On Synchronization of Pinning-Controlled Networks with Reducible and Asymmetric Coupling Matrix

Xuan Zhou, Hui Feng, Jianwen Feng, Yi Zhao

College of Mathematics and Computational Sciences, Shenzhen University, Shenzhen, China.
School of Mathematics and Statistics, Wuhan University, Wuhan, China.

DOI: 10.4236/cn.2011.32014                

Abstract

This paper investigates the synchronization of directed networks whose coupling matrices are reducible and asymmetrical by pinning-controlled schemes. A strong sufficient condition is obtained to guarantee that the synchronization of the kind of networks can be achieved. For the weakly connected network, a method is presented in detail to solve two challenging fundamental problems arising in pinning control of complex networks: (1) How many nodes should be pinned? (2) How large should the coupling strength be used in a fixed complex network to realize synchronization? Then, we show the answer to the question that why all the diagonal block matrices of Perron-Frobenius normal matrices should be pinned? Besides, we find out the relation between the Perron-Frobenius normal form of coupling matrix and the differences of two synchronization conditions for strongly connected networks and weakly connected ones with linear coupling configuration. Moreover, we propose adaptive feedback algorithms to make the coupling strength as small as possible. Finally, numerical simulations are given to verify our theoretical analysis.

Keywords

Globally Synchronization, Complex Networks, Pinning Control, Reducible and Asymmetric Matrix

