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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 $$ \left \{\alignedat2 \frac{\text dx_i(t)}{\text dt} &=-a_i(t)x_i(t)+\sum^m_{j=1}b_{ij}(t)f_j(x_j(t))+J_i(t), \qquad &&t > 0, \quad t\ne t_k,\\ \Delta x_i(t_k) &= x_i(t^+_k) - x_i(t^-_k)= - \gamma_{ik}x_i(t_k), \qquad &&i=1,\ldots,m,\quad k=1,2,\ldots \endalignedat \right .$$ where $a_i(t)>0, b_{ij}$, $J_i : \Bbb R \to \Bbb R$, $i,j=1,\ldots ,m, a_i, b_{ij}$, $J_i (i,j=1,\ldots,m)$ 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.

34A37Differential equations with impulses
34C25Periodic solutions of ODE
37N25Dynamical systems in biology
34D20Stability of ODE
82C32Neural nets (statistical mechanics)
Full Text: DOI
[1] Hopfield, J. J.: Proc. natl. Acad. sci. USA. 81, 3088 (1984)
[2] Gopalsamy, K.; He, X. Z.: Physica D. 76, 344 (1994)
[3] Gopalsamy, K.; Issic, K. C.; Leung, I. K. C.; Liu, P.: Appl. math. Comput.. 94, 171 (1998)
[4] Gulick, D.: Encounters with chaos. (1992) · Zbl 1253.37001
[5] Hertz, J.; Krogh, A.; Palmer, R. G.: Introduction to the theory of neural computation. (1991)
[6] Guan, Z.; Chen, G.; Qin, Y.: IEEE trans. Neural networks. 2, 534 (2000)
[7] Guan, Z. H.; James, L.; Chen, G.: Neural networks. 13, 63 (2000)
[8] Akca, H.; Alassar, R.; Covachev, V.; Covacheva, Z.; Al-Zahrani, E.: J. math. Anal. appl.. 290, 436 (2004)
[9] Mohamad, S.; Gopalsamy, K.: Math. comput. Simulation. 53, 1 (2000)
[10] S. Mohamad, Continuous and discrete dynamical systems with applications, Ph.D. thesis, The Flinders University of South Australia, April 17, 2000
[11] Gaines, R. E.; Mawhin, J. L.: Coincidence degree and nonlinear differential equations. (1977) · Zbl 0339.47031
[12] Bainov, D.; Simeonov, P. S.: Systems with impulse effect: stability, theory and applications. (1989) · Zbl 0676.34035
[13] Lakshmikantham, V.; Bainov, D. D.; Simeonov, P. S.: Theory of impulsive differential equations. Series in modern applied mathematics 6 (1989) · Zbl 0719.34002