Exponential Stability of Periodic Solutions in Inertial Neural Networks with Unbounded Delay

Authors: Yunquan Ke, Chunfang Miao

Abstract:

In this paper, the exponential stability of periodic solutions in inertial neural networks with unbounded delay are investigated. First, using variable substitution the system is transformed to first order differential equation. Second, by the fixed-point theorem and constructing suitable Lyapunov function, some sufficient conditions guaranteeing the existence and exponential stability of periodic solutions of the system are obtained. Finally, two examples are given to illustrate the effectiveness of the results.

Keywords: Inertial neural networks, unbounded delay, fixed-point theorem, Lyapunov function, periodic solutions, exponential stability.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1088194

References:

[1] L.O. Chua, L. Yang, Cellular neural networks: Theory, IEEE Trans. Circuits Syst. 35 (1988) 1257-1272.
[2] L.O. Chua , L. Yang, Cellular neural networks: Applications, IEEE Trans. Circuits Syst. 35 (1988) 1273-1290.
[3] L.O. Chua, CNN: A Paradigm for Complexity, World Scientific, Singapore, 1998.
[4] T. Fariaa, J. Oliveirab, General criteria for asymptotic and exponential stabilities of neural network models with unbounded delays, Applied Mathematics and Computation 217(23)(2011) 9646-9658.
[5] W. Zhaoa, A. Yan, Stability analysis of neural networks with both variable and unbounded delays, Chaos, Solitons , Fractals, 40( 2009) 697-707.
[6] J.Y. Zhang, Y. Suda, T. Iwas, Absolutely exponential stability of a class of neural networks with unbounded delay, Neural Networks, 17(3)( 2004) 391-397.
[7] Z.G. Zeng, J.Wang, X.X.Liao, Global asymptotic stability and global exponential stability of neural networks with unbounded time-varying delays, IEEE Trans Circuits Syst II: Express Briefs, 52(3)(2005) 168-73.
[8] Z.G. Zeng , X.X. Liao, Global stability for neural networks with unbounded time varying delays, Acta Mathematica Scientia, 25A(5)(2005) 621-626.
[9] Y.Zhang, P.A. Heng, K.S.Leung, Convergence analysis of cellular neural networks with unbounded delay, IEEE Trans Circuits Syst I: Fundam Theor Appl, 48(6)( 2001) 680-687.
[10] Q. Zhang, X.P.Wei, J. Xu, Global exponential stability for nonautonomous cellular neural networks with unbounded delays, em Chaos, Solitons Fractals, 39(3)( 2009) 1144-1151.
[11] W.R.Zhao, A.Z.Yan, Stability analysis of neural networks with both variable and unbounded delays, Chaos, Solitons Fractals, 40(2)( 2009) 697-707.
[12] J.Zhang, Absolutely stability analysis in cellular neural networks with variable delays and unbounded delay, Comput Math. Appl. 47(2004) 183- 194.
[13] Z. Zeng, J. Wang, X. Liao, Global asymptotic stability and global exponential stability of neural networks with unbounded time-varying delays, IEEE Transactions on Circuits Systems II , 52 (3)( 2005) 168- 173.
[14] J. Zhang, Y. Suda, T. Iwasa, Absolutely exponential stability of a class of neural networks with unbounded delay, Neural Networks, 17( 2004) 391-397.
[15] K.L. Badcock, R.M. Westervelt, Dynamics of simple electronic neural networks, Physica D. 28( 1987) 305-316.
[16] J. Tani, Proposal of chaotic steepest descent method for neural networks and analysis of their dynamics, Electron. Comm. Japan, 75(4)(1992) 62- 70.
[17] J. Tani, M. Fujita, Coupling of memory search and mental rotation by a nonequilibrium dynamics neural network, IEICE Trans. Fund. Electron., Commun. Comput. Sci. E75-A (5)(1992) 578-585.
[18] J. Tani, Model-based learning for mobile robot navigation from the dynamical systems perspective, IEEE Trans. Systems Man Cybernet 26 (3)( 1996) 421-436.
[19] C.G.Li, G.R.Chen, X.F. Liao, J.B. Yu, Hopf bifurcation and chaos in a single inertial neuron model with time delay, Eur.Phys. J. B, 41( 2004) 337-343.
[20] Q. Liu, X.F. Liao, G.Y. Wang, Y. Wu, Research for Hopf bifurcation of an inertial two-neuron system with time delay, in: IEEE Proceeding GRC, (2006) 420-423.
[21] Q. Liu, X.F. Liao, D.G. Yang, S.T. Guo, The research for Hopf bifurcation in a single inertial neuron model with external forcing, in: IEEE Proceeding GRC, ( 2007) 528-533.
[22] Q. Liu, X. Liao, Y. Liu, S. Zhou , S. Guo, Dynamics of an inertial two-neuron system with time delay, Nonlinear Dyn. 58( 2009) 573-609.
[23] W. Diek, W.C. Wheeler, Schieve.Stability and chaos in an inertial twoneuron system, Physica D, 105 (1997) 267-284.
[24] Q. Liu , X.F. Liao, S.T. Guo, Y. Wu, Stability of bifurcating periodic solutions for a single delayed inertial neuron model under periodic excitation, Nonlinear Analysis: Real World Applications, 10(2009) 2384- 2395.
[25] Y.Q. Ke, C.F. Miao, Stability analysis of BAM neural networks with inertial term and time delay, WSEAS. Transactions on System , 10(12)( 2011) 425-438.
[26] Y.Q. Ke, C.F. Miao, Stability and existence of periodic solutions in inertial BAM neural networks with time delay, Neural computing and Applications, (2012, DOI: 10.1007/s00521-012-1037-8, Online First, 15 July 2012).