Asymptotic stability of delayed fractional-order neural networks with impulsive effects

This paper has investigated the existence, uniqueness and the global asymptotic stability of equilibrium point for delayed fractional-order neural networks with impulsive effects. A lemma has been given based on Riemann-Liouville operator, which plays an important role in the stability analysis. Some sufficient conditions are derived to ensure the existence, uniqueness and the asymptotic stability of the fractional-order neural networks. An illustrative example is given to show the effectiveness of the obtained results by using a new numerical method of fractional-order differential equations.

[1]  Stephen Michael Zemyan.  The Classical Theory of Integral Equations: A Concise Treatment , 2012 .

[2]  Jinde Cao,et al.  Exponential lag synchronization of a class of chaotic delayed neural networks with impulsive effects , 2007 .

[3]  Lan Xu,et al.  Adaptive impulsive synchronization for a class of fractional-order chaotic and hyperchaotic systems , 2014 .

[4]  Liping Chen,et al.  Synchronization of a Class of Fractional-Order Chaotic Neural Networks , 2013, Entropy.

[5]  Arvet Pedas,et al.  Numerical solution of nonlinear fractional differential equations by spline collocation methods , 2014, J. Comput. Appl. Math..

[6]  O. Agrawal,et al.  Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering , 2007 .

[7]  Haijun Jiang,et al.  Projective synchronization for fractional neural networks , 2014, Neural Networks.

[8]  Shangbo Zhou,et al.  Chaos control and synchronization in a fractional neuron network system , 2008 .

[9]  Mohamed Ali Hajji,et al.  An Efficient Series Solution for Fractional Differential Equations , 2014 .

[10]  Weihua Deng,et al.  Remarks on fractional derivatives , 2007, Appl. Math. Comput..

[11]  A. Luo,et al.  Fractional Dynamics and Control , 2011 .

[12]  Jinde Cao,et al.  Stability and periodicity in delayed cellular neural networks with impulsive effects , 2007 .

[13]  R. Magin.  Fractional Calculus in Bioengineering , 2006 .

[14]  Wei Zhang,et al.  Fractional Calculus Approach to Dynamic Problems of Viscoelastic Materials , 1999 .

[15]  Ligang Wu,et al.  New global exponential stability criteria for interval-delayed neural networks , 2011 .

[16]  K. Diethelm.  The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type , 2010 .

[17]  Ivo Petras,et al.  Fractional-Order Nonlinear Systems , 2011 .

[18]  Ivanka M. Stamova,et al.  Stability analysis of impulsive functional systems of fractional order , 2014, Commun. Nonlinear Sci. Numer. Simul..

[19]  吴然超,et al.  Finite-Time Stability of Fractional-Order Neural Networks with Delay＊ , 2013 .

[20]  Daoyi Xu,et al.  Impulsive effects on stability of Cohen-Grossberg neural networks with variable delays , 2006, Appl. Math. Comput..

[21]  Huijun Gao,et al.  Novel Robust Stability Criteria for Stochastic Hopfield Neural Networks With Time Delays , 2009, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics).

[22]  Weiping Li,et al.  Applied Nonlinear Control , 1991 .

[23]  F. Zhu.  Full-order and reduced-order observer-based synchronization for chaotic systems with unknown disturbances and parameters☆ , 2008 .

[24]  Chunhai Kou,et al.  Stability Analysis for Fractional Differential Equations and Their Applications in the Models of HIV-1 Infection , 2009 .

[25]  Choon Ki Ahn,et al.  New sets of criteria for exponential L2-L∞ stability of Takagi-Sugeno fuzzy systems combined with hopfield neural networks , 2013 .

[26]  Stephen M. Zemyan,et al.  The Classical Theory of Integral Equations , 2012 .

[27]  Degang Chen,et al.  Asymptotic stability theorem for autonomous systems , 1993 .

[28]  Daoyi Xu,et al.  Stability Analysis of Delay Neural Networks With Impulsive Effects , 2005, IEEE Trans. Circuits Syst. II Express Briefs.

[29]  Tiedong Ma,et al.  Dynamic analysis of a class of fractional-order neural networks with delay , 2013, Neurocomputing.

[30]  P. Arena,et al.  Chaotic behavior in noninteger-order cellular neural networks , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[31]  Li Sheng,et al.  Exponential synchronization of a class of neural networks with mixed time-varying delays and impulsive effects , 2008, Neurocomputing.

[32]  Richard L. Magin,et al.  Fractional calculus models of complex dynamics in biological tissues , 2010, Comput. Math. Appl..

[33]  N. Ford,et al.  Analysis of Fractional Differential Equations , 2002 .

[34]  J. Klafter,et al.  The random walk's guide to anomalous diffusion: a fractional dynamics approach , 2000 .

[35]  I. Petráš.  Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation , 2011 .

[36]  O. Agrawal,et al.  Advances in Fractional Calculus , 2007 .

[37]  Nasser Hassan Sweilam,et al.  Two computational algorithms for the numerical solution for system of fractional differential equations , 2015 .

[38]  Yuanqing Xia,et al.  Stochastic Barbalat's Lemma and Its Applications , 2012, IEEE Transactions on Automatic Control.

[39]  N. Laskin.  Fractional market dynamics , 2000 .