### Citations

7682 | Matrix Analysis - Horn, Johnson - 1985 |

1555 |
Deterministic Nonperiodic Flows
- Lorenz
(Show Context)
Citation Context ...network (LN). Then the outer-coupling matrices are AGCN, ASN, and ALN, respectively, where AGCN =( −49 1 1 ⋅ ⋅ ⋅ 1 1 −49 1 ⋅ ⋅ ⋅ 1 1 1 −49 ⋅ ⋅ ⋅ 1 ⋅ ⋅ ⋅ 1 1 1 ⋅ ⋅ ⋅ −49 ) , ASN =( −49 1 1 ⋅ ⋅ ⋅ 1 1 −1 0 ⋅ ⋅ ⋅ 0 1 0 −1 ⋅ ⋅ ⋅ 0 ⋅ ⋅ ⋅ 1 0 0 ⋅ ⋅ ⋅ −1 ) , ALN =( −2 1 0 ⋅ ⋅ ⋅ 0 1 1 −2 1 ⋅ ⋅ ⋅ 0 0 0 1 −2 ⋅ ⋅ ⋅ 0 0 ⋅ ⋅ ⋅ 1 0 0 ⋅ ⋅ ⋅ 1 −2 ). (17) Lorenz system is a typical benchmark chaotic system, which is a simplified mathematical model first developed by Lorenz in 1963 to describe atmospheric convection. The model is a systemof three ordinary differential equations now known as the Lorenz equations [31]: x = ( −... |

1103 | Consensus problems in networks of agents with switching topology and time-delays - Saber, Murray - 2003 |

668 | Synchronization in chaotic systems - Pecora, Carroll - 1990 |

155 |
Master stability functions for synchronized coupled systems
- Pecora, Carroll
- 1998
(Show Context)
Citation Context ...etworks have received increasing attention from different fields in the past two decades. So far, the dynamics of complex networks has been extensively investigated, in which synchronization is a typical topic which has attracted lots of concern [1–17]. As an interesting phenomenon that enables coherent behavior in networks as a result of coupling, synchronization and the discussion upon its sufficient or necessary condition are fundamental and valuable. Pecora and his colleagues used the so-called Master Stability Function (MSF) approach to determine the synchronous region in coupled systems [18, 19], in which the negativeness of Lyapunov Exponent for master stability equation ensures synchronization. Combining MSF approach with Gershorin disk theory, Chen et al. imposed constraints on the coupling strengths to guarantee stability of the synchronous states in coupled dynamical network [20]. These methods, however, obtain just necessary conditions for synchronization due to the fact that Lyapunov Exponent is employed to judge the stability of system. Zhou et al. and Li and Chen investigated synchronization in general dynamical networks by integrating network models and an adaptive techniq... |

115 |
2002] “Synchronization in small-world systems
- Barahona, Pecora
(Show Context)
Citation Context ...etworks have received increasing attention from different fields in the past two decades. So far, the dynamics of complex networks has been extensively investigated, in which synchronization is a typical topic which has attracted lots of concern [1–17]. As an interesting phenomenon that enables coherent behavior in networks as a result of coupling, synchronization and the discussion upon its sufficient or necessary condition are fundamental and valuable. Pecora and his colleagues used the so-called Master Stability Function (MSF) approach to determine the synchronous region in coupled systems [18, 19], in which the negativeness of Lyapunov Exponent for master stability equation ensures synchronization. Combining MSF approach with Gershorin disk theory, Chen et al. imposed constraints on the coupling strengths to guarantee stability of the synchronous states in coupled dynamical network [20]. These methods, however, obtain just necessary conditions for synchronization due to the fact that Lyapunov Exponent is employed to judge the stability of system. Zhou et al. and Li and Chen investigated synchronization in general dynamical networks by integrating network models and an adaptive techniq... |

89 | Synchronization in an array of linearly coupled dynamical systems - Wu, Chua - 1995 |

89 | 2002] “Synchronization in scale-free dynamical networks: Robustness and fragility - Wang, Chen |

61 |
The double scroll,”
- Matsumoto, Chua, et al.
- 1985
(Show Context)
Citation Context ...ransverse eigenvectors. Therefore, the synchronous solution of dynamical network (1) is asymptotically stable if the following system is stable: ... |

48 | On pinning synchronization of complex dynamical networks - Yu, Chen, et al. - 2009 |

