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Dynamical Nonlinear Neural Networks with Perturbations Modeling and Global Robust Stability Analysis

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International Journal of Computer Applications
© 2014 by IJCA Journal
Volume 85 - Number 15
Year of Publication: 2014
Authors:
Gamal A. Elnashar
10.5120/14917-3479

Gamal A Elnashar. Article: Dynamical Nonlinear Neural Networks with Perturbations Modeling and Global Robust Stability Analysis. International Journal of Computer Applications 85(15):14-21, January 2014. Full text available. BibTeX

@article{key:article,
	author = {Gamal A. Elnashar},
	title = {Article: Dynamical Nonlinear Neural Networks with Perturbations Modeling and Global Robust Stability Analysis},
	journal = {International Journal of Computer Applications},
	year = {2014},
	volume = {85},
	number = {15},
	pages = {14-21},
	month = {January},
	note = {Full text available}
}

Abstract

This paper is devoted to studying both the global and local stability of dynamical neural networks. In particular, it has focused on nonlinear neural networks with perturbation. Properties relating to asymptotic and exponential stability and instability have been detailed. This paper also looks at the robustness of neural networks to perturbations and examines if the related properties have been preserved. Circumstances for global and local exponential stability of nonlinear neural network dynamics have been studied. We mentioned that the local exponential stability of any equilibrium point of dynamical neural networks is equivalent to the stability of the linearized system around that equilibrium point. From this, some well-known and new sufficient conditions for local exponential stability of neural networks have been obtained. The Lyapunov's procedure has been used to analyze the stability property of nonlinear dynamical systems and many outcomes have been combined and generalized. A kind of Lyapunov's stability of the stable points of Hopfield neural network (HNN) have been proven, which means that if the initial state of the network is close enough to a stable point, then the network state will remain in a small neighborhood of the stable point. These stability results indicate the convergence of the memory process of HNN. The theoretical results are illustrated through a few problem cases for a nonlinear dynamical system with perturbation behavior.

References

  • L. H. Tsoukalas, and R. E. Uhrig, , 1997. fuzzy and neural approaches in engineering, john Wiley &sons, Inc,NY
  • T. Kohonen, 1984. Self-Organization and Associative Memory, New York: Springer-Verlag.
  • Borisyuk A. , Friedman A. , Ermentrout B. and Terman D. 2005. Tutorials in Mathematical Biosciences I: Mathematical Neuroscience, Lecture Notes in Mathematics 1860, Springer- Verlag, Berlin.
  • L. T. Grujic and A. N. Michel, ,1991. "Exponential stability and trajectory bounds of neural networks under structural variations," IEEE Trans. Circuit Syst. , Vol. 38, no. 10. pp. 1182-1192.
  • J. H. Li, A. N. Michel, and W. Porod, ,1988 . "Qualitative analysis and synthesis of a class of neural networks," IEEE Trans. Circuit Syst. , vol. 35, no. 8, pp. 976-985.
  • V. Singh, , 2007. "On global robust stability of interval Hopfield neural networks with delay," Chaos, Solitons and Fractals, vol. 33, no. 4, pp. 1183–1188. [ 7] Z. Wang, Y. Liu, K. Fraser, and X. Liu, , 2006. "Stochastic stability of uncertain Hopfield neural networks with discrete and distributed delays," Physics Letters A, vol. 354, no. 4, pp. 288–297. [ 8] M. W. Hirsch, 1989. "Convergent activation dynamics in continuous timenetwork," Neural Networks, vol. 2, pp. 331-349. [9 ] K. Matsuoka , 1992. "Stability conditions for nonlinear continuous neural networks with with asymmetric connection weights," Neural Networks, vol. 5, pp. 495- 500. [10 ] D. G. Kelly 1990. "Stability in contractive nonlinear neural networks" IEEE Trans. Biomed. Eng. , vol. 3, pp. 231-242.
  • H. yang and T. S. Dillon, 1994. "Exponential stability and oscillation of Hopfield graded response neural network," IEEE trans. Neural Networks, vol. 5, no. 5, pp. 719-729.
  • S. Hui and S. H. ? Zak. August 1993. Robust output feedback stabilization of uncertain dynamic systems with bounded controllers. International Journal of Robust and Nonlinear Control, 3(2):115–132.
  • J. J. Hopfield , , May 1984. Neurons with graded response have collective computational properties like those of two-state neurons. Proceeding of the National Academy of Sciences of the USA, biophysics.
  • R. K. Miller and A. N. Michel, 1982. Ordinary Differential Equations, New York: Academic, [ 15] F. N. Bailey. The application of Lyapunov's second method to interconnected systems. Journal of SIAM on Control, 3(3):443–462, 1966 [16 ] A. N. Michel, J. A. Farrell, and W. , February 1989. Porod. Qualitative analysis of neural networks IEEE Transactions on Circuits and Systems, 36(2):229– 243. [17 ] K. Wang and A. N. Michel, 1994. Robustness and Perturbation Analysis of a Class of Nonlinear Systems with Applications to Neural Networks, IEEE Trans. Circ. Sys, Vol. 41, No. 1, 24–32. [18 ] N. Michel, J. A. Farrell, W. Porod, 1989. Qualitative Analysis of Neural Networks, IEEE Trans. Circ. Sys, Vol. 36, No. 2, 229–243. [19 ] A. N. Michel and R. K. Miller, 1977. Qualitative Analysis of Large Scale Dynamical Systems, New York: Academic, [20 ] Stanislaw H. Zak, 2003. Systems and Control, Oxford University Press, Inc