[ Log In | Register]
! Skip Navigation Links
Skip Navigation Links
Image of Discontinuity, Nonlinearity and Complexity
ISSN:2164-6376 (print)
ISSN:2164-6414 (online)
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania

Email: dumitru.baleanu@gmail.com

Almost Periodic Solutions of Recurrently Structured Impulsive Neural Networks

Discontinuity, Nonlinearity, and Complexity 11(3) (2022) 373--385 | DOI:10.5890/DNC.2022.09.001

Marat Akhmet, Gulbahar Erim

Department of Mathematics, Middle East Technical University, 06800 Ankara, Turkey

Download Full Text PDF

 

Abstract

The model under discussion is an elaborated recurrent impulsive neural network. This is the first time in literature that the impacts are structured completely as the original neural network, such that physical sense of impacts has been explained. Moreover, the impact part comprises all types of impacts in neural networks, which were traditionally studied in conservative models. In the research, neuron membranes with negative as well as positive capacitance, are considered newly as parts of the neural networks. This was not studied before. The system is analyzed in matrix form to facilitate more transparent presentation. The existence and uniqueness of asymptotically stable discontinuous almost periodic solutions are investigated. An example with simulations is provided to illustrate the results.

Acknowledgments

Marat Akhmet is supported by 2247-A National Leading Researchers Program of TUBITAK, Turkey, N 120C138.

