,...,N 1}) (k,l) S u i v j, i = 1,2,...,n, j = 1,2,...,m x 1 = x 2 = = x N L(G) = L(G 1 G 2 ) N 1 (6) λ 2 (L(G 1 G 2 )) = min(λ 2 (L(G 1 )),λ 2 (L(G 2 ))) M(< N 1) I M I M I I M = M i I M S i (k,τ) i I M S i ē i = 0 (i I M ) M ē i = (µ T i I n )x(t) = 0 for i I M (7) G 1,G 2 µ i L(G) λ i (7) { j,k I x j (t) = x k (t) u i = L(G 1 ){y i (t l) y j (t l)} for G 1 ij u i (t) = k 2 L(G 2 ){y i (t l) y j (t l)} for G 2 3 G 1, G 2 ẋ =(I nm A)x(t)+F(x) (9) ((k 1 L(G 1 ) k 2 L(G 2 )) BC)x(t l) (10) (1) (2) x T = [x T (1,1) G V,...,xT (1,m),...,xT (n,1),...,xt (n,m) ] E G = (V,E) G 1 = (V 1,E 1 )G 2 = (V 2,E 2 ) G = G 1 G 2 G V(G) = V 1 V 2 V(G) (v 1,v 2 )(v 1,v 2) S K2 v 1 = v 1 (v 2,v 2) E 2 v 2 = v 2 (v 1,v 1 ) E 1 S (i,j) (G) = {(,k 2,l) ( λ i +k 2 µ j,l) S K2 (11) 2 G 1 G 2 i,j S (i,j)

5 4 4.1 G 1 G Hindmarsh-Rose ẏ i (t) = yi 3(t)+3y2 i (t)+z i,1(t) z i,2 (t)+a+u i (t) ż i,1 (t) = 1 5y ( i 2(t) z i,1(t) ) ż i,2 (t) = b 4(y i (t)+c) z i2 (t) L(G 1 ) = (12) a = 3.25,b = 0.005,c = Fig. 2 sandglass (13) networkg 1 2 K 2 λ 1 = 0, λ 2 = 3 3, λ 3 = 4 6, λ 4 = 2, ½ λ 5 = 4, λ 6 = 3+ 3, λ 7 = 6, λ 8 = 4+ 6 ¾ (7) Fig. 5 k 2 k λ 2 2 k λ 2 > k 2 k λ 3 Fig. 2: Simplified Sandglass Network { u i (t) = L(G 1 ){y i (t l) y j (t l)} for G 1 x 1 = x 3 x 4 = x 5 x 6 = x 8 x 2 x 7 (14) kk 2 k = 0.52 u i (t) = k 2 L(K 2 ){y i (t l) y j (t l)} for K 2 Ë Ë Ë Ë ¼ ¾ K 2 ¾ ¾ Fig.3 4: Synchronization region forg 1 (k,l) 30 1 k G = G 1 K 2 (11) Fig. 5 l = 0 ¾ Wu-Chua Ë Ë Ë¾ 4 È˽ ÈË Ë Delayl Coupling Strengthk Fig. 3: Synchronization region for N = 2 ¼¾ ¼ ÆÓ ËÝÒ È˾ ¼ ¼¾ ÈË Fig. 5: Synchronization bifurcation diagram for G = G 1 K 2 PS1, PS3, PS4 Figs. 6-8 ½

6 (1,2 ) (1,1 ) (2,1 ) (3,1 ) (4,1 ) (5,1 ) (2,2 ) ( (3,2 ) (4,2 ) (6,1 ) (5,2 ) (7,1 ) (6,2 ) (7,2 ) (8,2 ) PS3,PS4, (,k 2 ) Figs PS3,PS4 x (1,1 ) x (2,1 ) x (2,1 ) x (7,1 ) (8,1 ) Fig. 6: Partial synchronization pattern :PS1 (1,1 ) (4,1 ) (2,1 ) (5,1 ) (3,1 ) (1,2 ) (2,2 ) ( (3,2 ) (4,2 ) (6,2 ) (6,1 ) (5,2 ) (7,1 ) (7,2 ) (8,2 ) k2 k2 x (1,2 ) x (2,2 ) x (2,2 ) x (7,2 ) k2 (8,1 ) Fig. 7: Partial synchronization pattern :PS3 Fig. 10: Success rate of synchronization between systems related with PS4 x (2,1 ) x (7,2 ) x (7,1 ) x (2,2 ) (1,2 ) (1,1 ) (2,2 ) (4,2 ) (6,2 ) k2 k2 (2,1 ) (4,1 ) (6,1 ) ( (3,2 ) (5,2 ) (7,2 ) (3,1 ) (5,1 ) (7,1 ) (8,1 ) (8,2 ) Fig. 11: Success rate of synchronization between systems related with FS Fig. 8: Partial synchronization pattern :PS4 4.2 k2 k2 x (1,1 ) x (3,2 )x (3,1 ) x (1,2 )x (4,1 ) x (5,2 ) x (5,1 ) x (4,2 )x (6,1 ) x (8,2 )x (8,1 ) x (6,2 ) Hindmarsh-Rose Eindhoven Fig (Fig. 13) Fig. 9: Success rate of synchronization between systems related with PS3 Fig. 12: An Electric Hindmarsh-Rose neuron

7 Fig. 14: Full Synchronization observed in Experiment PS4 5 Fig. 13: Experimental setup of electric neuron network system (,k 2 ) = (1.0,0.6) Fig. 14 Fig. 15(,k 2 ) = (1.0,0.5) G 1 G 1 PS4 C ) S. Strogatz, SYNC-The Emerging Science of Spontaneous Order, Penguin (2004) 2) T. Mimura and T. Oguchi, Synchronization and topology in networks of Lur e systems with delay couplings, Proc. of ENOC 2012, Rome, Italy (2011) 3) T. Mimura and T. Oguchi, Partial synchronization of Lur e type nonlinear systems with delay couplings, Proc. of 3rd IFAC Chaos, Cancún, Mexico (2012) 4) E. Steur, Synchronous behavior in networks of coupled systems- with applications to neuronal dynamics, PhD dissertation, Eindhoven University of Technology (2011) 5) E. Steur, T. Oguchi, C. van Leeuwen, and H. Nijmeijer, Partial Synchronization in difffusively time-delay coupled oscillator networks, Chaos (American Institute of Physics), Vol. 22, No. 4, (2012) 6). 55 pp (2012) 7) C.W. Wu. Synchronization in Complex Networks of Nonlinear Dynamical Systems, World Scientific (2007) 8) A. Pogromsky, G. Santoboni and H. Nijmeijer, Partial synchronization: from symmetry toward stability, Physica D, 172(1-4), 65/87 (2002) 9) E. Steur and H. Nijmeijer, Synchronization in networks of diffusively time-delay coupled (semi-)passive systems, IEEE Trans. Circ. Syst. I, 58-6, 1358/1371 (2011) 10) M. Mesbashi and M. Egerstedt, Graph Theoretic Methods in Multiagent Networks, Princeton Univ. Press (2010) Fig. 15: Partial Synchronization PS4 observed in Experiment N = 2 k k