## Global exponential stability and existence of periodic solution of Hopfield-type neural networks with impulses.(English)Zbl 1123.34303

Summary: We use the continuation theorem of coincidence degree theory and Lyapunov functions to study the existence and global exponential stability of periodic solution for Hopfield-type model of neural network with impulses
\left \{\begin{alignedat}{2}2 \frac{\text dx_i(t)}{\text dt} &=-a_i(t)x_i(t)+\sum^m_{j=1}b_{ij}(t)f_j(x_j(t))+J_i(t), \qquad &&t > 0, \quad t\neq t_k,\\ \Delta x_i(t_k) &= x_i(t^+_k) - x_i(t^-_k)= - \gamma_{ik}x_i(t_k), \qquad &&i=1,\ldots,m,\quad k=1,2,\ldots \end{alignedat} \right .
where $$a_i(t)>0, b_{ij}$$, $$J_i : \mathbb R \to \mathbb R$$, $$i,j=1,\ldots ,m, a_i, b_{ij}$$, $$J_i (i,j=1,\ldots,m)$$ are functions of period $$\omega> 0$$, and there exists a positive integer $$q$$, such that $$t_{k+q}=t_k+\omega, Y_{i(k+q)}=Y_{ik}>0$$. An illustrative example is given to demonstrate the effectiveness of the obtained results.

### MSC:

 34A37 Ordinary differential equations with impulses 34C25 Periodic solutions to ordinary differential equations 37N25 Dynamical systems in biology 34D20 Stability of solutions to ordinary differential equations 82C32 Neural nets applied to problems in time-dependent statistical mechanics
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