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Zbl 0990.34036
Xu, Daoyi; Zhao, Hongyong; Zhu, Hong
Global dynamics of Hopfield neural networks involving variable delays. (English)
[J] Comput. Math. Appl. 42, No.1-2, 39-45 (2001). ISSN 0898-1221

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.
[Ivan Ginchev (Varna)]

MSC 2000:: *34C11 Qualitative theory of solutions of ODE: Growth, etc.
92B20 General theory of neural networks

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

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