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**New conditions on synchronization of networks of linearly coupled dynamical systems with non-Lipschitz right-hand sides.**
*(English)*
Zbl 1258.93007

Summary: In this paper, we study synchronization of networks of linearly coupled dynamical systems. The node dynamics of the network can be very general, which may not satisfy the QUAD condition. We derive sufficient conditions for synchronization, which can be regarded as extensions of previous results. These results can be employed to networks of coupled systems, of which, in particular, the node dynamics have non-Lipschitz or even discontinuous right-hand sides. We also give several corollaries where the synchronization of some specific non-QUAD systems can be deduced. As an application, we propose a scheme to realize synchronization of coupled switching systems via coupling the signals which drive the switchings. Examples with numerical simulations are also provided to illustrate the theoretical results.

### MSC:

93A14 | Decentralized systems |

94C15 | Applications of graph theory to circuits and networks |

93A15 | Large-scale systems |

34H05 | Control problems involving ordinary differential equations |

05C82 | Small world graphs, complex networks (graph-theoretic aspects) |

### Keywords:

linearly coupled systems; synchronization; non-Lipschitz right-hand sides; discontinuous right-hand side
Full Text:
DOI

### References:

[1] | Aubin, J. P., Viability theory (1991), Birkhauser · Zbl 0755.93003 |

[2] | Aubin, J. P.; Frankowska, H., Set-valued analysis (1990), Birkhauser: Birkhauser Boston |

[3] | Belykh, I.; Belykh, V.; Hasler, M., Connection graph stability method for synchronized coupled chaotic systems, Physica D, 195, 159-187 (2004) · Zbl 1098.82622 |

[4] | Belykh, I.; Belykh, V.; Hasler, M., Blinking model and synchronization in small-world networks with a time-varying coupling, Physica D, 195, 188-206 (2004) · Zbl 1098.82621 |

[5] | Belykh, I.; Belykh, V.; Hasler, M., Synchronization in asymmetrically coupled networks with node balance, Chaos, 16, 015102 (2006) · Zbl 1144.37318 |

[7] | Chen, M. Y., Chaos synchronization in complex networks, IEEE Transactions on Circuits and Systems I, 55, 1335-1346 (2008) · Zbl 1452.62238 |

[8] | Clarke, F. H., Optimization and nonsmooth analysis (1983), Wiley and Sons: Wiley and Sons New York · Zbl 0727.90045 |

[9] | Collins, J. J.; Stewart, I., Coupled nonlinear oscillators and the symetries of animal gaits, Science, 3, 349-392 (1993) · Zbl 0808.92012 |

[10] | Duan, Z. S.; Chen, G. R., Global robust stability and synchronization of networks with Lorenz-Type nodes, IEEE Transactions on Circuits and Systems II, 56, 679-683 (2009) |

[11] | Duane, G. S.; Webster, P. J.; Weiss, J. B., Go-occurrence of Northern and Southern Hemisphere blocks as partially synchronized chaos, Journal of the Atmospheric Sciences, 56, 4183-4205 (1999) |

[12] | Filippov, A. F., Differential equations with discontinuous right-hand sides. Mathematics and its applications Soviet Series (1988), Kluwer Academic Publishers: Kluwer Academic Publishers Boston · Zbl 0664.34001 |

[13] | Honerkamp, J., The heat as a system of coupled nonlinear oscillators, Journal of Mathematical Biology, 19, 69-88 (1983) |

[14] | Hoppensteadt, F. C.; Izhikevich, E. M., Pattern recognition via synchronization in phase-locked loop neural networks, IEEE Transactions on Neural Networks, 11, 734-738 (2000) |

[15] | Horn, R. A.; Johnson, C. R., Topics in matrix analysis (1991), Cambridge University Press · Zbl 0729.15001 |

[16] | Kuramoto, Y., Chemical oscillations, waves and turbulence (1984), Berlin:Springer · Zbl 0558.76051 |

[17] | Liberzon, D., Switched systems (Control engineering series) (559-574) (2005), Birkhauser: Birkhauser Boston |

