Adaptive pinning synchronization in fractional-order uncertain complex dynamical networks with delay

Based on the stability theory of fractional-order systems, synchronization of general fractional-order uncertain complex networks with delay is investigated in this paper. By the inequality of the fractional derivative and the comparison principle of the linear fractional equation with delay, synchronization of complex networks with delay is realized under adaptive control. Some sufficient criteria ensuring local asymptotical synchronization under adaptive control and global asymptotical synchronization under adaptive pinning control are derived, respectively. Finally, numerical simulations are presented to demonstrate the validity and feasibility of the proposed synchronization criteria.

[1]  C. K. Michael Tse,et al.  Adaptive Feedback Synchronization of a General Complex Dynamical Network With Delayed Nodes , 2008, IEEE Transactions on Circuits and Systems II: Express Briefs.

[2]  Xinzhi Liu,et al.  Fractional order Lyapunov stability theorem and its applications in synchronization of complex dynamical networks , 2014, Commun. Nonlinear Sci. Numer. Simul..

[3]  N. Laskin.  Fractional quantum mechanics and Lévy path integrals , 1999, hep-ph/9910419.

[4]  Lada A. Adamic,et al.  Internet: Growth dynamics of the World-Wide Web , 1999, Nature.

[5]  Tianbo Wang,et al.  Proportional‐delay adaptive control for global synchronization of complex networks with time‐delay and switching outer‐coupling matrices , 2013 .

[6]  Diyi Chen,et al.  Synchronization and anti-synchronization of fractional dynamical networks , 2015 .

[7]  Junzhi Yu,et al.  Global stability analysis of fractional-order Hopfield neural networks with time delay , 2015, Neurocomputing.

[8]  Wuneng Zhou,et al.  Exponential Synchronization Analysis and Control for Discrete-Time Uncertain Delay Complex Networks with Stochastic Effects , 2012 .

[9]  Manuel A. Duarte-Mermoud,et al.  Lyapunov functions for fractional order systems , 2014, Commun. Nonlinear Sci. Numer. Simul..

[10]  Wei Xing Zheng,et al.  Generalized synchronization of complex dynamical networks via impulsive control. , 2009, Chaos.

[11]  Jiang-Wen Xiao,et al.  Robust synchronization of complex switched networks with parametric uncertainties and two types of delays , 2013 .

[12]  Elif Demirci,et al.  A fractional order SEIR model with vertical transmission , 2011, Math. Comput. Model..

[13]  I. Podlubny.  Fractional differential equations , 1998 .

[14]  Yixian Yang,et al.  Exponentially asymptotical synchronization in uncertain complex dynamical networks with time delay , 2010 .

[15]  R. Guimerà,et al.  Functional cartography of complex metabolic networks , 2005, Nature.

[16]  Jürgen Kurths,et al.  Synchronization of complex dynamical networks with time delays , 2006 .

[17]  R. Solé,et al.  Evolving protein interaction networks through gene duplication. , 2003, Journal of Theoretical Biology.

[18]  R. Hilfer.  Applications Of Fractional Calculus In Physics , 2000 .

[19]  You-Quan Li,et al.  Dynamical magnetoelectric effects in the distorted spiral multiferroic magnets , 2013 .

[20]  Wansheng Tang,et al.  Guaranteed cost control of synchronisation for uncertain complex delayed networks , 2012, Int. J. Syst. Sci..

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

[22]  Yi Chai,et al.  Adaptive pinning synchronization in fractional-order complex dynamical networks , 2012 .

[23]  Chunguang Li,et al.  Synchronization in general complex dynamical networks with coupling delays , 2004 .

[24]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations , 2006 .

[25]  Tiedong Ma,et al.  Cluster synchronization in fractional-order complex dynamical networks , 2012 .

[26]  Alessandro Vespignani,et al.  Epidemic spreading in scale-free networks. , 2000, Physical review letters.

[27]  Tao Fan,et al.  Robust decentralized adaptive synchronization of general complex networks with coupling delayed and uncertainties , 2014, Complex..

[28]  Xinchu Fu,et al.  Cluster synchronization in community networks with nonidentical nodes. , 2009, Chaos.

[29]  Xiaoji Liu,et al.  An Analog of the Adjugate Matrix for the Outer Inverse (2) , 2012 .

[30]  Guanrong Chen,et al.  A time-varying complex dynamical network model and its controlled synchronization criteria , 2004, IEEE Trans. Autom. Control..

[31]  Jinde Cao,et al.  On Pinning Synchronization of Directed and Undirected Complex Dynamical Networks , 2010, IEEE Transactions on Circuits and Systems I: Regular Papers.

[32]  Zheng-Guang Wu,et al.  Robust $\mathcal{H}_{\infty}$ decentralized dynamic control for synchronization of a complex dynamical network with randomly occurring uncertainties , 2012 .

[33]  Junan Lu,et al.  Structure identification of uncertain general complex dynamical networks with time delay , 2009, Autom..

[34]  Ping He,et al.  Finite-time mixed outer synchronization of complex networks with coupling time-varying delay. , 2012, Chaos.

[35]  Tianping Chen,et al.  Synchronization of coupled connected neural networks with delays , 2004, IEEE Transactions on Circuits and Systems I: Regular Papers.

[36]  Jinde Cao,et al.  Adaptive Q-S (lag, anticipated, and complete) time-varying synchronization and parameters identification of uncertain delayed neural networks. , 2006, Chaos.

[37]  Elena Grigorenko,et al.  Chaotic dynamics of the fractional Lorenz system. , 2003, Physical review letters.

[38]  Jing-Wen Yi,et al.  Adaptive pinning cluster synchronization of fractional-order complex dynamical networks , 2014, Appl. Math. Comput..

[39]  Zhong Chen,et al.  Impulsive synchronization of a general nonlinear coupled complex network , 2011 .

[40]  W. K. Wong,et al.  Robust synchronization of fractional-order complex dynamical networks with parametric uncertainties , 2012 .

[41]  Lixiang Li,et al.  Exponentially asymptotic synchronization of uncertain complex time-delay dynamical networks , 2013 .

[42]  Chunguang Li,et al.  Chaos in the fractional order Chen system and its control , 2004 .

[43]  Matthew R Whiteway,et al.  Local synchronization in complex networks of coupled oscillators. , 2011, Chaos.

[44]  T. Vicsek,et al.  Uncovering the overlapping community structure of complex networks in nature and society , 2005, Nature.

[45]  Jinde Cao,et al.  Adaptive synchronization and lag synchronization of uncertain dynamical system with time delay based on parameter identification , 2007 .

[46]  E. Ahmed,et al.  On fractional order differential equations model for nonlocal epidemics , 2007, Physica A: Statistical Mechanics and its Applications.

[47]  Jinhu Lü,et al.  Stability analysis of linear fractional differential system with multiple time delays , 2007 .

[48]  P. Butzer,et al.  AN INTRODUCTION TO FRACTIONAL CALCULUS , 2000 .

[49]  G. Dimirovski,et al.  Adaptive Exponential Synchronization of Uncertain Complex Dynamical Networks with Delay Coupling , 2008 .