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Global dynamics of Hopfield neural networks involving variable delays. (English) Zbl 0990.34036
The following system of delayed functional-differential equations used as a model describing the dynamics of Hopfield neural networks is considered $$\dot u(t)=-Bu(t)+Ag(u(t-\tau(t)))+J,\quad t\ge 0,\qquad u(t)=\phi(t),\quad -\tau\le t\le 0,$$ with $u(t)=\text{col}\{u_i(t)\} \in \bbfR^n$, $B=\text{diag}\{ b_i \}$, $A=(a_{ij})_{n\times n}$, $\tau(t)=(\tau_{ij}(t))$, $g(u)=\text{col}(g_i(u_i))$ 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.

MSC:
34C11Qualitative theory of solutions of ODE: growth, boundedness
92B20General theory of neural networks (mathematical biology)
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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] Den Driessche, P. Van; 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) · Zbl 0364.93002
[6] Horn, R. A.; Johnson, C. R.: Matrix analysis. (1991) · Zbl 0729.15001
[7] Xu, D. Y.: Integro-differential equations and delay integral inequalities. TĂ´hoki 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) · Zbl 0952.45004
[9] Li, S. Y.; Xu, D. Y.; Zhao, H. Y.: Stability region of nonlinear integro-differential equations. Appl. math. Lett. 13, No. 2, 77-82 (2000) · Zbl 0974.45006
[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, No. 5/6, 155-162 (1999) · Zbl 0939.39013
[11] Berman, A.; Plemmons, R. J.: Nonnegative matrices in the mathematical sciences. 133-137 (1979) · Zbl 0484.15016