Generalized Mutual Synchronization between Two Controlled Interdependent Networks

This paper mainly focuses on the generalized mutual synchronization between two controlled interdependent networks. First, we propose the general model of controlled interdependent networks and with time-varying internetwork delays coupling. Then, by constructing Lyapunov functions and utilizing adaptive control technique, some sufficient conditions are established to ensure that the mutual synchronization errors between the state variables of networks and can asymptotically converge to zero. Finally, two numerical examples are given to illustrate the effectiveness of the theoretical results and to explore potential application in future smart grid. The simulation results also show how interdependent topologies and internetwork coupling delays influence the mutual synchronizability, which help to design interdependent networks with optimal mutual synchronizability.

[1]  Hongtao Lu,et al.  Projective lag synchronization of the general complex dynamical networks with distinct nodes , 2012 .

[2]  Jiao Licheng,et al.  Adaptive Synchronization between Two Different Complex Networks with Time-Varying Delay Coupling , 2009 .

[3]  Xiaoping Xue,et al.  Outer synchronization of coupled networks using arbitrary coupling strength. , 2010, Chaos.

[4]  Hui Zhao,et al.  Fuzzy Sliding Mode Variable Structure Control of Chaotic Power System with Uncertainty , 2011 .

[5]  Harry Eugene Stanley,et al.  Cascade of failures in coupled network systems with multiple support-dependent relations , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[6]  H. Stanley,et al.  Networks formed from interdependent networks , 2011, Nature Physics.

[7]  Jürgen Kurths,et al.  Synchronization between two coupled complex networks. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[8]  Gang Feng,et al.  Controlling complex dynamical networks with coupling delays to a desired orbit , 2006 .

[9]  Yun Shang,et al.  Generalized synchronization of complex networks. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[10]  S. Strogatz Exploring complex networks , 2001, Nature.

[11]  Tao Zhou,et al.  Relations between average distance, heterogeneity and network synchronizability , 2006 .

[12]  Jürgen Kurths,et al.  Generalized synchronization between two different complex networks , 2012 .

[13]  Jin-Liang Wang,et al.  Local and global exponential output synchronization of complex delayed dynamical networks , 2012 .

[14]  Wenxu Wang,et al.  Enhanced synchronizability by structural perturbations. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[15]  E. Kuffel,et al.  High voltage engineering , 2006, 2006 Eleventh International Middle East Power Systems Conference.

[16]  Jinde Cao,et al.  Outer synchronization between two nonidentical networks with circumstance noise , 2010 .

[17]  Hongtao Lu,et al.  Outer synchronization of uncertain general complex delayed networks with adaptive coupling , 2012, Neurocomputing.

[18]  Harry Eugene Stanley,et al.  Catastrophic cascade of failures in interdependent networks , 2009, Nature.

[19]  Jian Xiao,et al.  H∞ control synthesis of switched discrete‐time fuzzy systems via hybrid approach , 2013 .

[20]  V. Latora,et al.  Complex networks: Structure and dynamics , 2006 .

[21]  J. Kurths,et al.  Single impulsive controller for globally exponential synchronization of dynamical networks , 2013 .

[22]  Xiang Li,et al.  Pinning a complex dynamical network to its equilibrium , 2004, IEEE Trans. Circuits Syst. I Regul. Pap..

[23]  Xiao Fan Wang,et al.  Synchronization in scale-free dynamical networks: robustness and fragility , 2001, cond-mat/0105014.

[24]  Song Zheng,et al.  Adaptive synchronization of two nonlinearly coupled complex dynamical networks with delayed coupling , 2012 .

[25]  W. Zheng,et al.  Generalized outer synchronization between complex dynamical networks. , 2009, Chaos.

[26]  S. Buldyrev,et al.  Interdependent networks with identical degrees of mutually dependent nodes. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[27]  Mohammad Bagher Menhaj,et al.  Fuzzy Complex Dynamical Networks and Its Synchronization , 2013, IEEE Transactions on Cybernetics.

[28]  Jian Xiao,et al.  Finite-time stability and stabilisation for switched linear systems , 2013, Int. J. Syst. Sci..

[29]  Chen Laijun Overviews and Prospects of the Cyber Security of Smart Grid from the View of Complex Network Theory , 2011 .

[30]  Jian Xiao,et al.  Stability analysis and control synthesis of switched impulsive systems , 2012 .

[31]  Beom Jun Kim,et al.  Synchronization in interdependent networks. , 2011, Chaos.

[32]  Jian Xiao,et al.  $\mathcal{H}_{\infty}$ Control Synthesis for Short-Time Markovian Jump Continuous-Time Linear Systems , 2013, Circuits Syst. Signal Process..

[33]  Ju H. Park,et al.  On synchronization criterion for coupled discrete-time neural networks with interval time-varying delays , 2013, Neurocomputing.

[34]  Jurgen Kurths,et al.  Synchronization in complex networks , 2008, 0805.2976.

[35]  Liang Chen,et al.  Adaptive synchronization between two complex networks with nonidentical topological structures , 2008 .

[36]  Duncan J. Watts,et al.  Collective dynamics of ‘small-world’ networks , 1998, Nature.

[37]  Albert-László Barabási,et al.  Scale-Free Networks: A Decade and Beyond , 2009, Science.

[38]  Alessandro Vespignani,et al.  Complex networks: The fragility of interdependency , 2010, Nature.

[39]  Hongxing Yao,et al.  Generalized Synchronization between Two Complex Dynamical Networks with Time-Varying Delay and Nonlinear Coupling , 2011 .

[40]  Xinzhi Liu,et al.  Robust impulsive synchronization of uncertain dynamical networks , 2005, IEEE Transactions on Circuits and Systems I: Regular Papers.

[41]  Mark E. J. Newman,et al.  The Structure and Function of Complex Networks , 2003, SIAM Rev..

[42]  E A Leicht,et al.  Suppressing cascades of load in interdependent networks , 2011, Proceedings of the National Academy of Sciences.

[43]  H. Stanley,et al.  Robustness of network of networks under targeted attack. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[44]  Jian Xiao,et al.  H∞ control for switched fuzzy systems via dynamic output feedback: Hybrid and switched approaches , 2013, Commun. Nonlinear Sci. Numer. Simul..

[45]  I. Grosu,et al.  Collection of master–slave synchronized chaotic systems , 2004 .