48 | A new chaotic attractor coined,” - Lü, Chen - 2002 |

45 | Adaptive synchronization of an uncertain complex dynamical network
- Zhou, Lu, et al.
- 2006
(Show Context)
Citation Context ...ynchronization. Combining MSF approach with Gershorin disk theory, Chen et al. imposed constraints on the coupling strengths to guarantee stability of the synchronous states in coupled dynamical network [20]. These methods, however, obtain just necessary conditions for synchronization due to the fact that Lyapunov Exponent is employed to judge the stability of system. Zhou et al. and Li and Chen investigated synchronization in general dynamical networks by integrating network models and an adaptive technique and proved that strong enough couplings will synchronize an array of identical cells [11, 12]. To overcome the difficulties caused by too many controllers in large scale complex networks, pinning mechanism is further applied to analyze network synchronization criteria in the works by Zhou et al. and Chen et al. [13, 14]. Research studies on network synchronizationmentioned above focused on sufficient conditions, but all of them are gained by introducing controllers. For general complex dynamical networks in the absence of control, we investigate their sufficient conditions for achieving network synchronization in the current work. Using Lyapunov direct method [21, 22] and matrix theor... |

39 | Synchronization in networks of nonlinear dynamical systems coupled via a directed graph”, - Wu - 2005 |

37 | Synchronization reveals topological scales in complex networks,” - Arenas, Dıaz-Guilera, et al. - 2006 |

34 | Pinning complex networks by a single controller
- Chen, Liu, et al.
- 2007
(Show Context)
Citation Context ...ever, obtain just necessary conditions for synchronization due to the fact that Lyapunov Exponent is employed to judge the stability of system. Zhou et al. and Li and Chen investigated synchronization in general dynamical networks by integrating network models and an adaptive technique and proved that strong enough couplings will synchronize an array of identical cells [11, 12]. To overcome the difficulties caused by too many controllers in large scale complex networks, pinning mechanism is further applied to analyze network synchronization criteria in the works by Zhou et al. and Chen et al. [13, 14]. Research studies on network synchronizationmentioned above focused on sufficient conditions, but all of them are gained by introducing controllers. For general complex dynamical networks in the absence of control, we investigate their sufficient conditions for achieving network synchronization in the current work. Using Lyapunov direct method [21, 22] and matrix theory [23– 27], a criterion for synchronization in generally coupled identical systems is proposed. We conclude that network synchronization will be reached when the coupling strength is larger than a threshold, given that the symme... |

34 | Bridge the gap between the Lorenz system and Chen system - Lu¨, Chen, et al. - 2002 |

32 | Network synchronization, diffusion, and the paradox of heterogeneity,” Phys - Motter, Zhou, et al. - 2005 |

30 | New approach to synchronization analysis of linearly coupled ordinary differential systems - Lu, Chen - 2006 |

27 | Global synchronization of linearly hybrid coupled networks with timevarying delay - Yu, Cao, et al. - 2008 |

26 |
Pinning adaptive synchronization of a general complex dynamical network
- Zhou, Lu, et al.
- 2008
(Show Context)
Citation Context ...ever, obtain just necessary conditions for synchronization due to the fact that Lyapunov Exponent is employed to judge the stability of system. Zhou et al. and Li and Chen investigated synchronization in general dynamical networks by integrating network models and an adaptive technique and proved that strong enough couplings will synchronize an array of identical cells [11, 12]. To overcome the difficulties caused by too many controllers in large scale complex networks, pinning mechanism is further applied to analyze network synchronization criteria in the works by Zhou et al. and Chen et al. [13, 14]. Research studies on network synchronizationmentioned above focused on sufficient conditions, but all of them are gained by introducing controllers. For general complex dynamical networks in the absence of control, we investigate their sufficient conditions for achieving network synchronization in the current work. Using Lyapunov direct method [21, 22] and matrix theory [23– 27], a criterion for synchronization in generally coupled identical systems is proposed. We conclude that network synchronization will be reached when the coupling strength is larger than a threshold, given that the symme... |

13 |
Robust adaptive synchronization of uncertain dynamical networks
- Li, Chen
(Show Context)
Citation Context ...ynchronization. Combining MSF approach with Gershorin disk theory, Chen et al. imposed constraints on the coupling strengths to guarantee stability of the synchronous states in coupled dynamical network [20]. These methods, however, obtain just necessary conditions for synchronization due to the fact that Lyapunov Exponent is employed to judge the stability of system. Zhou et al. and Li and Chen investigated synchronization in general dynamical networks by integrating network models and an adaptive technique and proved that strong enough couplings will synchronize an array of identical cells [11, 12]. To overcome the difficulties caused by too many controllers in large scale complex networks, pinning mechanism is further applied to analyze network synchronization criteria in the works by Zhou et al. and Chen et al. [13, 14]. Research studies on network synchronizationmentioned above focused on sufficient conditions, but all of them are gained by introducing controllers. For general complex dynamical networks in the absence of control, we investigate their sufficient conditions for achieving network synchronization in the current work. Using Lyapunov direct method [21, 22] and matrix theor... |

9 | Synchronization via pinning control on general complex networks,” - Yu, Chen, et al. - 2013 |

8 | Maximum performance at minimum cost in network synchronization - Nishikawa, Motter |