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	A. L. Barabási and R. Albert, “Emergence of Scaling in Random Networks,” Science, Vol. 286, No. 5439, 1999, pp. 509-512. doi:10.1126/science.286.5439.509
[2]	S. H. Strogatz, “Exploring Complex Networks,” Nature, Vol. 410, 2001, pp. 268-276. doi:10.1038/35065725
[3]	D. J. Watts and S. H. Strogatz, “Collective Dynamics of Small-World,” Nature, Vol. 393, 1998, pp. 440-442. doi:10.1038/30918
[4]	S. A. Pandit and R. E. Amritkar, “Characterization and Control of Small-World Networks,” Physical Review E, Vol. 60, No. 2, 1999, pp. 1119-1122. doi:10.1103/PhysRevE.60.R1119
[5]	Q. Song, J. D. Cao and W. W. Yu, “Second-Order Leader- Following Consensus of Nonlinear Multi-Agent Systems via Pinning Control,” Systems and Control Letters, Vol. 59, No. 9, 2010, pp. 553-562. doi:10.1016/j.sysconle.2010.06.016
[6]	Q. Song and J. D. Cao, “On Pinning Synchronization of Directed and Undirected Complex Dynamical Networks,” IEEE Transactions on Circuit System I, Vol. 57, No. 3, 2010, pp. 672-680. doi:10.1109/TCSI.2009.2024971
[7]	W. Lin and H. F. Ma, “Synchronization between Adaptively Coupled Systems with Discrete and Distributed Time- Delays,” IEEE Transactions on Automatic Control, Vol. 55, No. 4, 2010, pp. 819-830. doi:10.1109/TAC.2010.2041993
[8]	X. W. Liu and T. P. Chen, “Synchronization Analysis for Nonlinearly-Coupled Complex Networks with an Asym- Metrical Coupling Matrix,” Physica A, Vol. 387, No. 16- 17, 2008, pp. 4429-4439.
[9]	W. W. Yu, G. R. Chen and J. H. Lü, “On Pinning Syn- Chronization of Complex Dynamical Networks,” Automa- tica, Vol. 45, No. 2, 2009, pp. 429-435. doi:10.1016/j.automatica.2008.07.016
[10]	Z. G. Zhang and X. Z. Liu, “Observer-Based Impulsive Chaotic Synchronization of Discrete-Time Switched Systems,” Nonlinear Dynamics, Vol. 62, No. 4, 2010, pp. 781-789. doi:10.1007/s11071-010-9762-y
[11]	W. L. Guo, F. Austin and S. H. Chen, “Global Synchroni- zation of Nonlinearly Coupled Complex Networks with Non-Delayed and Delayed Coupling,” Communications in Nonlinear Science and Numerical Simulation, Vol. 15, No. 6, 2010, pp. 1631-1639. doi:10.1016/j.cnsns.2009.06.016
[12]	F. Liu, Q. Song and J. D. Cao, “Improvements and Application of Entrainment Control for Nonlinear Dynami- cal Systems,” Chaos, Vol. 18, No. 4, 2008, pp. 043120. doi:10.1063/1.3029670
[13]	T. P. Chen, X. W. Liu and W. L. Lu, “Pinning Complex Networks by a Single Controller,” IEEE Transactions on Circuits and Systems-I: Regular Papers, Vol. 54, 2007, pp. 1317-1326. doi:10.1109/TCSI.2007.895383
[14]	W. G. Xia and J. D. Cao, “Pinning Synchronization of Delayed Dynamical Networks via Periodically Intermitt- ent Control,” Chaos, Vol. 19, No. 1, 2009, Article ID: 013120.doi:10.1063/1.3071933
[15]	D. P. Lellis, D. M. Bernardo and F. Garofalo, “Synchroni- zation of Complex Networks through Local Adaptive Coupling,” Chaos, Vol. 18, No. 3, 2008, Article ID: 037110.doi:10.1063/1.2944236
[16]	D. P. Lellis, D. M. Bernardo and F. Garofalo, “Novel Decentralized Adaptive Strategeis for the Synchroniza- tion of Complex Networks,” Automatica, Vol. 45, No. 5, 2009, pp. 1312-1318. doi:10.1016/j.automatica.2009.01.001
[17]	M. Porfiri and D. M. Bernardo, “Criteria for Global Pinning- Controllability of Complex Networks,” Automatica, Vol. 44, No. 12, 2008, pp. 3100-3106.
[18]	T. Yanagita and A. S. Mikhailov, “Design of Easily Synchronizable Oscillator Networks Using the Monte Carlo Optimization Method,” Physical Review E, Vol. 81, No. 5, 2010, Article ID: 056204. doi:10.1103/PhysRevE.81.056204
[19]	J. C. Zhao, J. A. Lu and X. Q. Wu, “Pinning Control of General Complex Dynamical Networks with Optimiza- tion,” Science China-Information Sciences, Vol. 53, No. 4, 2010, pp. 813-822. doi:10.1007/s11432-010-0039-3
[20]	G. S. Duane, “A ‘Cellular Neuronal’ Approach to Optimi- zation Problems,” Chaos, Vol. 19, No. 3, 2009, Article ID: 033114.doi:10.1063/1.3184829
[21]	J. H. Park, S. M. Lee and H. Y. Jung, “LMI Optimization Approach to Synchronization of Stochastic Delayed Discrete- Time Complex Networks,” Journal of Optimization Theory and Application, Vol. 143, No. 2, 2009, pp. 357-367. doi:10.1007/s10957-009-9562-z
[22]	L. Z. Lu and C. E. M. Pearce, “Some New Bounds for Singular Values and Eigenvalues of Matrix Product,” Annals of Operations Research, Vol. 98, No. 1-4, 2000, pp. 141-148. doi:10.1023/A:1019200322441
[23]	W. W. Yu, J. D. Cao, K. W. Wong and J. H. Lü, “New Communication Schemes Based on Adaptive Synchroni- Zation,” Chaos, Vol. 17, No. 3, 2007, Article ID: 033114. doi:10.1063/1.2767407
[24]	X. F. Li, A. C. S. Leung and X. P. Han, “Complete (Anti-) Synchronization of Chaotic Systems with Fully Uncertain Parameters by Adaptive Control,” Nonlinear Dynamics, Vol. 63, No. 1-2, 2011, pp. 263-275. doi:10.1007/s11071-010-9802-7