Existence and exponential stability of periodic solution for stochastic Hopfield neural networks on time scales

In this paper, by using the contraction mapping theorem and Gronwall?s Inequality on time scales, we establish some sufficient conditions on the existence and exponential stability of periodic solutions for a class of stochastic neural networks on time scales. Moreover, we present an example to illustrate the feasibility of our results and to show that the continuous-time neural network and its discrete-time analogue have the same dynamical behaviors.

[1]  S. Hilger.  Analysis on Measure Chains — A Unified Approach to Continuous and Discrete Calculus , 1990 .

[2]  Legang Lin,et al.  Method of upper and lower solutions for fractional differential equations , 2012 .

[3]  R. Rakkiyappan,et al.  Global asymptotic stability of stochastic recurrent neural networks with multiple discrete delays and unbounded distributed delays , 2008, Appl. Math. Comput..

[4]  Xiaofei He,et al.  On inequalities of Lyapunov for linear Hamiltonian systems on time scales , 2011 .

[5]  Jinde Cao.  New results concerning exponential stability and periodic solutions of delayed cellular neural networks , 2003 .

[6]  Agacik Zafer,et al.  The stability of linear periodic Hamiltonian systems on time scales , 2013, Appl. Math. Lett..

[7]  Zaitang Huang,et al.  Existence and exponential stability of almost periodic solution for stochastic cellular neural networks with delay , 2009 .

[8]  Shiji Song,et al.  Research on synchronization of chaotic delayed neural networks with stochastic perturbation using impulsive control method , 2014, Commun. Nonlinear Sci. Numer. Simul..

[9]  Martin Bohner,et al.  The Kalman filter for linear systems on time scales , 2013 .

[10]  Xiaodi Li,et al.  Existence and global exponential stability of periodic solution for delayed neural networks with impulsive and stochastic effects , 2010, Neurocomputing.

[11]  Qiaoluan Li,et al.  Positive solutions for higher order nonlinear neutral dynamic equations on time scales , 2009 .

[12]  A. Peterson,et al.  Dynamic Equations on Time Scales: An Introduction with Applications , 2001 .

[13]  Shukai Duan,et al.  Exponential Stability of Discrete-Time Delayed Hopfield Neural Networks with Stochastic Perturbations and Impulses , 2012 .

[14]  Shouming Zhong,et al.  Robust stability analysis for discrete-time stochastic neural networks systems with time-varying delays , 2009, Appl. Math. Comput..

[15]  V. Lupulescu,et al.  RANDOM DYNAMICAL SYSTEMS ON TIME SCALES , 2012 .

[16]  Lihong Huang,et al.  Existence and exponential stability of almost periodic solutions for Hopfield neural networks with delays , 2005, Neurocomputing.

[17]  Yongkun Li,et al.  Weighted pseudo almost automorphic functions with applications to abstract dynamic equations on time scales , 2013 .

[18]  Yongkun Li,et al.  Sobolev’s spaces on time scales and its applications to a class of second order Hamiltonian systems on time scales , 2010 .

[19]  Weiwei Su,et al.  Global robust stability criteria of stochastic Cohen–Grossberg neural networks with discrete and distributed time-varying delays , 2009 .

[20]  Guowei Yang,et al.  Exponential stability of impulsive stochastic fuzzy reaction-diffusion Cohen-Grossberg neural networks with mixed delays , 2012, Neurocomputing.

[21]  Ping Liu,et al.  Existence and Exponential Stability of Periodic Solution of High-Order Hopfield Neural Network with Delays on Time Scales , 2009 .

[22]  Simon Haykin,et al.  Neural networks , 1994 .

[23]  Fen Wang,et al.  Mean square exponential stability and periodic solutions of stochastic interval neural networks with mixed time delays , 2010, Neurocomputing.

[24]  V. Volpert,et al.  On the solvability in the sense of sequences for some non-Fredholm operators , 2014 .

[25]  Xiaofeng Liao,et al.  Existence and stability of periodic solution in impulsive Hopfield neural networks with finite distributed delays , 2005 .

[26]  Everestus Eze.  Existence and Stability of Periodic Solutions... , 2017 .

[27]  Youssef N. Raffoul,et al.  Periodic solutions for a neutral nonlinear dynamical equation on a time scale , 2006 .

[28]  Daoyi Xu,et al.  Existence and global p-exponential stability of periodic solution for impulsive stochastic neural networks with delays , 2012 .

[29]  S. Sanyal,et al.  The quadratic variation of Brownian motion on a time scale , 2012 .

[30]  Hongjun Xiang,et al.  Existence and global exponential stability of periodic solution for delayed high-order Hopfield-type neural networks , 2006 .

[31]  David E. Grow,et al.  Brownian Motion Indexed by a Time Scale , 2011 .

[32]  O. Stanzhytskyi,et al.  STOCHASTIC DYNAMIC EQUATIONS ON GENERAL TIME SCALES , 2013 .

[33]  Zhigang Liu,et al.  Existence and global exponential stability of periodic solution of cellular neural networks with time-varying delays☆ , 2004 .

[34]  Lili Zhao,et al.  Existence and Exponential Stability of Anti-periodic Solutions of High-order Hopfield Neural Networks with Delays on Time Scales , 2011 .

[35]  Fuke Wu,et al.  Positive solution and its asymptotic behaviour of stochastic functional Kolmogorov-type system , 2010 .

[36]  Xuerong Mao,et al.  Stability of stochastic delay neural networks , 2001, J. Frankl. Inst..

[37]  Yongkun Li,et al.  Existence of positive periodic solutions for functional differential equations with impulse effects on time scales , 2009 .

[38]  A. Peterson,et al.  Dynamic Equations on Time Scales , 2001 .

[39]  Nan Ding,et al.  Dynamic analysis of stochastic Cohen-Grossberg neural networks with time delays , 2006, Appl. Math. Comput..

[40]  Tiedong Ma,et al.  Synchronization of multi-agent stochastic impulsive perturbed chaotic delayed neural networks with switching topology , 2015, Neurocomputing.

[41]  Yongkun Li,et al.  Global Exponential Stability of Fuzzy Interval Delayed Neural Networks with impulses on Time Scales , 2009, Int. J. Neural Syst..

[42]  Jinde Cao,et al.  Mean square exponential stability for discrete-time stochastic switched static neural networks with randomly occurring nonlinearities and stochastic delay , 2014, Neurocomputing.

[43]  Huaiping Zhu,et al.  Existence and roughness of exponential dichotomies of linear dynamic equations on time scales , 2010, Comput. Math. Appl..