Global dynamics of Hopfield neural networks involving variable delays. (English) Zbl 0990.34036

Xu, Daoyi; Zhao, Hongyong; Zhu, Hong

Comput. Math. Appl. 42, No. 1-2, 39-45 (2001).

The following system of delayed functional-differential equations used as a model describing the dynamics of Hopfield neural networks is considered \[ \dot u(t)=-Bu(t)+Ag(u(t-\tau(t)))+J,\quad t\geq 0,\qquad u(t)=\phi(t),\quad -\tau\leq t\leq 0, \] with \(u(t)=\text{col}\{u_i(t)\} \in \mathbb{R}^n\), \(B=\text{diag}\{ b_i \}\), \(A=(a_{ij})_{n\times n}\), \(\tau(t)=(\tau_{ij}(t))\), \(g(u)=\text{col}(g_i(u_i))\) with \(g(0)=0\) continuous, \(J=\text{col}\{J_i\}\), \(\varphi=\text{col}\{\varphi_i\}\). Conditions for the uniform boundedness of the solutions are given. Existence and uniqueness of an equilibrium point under general conditions are established. Further, sufficient criteria for the global asymptotic stability are derived using a technique based on properties of nonnegative matrices and matrix inequalities. In particular, sufficient criteria for the global asymptotic stability independent of the delay are obtained.

Reviewer: Ivan Ginchev (Varna)

MSC:

34C11	Growth and boundedness of solutions to ordinary differential equations
92B20	Neural networks for/in biological studies, artificial life and related topics

Keywords:

Hopfield neural networks; delayed system of differential equations; equilibrium points; boundedness; global asymptotic stability

References:

[1]	Baldi, P.; Atiya, A. F., How delays affect neural dynamics and learning, IEEE Trans. Neural Networks, 5, 612-621 (1994)
[2]	Marcus, C. M.; Westervelt, R. M., Stability of analog neural networks with delay, Phys. Rev. A, 39, 347-359 (1989)
[3]	Van Den Driessche, P.; Zou, X. F., Global attractivity in delayed Hopfield neural networks models, SIAM J. Appl. Math., 58, 1878-1890 (1998) · Zbl 0917.34036
[4]	Hou, C. H.; Qian, J. E., Stability analysis for neural dynamics with time-varying delays, IEEE Transactions on Neural Networks, 9, 221-223 (1998)
[5]	Lasalle, J. P., The Stability of Dynamical System (1976), SIAM: SIAM Philadelphia, PA · Zbl 0364.93002
[6]	Horn, R. A.; Johnson, C. R., Matrix Analysis (1991), World Publishing: World Publishing Beijing · Zbl 0729.15001
[7]	Xu, D. Y., Integro-differential equations and delay integral inequalities, Tôhoki Math. J., 44, 365-378 (1992) · Zbl 0760.34059
[8]	Ma, Z. X.; Guo, Q. Y.; Xu, D. Y., Stability and domain of attraction for a class of integral equations, J. of Math., 19, 299-303 (1999), (Chinese)
[9]	Li, S. Y.; Xu, D. Y.; Zhao, H. Y., Stability region of nonlinear integro-differential equations, Appl. Math. Lett., 13, 2, 77-82 (2000)
[10]	Xu, D. Y.; Li, S. Y.; Pu, Z. L.; Guo, Q. Y., Domain of attraction of nonlinear discrete systems with delays, Computers Math. Applic., 38, 5/6, 155-162 (1999) · Zbl 0939.39013
[11]	Berman, A.; Plemmons, R. J., (Nonnegative Matrices in the Mathematical Sciences (1979), Academic Press: Academic Press New York), 133-137