References

  1. [1]  Yang, W., Yu, W., Cao, J., Alsaadi, F.E., and Hayat, T. (2017), Almost automorphic solution for neutral type high-order Hopfield BAM neural networks with time-varying leakage delays on time scales, Neurocomputing, 267.
  2. [2]  Li, Y.K. and Yang, L. (2014), Almost automorphic solution for neutral type high-order Hopfield neural networks with delays in leakage terms on time scales, Applied Mathematics and Computation, 242, 679-693.
  3. [3]  Akhmet, M., Fen, M.O., and Kirane, M. (2016), Almost periodic solutions of retarded SICNNs with functional response on piecewise constant argument, Neural Computing and Applications, 27(8), 2483-2495.
  4. [4]  Chen, W.H., Luo, S., and Zheng, W.X. (2016), Impulsive stabilization of periodic solutions of recurrent neural networks with discrete and distributed delays, 2016 IEEE International Symposium on Circuits and Systems (ISCAS).
  5. [5]  Chua, L.O. and Yang, L. (1988), Cellular neural networks: applications, IEEE Transactions on Circuits and Systems, 35(10), 1273-1290.
  6. [6]  Chua, L.O. and Yang, L. (1988), Cellular neural networks: theory, IEEE Transactions on Circuits and Systems, 35(10), 1257-1272.
  7. [7]  Wang, L. and Zou, X.F. (2002), Exponential stability of Cohen-Grossberg neural networks, Neural Networks, 15(3), 415-422.
  8. [8]  Xiang, H.J. and Cao, J.W. (2009), Exponential stability of periodic solution to Cohen-Grossberg-type BAM networks with time-varying delays, Neurocomputing, 72, 1702-1711.
  9. [9]  Zhang, Z.Q. and Zheng, T. (2018), Global asymptotic stability of periodic solutions for delayed complex-valued Cohen-Grossberg neural networks by combining coincidence degree theory with LMI method, Neurocomputing, 289.
  10. [10]  Akhmet, M. and Y\i lmaz, E. (2012), Global exponential stability of neural networks with non-smooth and impact activations, Neural networks: the official journal of the International Neural Network Society, 34, 18-27.
  11. [11]  Oliveira, J.J. (2011), Global stability of a Cohen-Grossberg neural network with both time-varying and continuous distributed delays, Nonlinear Analysis: Real World Applications, 12(5), 2861-2870.
  12. [12]  Akhmet, M. and Y\i lmaz, E. (2013), Neural Networks with Discontinuous/Impact Activations, Springer Publishing Company, Incorporated.
  13. [13]  Liu, Q.M. and Xu, R. (2011), Periodic solutions of high-order Cohen-Grossberg neural networks with distributed delays, Communications in Nonlinear Science and Numerical Simulation 16, 2887-2893.
  14. [14]  Akhmet, M. and Fen, M.O. (2013), Shunting inhibitory cellular neural networks with chaotic external inputs, Chaos (Woodbury, N.Y.), 23, 023112.
  15. [15]  Meng, Y.M., Huang, L.H., Guo, Z.Y., and Hu, Q.W. (2010), Stability analysis of Cohen-Grossberg neural networks with discontinuous neuron activations, Applied Mathematical Modelling, 34, 358-365.
  16. [16]  Cao, J.D. and Wang, J. (2005), Global exponential stability and periodicity of recurrent neural networks with time delays, IEEE Transactions on Circuits and Systems I: Regular Papers, 52(5), 920-931.
  17. [17]  Ak\c ca, H., Alassar, R., Covachev, V., Covacheva, Z., and Al-Zahrani, E. (2004), Continuous-time additive Hopfield-type neural networks with impulses, Journal of Mathematical Analysis and Applications, 290(2), 436-451.
  18. [18]  Aouiti, C., M'hamdi, M.S., Cherif, F., and Alimi, A.M. (2018), Impulsive generalized high-order recurrent neural networks with mixed delays: Stability and periodicity, Neurocomputing, 321, 296-307.
  19. [19]  Zhou, T.J., Wang, M., and Li, C. (2015), Almost periodic solution for multidirectional associative memory neural network with distributed delays, Mathematics and Computers in Simulation, 107, 52-60.
  20. [20]  Xia, Y.H., Huang, Z.K., and Han, M.A. (2008), Existence and globally exponential stability of equilibrium for BAM neural networks with impulses, Chaos, Solitons $\&$ Fractals, 37(2), 588-597.
  21. [21]  Wang, J.F. and Huang, L.H. (2012), Almost periodicity for a class of delayed Cohen-Grossberg neural networks with discontinuous activations, Chaos, Solitons $\&$ Fractals, 45(9), 1157-1170.
  22. [22]  Li, Y.K. and Fan, X.L. (2009), Existence and globally exponential stability of almost periodic solution for Cohen-Grossberg BAM neural networks with variable coefficients, Applied Mathematical Modelling, 33(4), 2114-2120.
  23. [23]  Qin, S.T., Xue, X.P., and Wang, P. (2013), Global exponential stability of almost periodic solution of delayed neural networks with discontinuous activations, Inf. Sci., 220, 367-378.
  24. [24]  Liu, Y.G. and You, Z.S. (2007), Multi-stability and almost periodic solutions of a class of recurrent neural networks, Chaos, Solitons $\&$ Fractals, 33, 554-563.
