Li, Yongkun; Zhao, Lili; Liu, Ping Existence and exponential stability of periodic solution of high-order Hopfield neural network with delays on time scales. (English) Zbl 1187.34129 Discrete Dyn. Nat. Soc. 2009, Article ID 573534, 18 p. (2009). A class of higher-order Hopfield neural networks with delays on time scales is studied. Sufficient conditions for existence and exponential stability of the periodic solutions of continuous and discrete systems are obtained by the authors. An example is provided as well. Reviewer: Angela Slavova (Sofia) Cited in 9 Documents MSC: 34N05 Dynamic equations on time scales or measure chains 92B20 Neural networks for/in biological studies, artificial life and related topics 34K20 Stability theory of functional-differential equations 34K13 Periodic solutions to functional-differential equations Keywords:Hopfield neural network; delay; periodic solutions; exponential stability PDFBibTeX XMLCite \textit{Y. Li} et al., Discrete Dyn. Nat. Soc. 2009, Article ID 573534, 18 p. (2009; Zbl 1187.34129) Full Text: DOI EuDML References: [1] Z. Wang, J. Fang, and X. Liu, “Global stability of stochastic high-order neural networks with discrete and distributed delays,” Chaos, Solitons & Fractals, vol. 36, no. 2, pp. 388-396, 2008. · Zbl 1141.93416 · doi:10.1016/j.chaos.2006.06.063 [2] B. Xiao and H. Meng, “Existence and exponential stability of positive almost periodic solutions for high-order Hopfield neural networks,” Applied Mathematical Modelling, vol. 33, no. 1, pp. 532-542, 2009. · Zbl 1167.34385 · doi:10.1016/j.apm.2007.11.027 [3] X. Yi, J. Shao, Y. Yu, and B. Xiao, “New convergence behavior of high-order Hopfield neural networks with time-varying coefficients,” Journal of Computational and Applied Mathematics, vol. 219, no. 1, pp. 216-222, 2008. · Zbl 1161.34050 · doi:10.1016/j.cam.2007.07.011 [4] B. Xu, Q. Wang, and X. Liao, “Stability analysis of high-order Hopfield type neural networks with uncertainty,” Neurocomputing, vol. 71, no. 4-6, pp. 508-512, 2008. · Zbl 05718489 · doi:10.1016/j.neucom.2007.03.014 [5] X.-Y. Lou and B.-T. Cui, “Novel global stability criteria for high-order Hopfield-type neural networks with time-varying delays,” Journal of Mathematical Analysis and Applications, vol. 330, no. 1, pp. 144-158, 2007. · Zbl 1111.68104 · doi:10.1016/j.jmaa.2006.07.058 [6] B. Xu, X. Liu, and X. Liao, “Global exponential stability of high order Hopfield type neural networks,” Applied Mathematics and Computation, vol. 174, no. 1, pp. 98-116, 2006. · Zbl 1099.34051 · doi:10.1016/j.amc.2005.03.020 [7] B. Xiao and H. Meng, “Existence and exponential stability of positive almost periodic solutions for high-order Hopfield neural networks,” Applied Mathematical Modelling, vol. 33, no. 1, pp. 532-542, 2009. · Zbl 1167.34385 · doi:10.1016/j.apm.2007.11.027 [8] B. Liu and L. Huang, “Existence and exponential stability of periodic solutions for a class of Cohen-Grossberg neural networks with time-varying delays,” Chaos, Solitons & Fractals, vol. 32, no. 2, pp. 617-627, 2007. · Zbl 1145.34049 · doi:10.1016/j.chaos.2005.11.009 [9] F. Zhang and Y. Li, “Almost periodic solutions for higher-order Hopfield neural networks without bounded activation functions,” Electronic Journal of Differential Equations, vol. 2007, no. 99, pp. 1-10, 2007. · Zbl 1138.34346 [10] M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser, Boston, Mass, USA, 2001. · Zbl 0978.39001 [11] M. Bohner and A. Peterson, Eds., Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, Mass, USA, 2003. · Zbl 1025.34001 [12] S. Hilger, “Analysis on measure chains-a unified approach to continuous and discrete calculus,” Results in Mathematics, vol. 18, no. 1-2, pp. 18-56, 1990. · Zbl 0722.39001 · doi:10.1007/BF03323153 [13] M. Bohner, M. Fan, and J. Zhang, “Existence of periodic solutions in predator-prey and competition dynamic systems,” Nonlinear Analysis: Real World Applications, vol. 7, no. 5, pp. 1193-1204, 2006. · Zbl 1104.92057 · doi:10.1016/j.nonrwa.2005.11.002 [14] F.-H. Wong, C.-C. Yeh, S.-L. Yu, and C.-H. Hong, “Young’s inequality and related results on time scales,” Applied Mathematics Letters, vol. 18, no. 9, pp. 983-988, 2005. · Zbl 1080.26025 · doi:10.1016/j.aml.2004.06.028 [15] J. Mawhin, Topological Degree Methods in Nonlinear Boundary Value Problems, vol. 40 of CBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence, RI, USA, 1979. · Zbl 0414.34025 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.