×

Maximum performance at minimum cost in network synchronization. (English) Zbl 1117.34048

The subject of the paper is the network of coupled identical oscillators
\[ \dot x_i = F(x_i) + \sigma \sum_{j=1}^{n} A_{ij} [H(x_j)-H(x_i)], \quad i=1,\dots,n, \]
where \(\sigma\) is the parameter which controls the overall coupling, \(x_i \in \mathbb{R}^m\) is the phase variable of \(i\)-th oscillator, \(A_{ij}\geq 0,\) is the relative strength of the coupling, and \(H: \mathbb{R}^m\to \mathbb{R}^m\) is the output function.
The following optimization problems are considered: 1. Maximization of synchronizability, i.e. the maximization of the region of parameter \(\sigma\), for which stable synchronization occurs. 2. Minimization of the synchronization cost, which is defined as the sum of the total input strength of all nodes at the lower synchronization threshold \(\sigma_{\min}\): \(C=\sigma_{\min} \sum_{i,j=1}^{n}A_{ij}\).
The main result shows that the solution sets for these two optimization problems coincide and are characterized by the following condition on the Laplacian (i.e. the matrix \((A_{ij})\)) eigenvalues: \[ 0=\lambda_1<\lambda_2=\cdots=\lambda_n. \] The authors study such optimal networks and drive some useful conclusions about their structure.

MSC:

34D05 Asymptotic properties of solutions to ordinary differential equations
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
34C30 Manifolds of solutions of ODE (MSC2000)
34C28 Complex behavior and chaotic systems of ordinary differential equations
34D08 Characteristic and Lyapunov exponents of ordinary differential equations
49N99 Miscellaneous topics in calculus of variations and optimal control
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] Zhou, C.S.; Motter, A.E.; Kurths, J., Phys. rev. lett., 96, 034101, (2006)
[2] Li, C.; Chen, L.; Aihara, K., Phys. biol., 3, 37, (2006)
[3] Stilwell, D.J.; Bollt, E.M.; Roberson, D.G., SIAM J. appl. dyn. syst., 5, 140, (2006)
[4] Arenas, A.; Díaz-Guilera, A.; Pérez-Vicente, C.J., Phys. rev. lett., 96, 114102, (2006)
[5] Wiley, D.A.; Strogatz, S.H.; Girvan, M., Chaos, 16, 015103, (2006)
[6] Park, S.M.; Kim, B.J., Phys. rev. E, 74, 026114, (2006)
[7] Boccaletti, S.; Hwang, D.-U.; Chavez, M.; Amann, A.; Kurths, J.; Pecora, L.M., Phys. rev. E, 74, 016102, (2006)
[8] Hasegawa, H., Phys. rev. E, 72, 056139, (2005)
[9] Donetti, L.; Hurtado, P.I.; Muñoz, M.A., Phys. rev. lett., 95, 188701, (2005)
[10] Morelli, L.G.; Cerdeira, H.; Zanette, D.H., Eur. phys. J. B, 43, 243, (2005)
[11] Restrepo, J.G.; Ott, E.; Hunt, B.R., Phys. rev. E, 71, 036151, (2005)
[12] Oh, E.; Rho, K.; Hong, H.; Kahng, B., Phys. rev. E, 72, 047101, (2005)
[13] Grinstein, G.; Linsker, R., Proc. natl. acad. sci., 102, 9948, (2005)
[14] Wu, C.W., Nonlinearity, 18, 1057, (2005)
[15] Masoller, C.; Martí, A.C., Phys. rev. lett., 94, 134102, (2005)
[16] Jalan, S.; Amritkar, R.E.; Hu, C.-K., Phys. rev. E, 72, 016211, (2005)
[17] Lind, P.G.; Gallas, J.A.C.; Herrmann, H.J., Phys. rev. E, 70, 056207, (2004)
[18] Atay, F.M.; Jost, J.; Wende, A., Phys. rev. lett., 92, 144101, (2004)
[19] Belykh, V.N.; Belykh, I.V.; Hasler, M., Physica D, 195, 159, (2004)
[20] Timme, M.; Wolf, F.; Geisel, T., Phys. rev. lett., 92, 074101, (2004)
[21] Wang, B.; Tang, H.; Zhou, T.; Xiu, Z.
[22] Donetti, L.; Hurtado, P.I.; Muñoz, M.A.
[23] Barahona, M.; Pecora, L.M., Phys. rev. lett., 89, 054101, (2002)
[24] Nishikawa, T.; Motter, A.E.; Lai, Y.-C.; Hoppensteadt, F.C., Phys. rev. lett., 91, 014101, (2003)
[25] McGraw, P.N.; Menzinger, M., Phys. rev. E, 72, 015101(R), (2005)
[26] Kocarev, L.; Amato, P., Chaos, 15, 024101, (2005)
[27] Motter, A.E.; Zhou, C.S.; Kurths, J., Europhys. lett., 69, 334, (2005)
[28] Motter, A.E.; Zhou, C.S.; Kurths, J., Phys. rev. E, 71, 016116, (2005)
[29] Motter, A.E.; Zhou, C.S.; Kurths, J., AIP conf. proc., 776, 201, (2005)
[30] Winfree, A.T., The geometry of biological time, (2001), Springer-Verlag New York · Zbl 0856.92002
[31] Rodriguez, E.; George, N.; Lachaux, J.-P.; Martinerie, J.; Renault, B.; Varela, F.J., Nature, 397, 430, (1999)
[32] Stopfer, M.; Bhagavan, S.; Smith, B.H.; Laurent, G., Nature, 390, 70, (1997)
[33] Hwang, D.-U.; Chavez, M.; Amann, A.; Boccaletti, S., Phys. rev. lett., 94, 138701, (2005)
[34] Nishikawa, T.; Motter, A.E., Phys. rev. E, 73, 065106, (2006)
[35] Fischer, E.; Sauer, U., Nat. genet., 37, 636, (2005)
[36] Korniss, C.; Novotny, M.A.; Guclu, H.; Toroczkai, Z.; Rikvold, P.A., Science, 299, 677, (2003)
[37] Pecora, L.M.; Carroll, T.L., Phys. rev. lett., 80, 2109, (1998)
[38] Lu, W.; Chen, T., Physica D, 213, 214, (2006)
[39] Lu, W.; Chen, T., Physica D, 198, 148, (2004)
[40] Lu, W.; Chen, T., IEEE trans. circuits syst. I, 51, 2491, (2004)
[41] Fink, K.S.; Johnson, G.; Carroll, T.; Mar, D.; Pecora, L., Phys. rev. E, 61, 5080, (2000)
[42] Heagy, J.F.; Pecora, L.M.; Carroll, T.L., Phys. rev. lett., 74, 4185, (1995)
[43] Wu, C.W., Linear algebra appl., 402, 207, (2005)
[44] S. Boccaletti, private communication
[45] Braess, D., Unternehmensforschung, 12, 258, (1968)
[46] Irvine, A.D., Int. stud. philos. sci., 7, 145, (1993)
[47] Toroczkai, Z.; Bassler, K.E., Nature, 428, 716, (2004)
[48] Durand, M.
[49] Yook, S.-H.; Meyer-Ortmanns, H.
[50] Zheng, Z.; Hu, G.; Hu, B., Phys. rev. E, 62, 7501, (2000)
[51] Jing, J.; Weiss, K.R., Curr. biol., 15, 1712, (2005)
[52] Quiroga, R.Q.; Reddy, L.; Kreiman, G.; Koch, C.; Fried, I., Nature, 435, 1102, (2005)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.