  25. [25]  Zhao, W.R. and Zhang, H.S. (2009), New results of almost periodic solutions for cellular neural networks with mixed delays, Chaos Solitons $\&$ Fractals, 40, 831-838.
  26. [26]  Xia, Y.H., Cao, J.D., and Lin, M.R. (2007), New results on the existence and uniqueness of almost periodic solution for bam neural networks with continuously distributed delays, Chaos, Solitons $\&$ Fractals, 31, 928-936.
  27. [27]  Akhmet, M. (2011), Nonlinear Hybrid Continuous/Discrete-Time Models, Atlantis Press.
  28. [28]  Samoilenko, A.M. and Perestyuk, N.A. (1995), Impulsive Differential Equations, World Scientific.
  29. [29]  Lakshmikantham, V., Bainov, D.D., and Simeonov, P.S. (1989), Theory of Impulsive Differential Equations, World Scientific.
  30. [30]  Akhmet, M. and Kashkynbayev, A. (2013), Non-autonomous bifurcation in impulsive systems, Electronic Journal of Qualitative Theory of Differential Equations, 74, 1-23.
  31. [31]  Kashkynbayev, A., Cao, J.D., and Damiyev, Z. (2019), Stability analysis for periodic solutions of fuzzy shunting inhibitory CNNs with delays, Advances in Difference Equations, 384.
  32. [32]  Akhmet, M. and Kashkynbayev, A. (2013), Nonautonomous transcritical and Pitchfork bifurcations in impulsive systems, Miskolc Mathematical Notes, 14(3), 737-748.
  33. [33]  Zhang, H.Y. and Xia, Y.H. (2008), Existence and exponential stability of almost periodic solution for Hopfield-type neural networks with impulse, Chaos Solitons $\&$ Fractals, 37, 1076-1082.
  34. [34]  Stamov, G.Tr. and Stamova, I.M. (2007), Almost periodic solutions for impulsive neural networks with delay, Applied Mathematical Modelling, 31(7), 1263-1270.
  35. [35]  Allegretto, W., Papini, D., and Forti, M. (2010), Common asymptotic behavior of solutions and almost periodicity for discontinuous, delayed, and impulsive neural networks, IEEE Trans. Neural Networks, 21(7), 1110-1125.
  36. [36]  Xia, Y.H., Jinde Cao, J.D., and Huang, Z.K. (2007), Existence and exponential stability of almost periodic solution for shunting inhibitory cellular neural networks with impulses, Chaos, Solitons $\&$ Fractals, 34(5), 1599-1607.
  37. [37]  Pinto, M. and Robledo, G. (2010), Existence and stability of almost periodic solutions in impulsive neural network models, Applied Mathematics and Computation, 217(8), 4167-4177.
  38. [38]  Brahmi, H., Ammar, B., Alimi, A.M., and Ch{{e}}rif, F. (2016), Pseudo almost periodic solutions of impulsive recurrent neural networks with mixed delays, IEEE-IJCNN, 464-470.
  39. [39]  Li, Y.K., Zhang, T.W., and Xing, Z.W. (2010), The existence of nonzero almost periodic solution for Cohen-Grossberg neural networks with continuously distributed delays and impulses, Neurocomputing, 73(16-18), 3105-3113.
  40. [40]  Akhmet, M. (2008), Almost periodic solutions of differential equations with piecewise constant argument of generalized type, Nonlinear Analysis: Hybrid Systems, 2(2), 456-467.
  41. [41]  Liu, Y.G., Huang, Z.X., and Chen, L.P. (2012), Almost periodic solution of impulsive Hopfield neural networks with finite distributed delays, Neural Computing and Applications, 21(5), 821-831.
  42. [42]  Shi, P.L. and Dong, L.Z. (2010), Existence and exponential stability of anti-periodic solutions of Hopfield neural networks with impulses, Applied Mathematics and Computation, 216(2), 623-630.
  43. [43]  Khan, A. and Salahuddin, S. (2015), Negative capacitance in ferroelectric materials and implications for steep transistors, 2015 IEEE SOI-3D-Subthreshold Microelectronics Technology Unified Conference (S3S), 1-3.
  44. [44]  Khan, A.I., Chatterjee, K., Duarte, J.P., Lu, Z., Sachid, A., Khandelwal, S., Ramesh, R., Hu, C., and Salahuddin, S. (2016), Negative capacitance in short-channel finfets externally connected to an epitaxial ferroelectric capacitor, IEEE Electron Device Letters, 37(1), 111-114.
  45. [45]  Si, M., Su, C.J., Jiang, C., Conrad, N.J., Zhou, H., Maize, K.D., Qiu, G., Wu, C.T., Shakouri, A., Alam, M.A., and Ye, P.D. (2018), Steep-slope hysteresis-free negative capacitance MoS$ _2 $ transistors, Nature Nanotechnology.
  46. [46]  Gopalsamy, K. and He, X.Z. (1994), Stability in asymmetric Hopfield nets with transmission delays, Physica D: Nonlinear Phenomena, 76, 344-358.
  47. [47]  Akhmet, M. (2010), Principles of Discontinuous Dynamical Systems, Springer-Verlag New York.
  48. [48]  Akhmet, M. (2020), Almost Periodicity, Chaos, and Asymptotic Equivalence, Nonlinear Systems and Complexity 27, Springer.
  49. [49]  Fink, A.M. (2006), Almost Periodic Differential Equations, Lecture Notes in Mathematics, Springer, Berlin Heidelberg.

Copyright © 2011-2024 L & H Scientific Publishing. All rights reserved. Wildcard SSL Certificates