[18] | Liberzon, D.; Morse, A. S., Basic problems in stability and design of switched systems, IEEE Control Systems Magazine, 19, 59-70 (1999) · Zbl 1384.93064 |

[19] | Liu, X. W.; Chen, T. P., Boundedness and synchronization of y-coupled Lorenz systems with or without controllers, Physica D, 237, 630-639 (2008) · Zbl 1168.34338 |

[20] | Lu, W. L.; Chen, T. P., Synchronization of coupled connected neural networks with delays, IEEE Transactions on Circuits and Systems I, 51, 12, 2491-2503 (2004) · Zbl 1371.34118 |

[21] | Lu, W. L.; Chen, T. P., New approach to synchronization analysis of linearly coupled ordinary differential systems, Physica D, 213, 214-230 (2006) · Zbl 1105.34031 |

[22] | Mirollo, R.; Strogatz, S., Synchronization of pulsed-coupled biological oscillators, SIAM Journal on Applied Mathematics, 50, 1645-1662 (1990) · Zbl 0712.92006 |

[23] | Netoff, T. I.; Schiff, S. J., Decreased neuronal synchronization during experimental seizures, Journal of Neuroscience, 22, 7297-7307 (2002) |

[24] | Ohtsubop, J., Feedback induced instability and chaos in semiconductor lasers and their applications, Optical Review, 6, 1-15 (1999) |

[25] | Pavlov, A. V.; van de Wouw, N.; Nijmeijer, H., On convergence properties of piecewise affine systems, International Journal of Control, 80, 1233-1247 (2007) · Zbl 1133.93021 |

[26] | Pecora, L. M.; Carroll, T. L., Synchronization in chaotic systems, Physical Review Letters, 64, 821-824 (1990) · Zbl 0938.37019 |

[27] | Petreczky, M., Realization theory for linear switched systems:formal power series approach, Systems and Control Letters, 56, 588-595 (2007) · Zbl 1155.93335 |

[28] | Porfiri, M.; Stilwell, D. J.; Bollt, E. M., Synchronization in random weighted directed networks, IEEE Transactions on Circuits and Systems I, 55, 3170-3177 (2008) |

[29] | Russo, G.; di Bernardo, M., Contraction theory and master stability function: linking two approaches to study synchronization of complex networks, IEEE Transactions on Circuits and Systems II, 56, 177-181 (2009) |

[30] | Strogatz, S., Exploring complex networks, Nature, 410, 268-276 (2001) · Zbl 1370.90052 |

[31] | Torre, V., A theory of synchronization of two heart pacemaker cells, Journal of Theoretical Biology, 61, 55-71 (1976) |

[32] | Van de Wouw, N.; Pavlov, A., Tracking and synchronisation for a class of PWA systems, Automatica, 44, 2909-2915 (2008) · Zbl 1152.93389 |

[33] | VanWiggeren, G. D.; Roy, P., Communication with laser, Science, 279, 1198-1200 (1998) |

[34] | Vieria, M.de S., Chaos and synchronized chaos in an earthquake model, Physical Review Letters, 82, 201-204 (1999) |

[35] | Wu, C. W., Synchronization in coupled arrays of chaotic oscillators with nonreciprocal coupling, IEEE Transactions on Circuits and Systems I, 50, 294-297 (2003) · Zbl 1368.34051 |

[36] | Wu, C. W., Synchronization in arrays of coupled nonlinear systems with delay and nonreciprocal time-varying coupling, IEEE Transactions on Circuits and Systems II, 52, 282-286 (2005) |

[37] | Wu, C. W.; Chua, L. O., Synchronization in an array of linearly coupled dynamical systems, IEEE Transactions on Circuits and Systems I, 42, 430-447 (1995) · Zbl 0867.93042 |

[38] | Yang, T.; Chua, L., Impulsive control and synchronization of nonlinear dynamical systems and application to secure communication, International Journal of Bifurcation and Chaos, 7, 645-664 (1997) · Zbl 0925.93374 |

[39] | Zhao, J.; Hill, D. J., Passivity and stability of switched systems: a multiple storage function method, Systems and Control Letters, 57, 158-164 (2008) · Zbl 1137.93051 |

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