8 |
General stability analysis of synchronized dynamics in coupled systems”,
- Chen, Rangarajan, et al.
- 2003
(Show Context)
Citation Context ...rent behavior in networks as a result of coupling, synchronization and the discussion upon its sufficient or necessary condition are fundamental and valuable. Pecora and his colleagues used the so-called Master Stability Function (MSF) approach to determine the synchronous region in coupled systems [18, 19], in which the negativeness of Lyapunov Exponent for master stability equation ensures synchronization. Combining MSF approach with Gershorin disk theory, Chen et al. imposed constraints on the coupling strengths to guarantee stability of the synchronous states in coupled dynamical network [20]. These methods, however, obtain just necessary conditions for synchronization due to the fact that Lyapunov Exponent is employed to judge the stability of system. Zhou et al. and Li and Chen investigated synchronization in general dynamical networks by integrating network models and an adaptive technique and proved that strong enough couplings will synchronize an array of identical cells [11, 12]. To overcome the difficulties caused by too many controllers in large scale complex networks, pinning mechanism is further applied to analyze network synchronization criteria in the works by Zhou et ... |

7 | Exponential stability of discrete-time genetic regulatory networks with delays,” - Cao, Ren - 2008 |

5 | Nonlinear Systems,
- Hassan
- 1996
(Show Context)
Citation Context ... identical cells [11, 12]. To overcome the difficulties caused by too many controllers in large scale complex networks, pinning mechanism is further applied to analyze network synchronization criteria in the works by Zhou et al. and Chen et al. [13, 14]. Research studies on network synchronizationmentioned above focused on sufficient conditions, but all of them are gained by introducing controllers. For general complex dynamical networks in the absence of control, we investigate their sufficient conditions for achieving network synchronization in the current work. Using Lyapunov direct method [21, 22] and matrix theory [23– 27], a criterion for synchronization in generally coupled identical systems is proposed. We conclude that network synchronization will be reached when the coupling strength is larger than a threshold, given that the symmetric part of the inner-couplingmatrix is positive definite. It is analytically derived in our paper that a network belongs to Type I with respect to synchronized region [28], provided with a positive definite inner-coupling matrix. For discriminating network synchronizability, it is well known that a dilemma is usually encountered in the process 2 Abstr... |

3 | Generalized synchronization between two different complex networks,” - Wu, Wu, et al. - 2012 |

3 |
Introduction to Complex Networks: Models, Structures and Dynamics, Higher Education Press, 1st edition,
- Chen, Wang, et al.
- 2012
(Show Context)
Citation Context ... dynamical networks in the absence of control, we investigate their sufficient conditions for achieving network synchronization in the current work. Using Lyapunov direct method [21, 22] and matrix theory [23– 27], a criterion for synchronization in generally coupled identical systems is proposed. We conclude that network synchronization will be reached when the coupling strength is larger than a threshold, given that the symmetric part of the inner-couplingmatrix is positive definite. It is analytically derived in our paper that a network belongs to Type I with respect to synchronized region [28], provided with a positive definite inner-coupling matrix. For discriminating network synchronizability, it is well known that a dilemma is usually encountered in the process 2 Abstract and Applied Analysis of applyingMSFmethod.That is, we have no prior knowledge of the network type that the synchronous region belongs to. Stemmed from our results, the eigenvalue of the outercoupling matrix nearest 0 can be used for judging synchronizability of a dynamical networkwith positive definite innercoupling matrix. Even though we cannot gain the necessary and sufficient conditions for synchronizing a n... |

1 | Sendina-Nadal et al., “Explosive transitions to synchronization in networks of phase oscillators,” - Leyva, Navas, et al. - 2013 |

1 | Effective dynamics for chaos synchronization in networks with timevarying topology,” - Szmoski, Pereira, et al. - 2013 |

1 |
Nonlinear Control Systems II, Electronic Industry Press, 1st edition,
- Isidori
- 2012
(Show Context)
Citation Context ... identical cells [11, 12]. To overcome the difficulties caused by too many controllers in large scale complex networks, pinning mechanism is further applied to analyze network synchronization criteria in the works by Zhou et al. and Chen et al. [13, 14]. Research studies on network synchronizationmentioned above focused on sufficient conditions, but all of them are gained by introducing controllers. For general complex dynamical networks in the absence of control, we investigate their sufficient conditions for achieving network synchronization in the current work. Using Lyapunov direct method [21, 22] and matrix theory [23– 27], a criterion for synchronization in generally coupled identical systems is proposed. We conclude that network synchronization will be reached when the coupling strength is larger than a threshold, given that the symmetric part of the inner-couplingmatrix is positive definite. It is analytically derived in our paper that a network belongs to Type I with respect to synchronized region [28], provided with a positive definite inner-coupling matrix. For discriminating network synchronizability, it is well known that a dilemma is usually encountered in the process 2 Abstr... |