##
**The changes on synchronizing ability of coupled networks from ring networks to chain networks.**
*(English)*
Zbl 1169.34327

Summary: In this paper, two different ring networks with unidirectional couplings and with bidirectional couplings were discussed by theoretical analysis. It was found that the effects on synchronizing ability of the two different structures by cutting a link are completely opposite. The synchronizing ability will decrease if the change is from bidirectional ring to bidirectional chain. Moreover, the change on synchronizing ability will be four times if the number of \(N\) is large enough. However, it will increase obviously from unidirectional ring to unidirectional chain. It will be \(N^{2}/(2\pi ^{2})\) times if the number of \(N\) is large enough. The numerical simulations confirm the conclusion in quality. This paper also discusses the effects on synchronization by adding one link with different length d to these two different structures. It can be seen that the effects are different. Theoretical results are accordant to numerical simulations. Synchronization is an essential physics problem. These results proposed in this paper have some important reference meanings on the real world networks, such as the bioecological system networks, the designing of the circuit, etc.

### MSC:

34D05 | Asymptotic properties of solutions to ordinary differential equations |

05C50 | Graphs and linear algebra (matrices, eigenvalues, etc.) |

05C90 | Applications of graph theory |

15A18 | Eigenvalues, singular values, and eigenvectors |

### Keywords:

complex networks; synchronizing ability; unidirectional couplings; bidirectional couplings; ring; chain; coupling matrix
PDFBibTeX
XMLCite

\textit{X. Han} and \textit{J. Lu}, Sci. China, Ser. F 50, No. 4, 615--624 (2007; Zbl 1169.34327)

Full Text:
DOI

### References:

[1] | Watts D J, Strogatz S H. Collective dynamics of ’small world’ networks. Nature, 1998, 393: 440–442 · Zbl 1368.05139 · doi:10.1038/30918 |

[2] | Barabasi A L, Albert R. Emergence of scaling in random networks. Science, 1999, 286: 509–512 · Zbl 1226.05223 · doi:10.1126/science.286.5439.509 |

[3] | Wang X F, Li X, Chen G R. Theory and Application of Complex Network. (In Chinese) Beijing: Tsinghua University Press, 2006 |

[4] | Zheng Z G Spatiotemporal Dynamics and Collective Behaviors in Coupled Nonlinear Systems (in Chinese). Beijing: Higher Education Press, 2004 |

[5] | Hong H, Kim B J, Choi M Y, et al. Factors that predict better synchronizability on complex networks. Phys Rev E, 2004, 69: 067105-1–067105-4 · doi:10.1103/PhysRevE.69.067105 |

[6] | Barahona M, Pecora L M. Synchronization in small-world systems. Phys Rev Lett, 2002, 89(5): 054101-1–054101-4 · doi:10.1103/PhysRevLett.89.054101 |

[7] | Wang X F, Chen G R. Synchronization in scale-free dynamical networks: robustness and fragility. IEEE Trans Circuits Systems-I, 2002, 49(1): 54–62 · Zbl 1368.93576 · doi:10.1109/81.974874 |

[8] | Wu C W, Chua L O. Application of Graph Theory to the Synchronization in an array of coupled nonlinear oscillators. IEEE Trans Circuits Systems-I, 1995, 42(8): 494–497 · doi:10.1109/81.404064 |

[9] | Wang X F, Chen G R. Synchronization in small-world dynamical networks. Int J Bifur Chaos, 2002, 12(1): 187–192 · doi:10.1142/S0218127402004292 |

[10] | Matías M A, Pére-Muńuzuri V, Lorenzo M N, et al. Observation of a fast rotating wave in rings of coupled chaotic oscillators. Phys Rev Lett, 1997, 78(2): 219–222 · doi:10.1103/PhysRevLett.78.219 |

[11] | Lorenzo M N, Marińo I P, Pére-Muńuzuri V, et al. Synchronization waves in arrays of driven chaotic systems. Phys Rev E, 1996, 54(4): 3094–3097 · doi:10.1103/PhysRevE.54.R3094 |

[12] | Cuomo K M, Oppenhein A V. Circuit implementation of synchronized chaos with applications to communications. Phys Rev Lett, 1993, 71(1): 65–68 · Zbl 1051.37509 · doi:10.1103/PhysRevLett.71.65 |

[13] | Chua L O, Yang L. Cellular neural networks: Theory and applications. IEEE Trans Circuits Systems, 1988, 35: 1257–1272 · Zbl 0663.94022 · doi:10.1109/31.7600 |

[14] | Néry L, Lefebvre H, Fradet A. Kinetic and mechanistic studies of carboxylic acid-bisoxazoline chain-coupling reactions. Macromol Chem Phys, 2003, 204: 1755–1764 · doi:10.1002/macp.200350036 |

[15] | Jund P, Kim S G, Toma’nek D, et al. Stability and fragmentation of complex structures in ferrofluids. Phys Rev Lett, 1995, 74(15): 3049–3052 · doi:10.1103/PhysRevLett.74.3049 |

[16] | Collins J J E, Stewart I N. A group-theoretic approach to rings of coupled biological oscillators. Biol Cybem, 1994, 71: 95–103 · Zbl 0804.92009 · doi:10.1007/BF00197312 |

[17] | Abarbanel H D I, Rabinovich M I, Selverston A, et al. Synchronization in neural networks. Phys Uspek, 1996, 39: 337–362 · doi:10.1070/PU1996v039n04ABEH000141 |

[18] | Wu C W, Chua L O. Synchronization in an array of linearly coupled dynamical systems. IEEE Trans Circuits Systems-I, 1995, 42(8): 430–447 · Zbl 0867.93042 · doi:10.1109/81.404047 |

[19] | Ranu J, Elizabeth J B, James J A. Real-time interactions between a neuromorphic electronic circuit and spinal cord. IEEE Trans Neural Systems Rehab Eng, 2001, 9(3): 319–326 · doi:10.1109/7333.948461 |

[20] | Sahakian A V, Myers G A, Maglaveras N. Unidirectional block in cardiac fibers: Effects of discontinuties in coupling resistance and spatial changes in resting membrane potential in a computer simulation study. IEEE Trans on Biomed Eng, 1992, 39(5): 510–522 · doi:10.1109/10.135545 |

[21] | Ralf M, Laura C, Andreas P, et al. Directionality of coupling of physiological subsystems: age-related changes of cardiorespiratory interaction during different sleep stages in babies. Am J Physiol Regulatory Integrative Comp Physiol, 2003, 285: 195–1401 |

[22] | Otsuka K, Ikeda K. Cooperative dynamics and functions in a collective nonlinear optical element system. Phys Rev A, 1989, 39(10): 5209–5228 · doi:10.1103/PhysRevA.39.5209 |

[23] | Wang W, Slotine J J E. On partial contraction analysis for coupled nonlinear oscillators. Biol Cybern, 2004, 92(1): 38–53 · Zbl 1096.92001 · doi:10.1007/s00422-004-0527-x |

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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.