Skip Navigation Links
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain


Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email:

Dynamical Behavior of Cooperative Supportive System Involving Intra-Network Delays in Information Propagation

Journal of Applied Nonlinear Dynamics 11(3) (2022) 719--739 | DOI:10.5890/JAND.2022.09.012

Shirisha G., Venkata Ratnam K.

Department of Mathematics, Birla Institute of Technology and Science-Pilani-Hyderabad Campus, Jawahar

Nagar, Hyderabad-500078, Telangana State, India

Download Full Text PDF



A Cooperative Supportive network system consisting of the main system supported by a subsystem whose dynamics are described in two separate neuronal fields is considered. Time delays are introduced in intra-network propagation (communication) of information in the main system. The qualitative behavior of the solutions of the system is analyzed. The existence and uniqueness of equilibria and their stability is investigated. Several independent criteria on the parameters and the functional relations of the systems are obtained, to establish the global stability of the equilibrium. Estimates on time delay parameter for which the system remains stable are also obtained, using Lyapunov functional. Several numerical examples are provided to illustrate the results. Levenberg-Marquardt Algorithm is utilized for training the network and obtaining simulation results for the examples provided.


  1. [1]  Hicham, G.T. and Lotfi, E. (2017), Comparative study of neural networks algorithms for cloud computing CPU scheduling, International Journal of Electrical and Computer Engineering, 7(6), 3570-3577,
  2. [2]  Patel, J.L. and Goyal, R.K. (2007), Applications of artificial neural networks in medical science, Current Clinical Pharmacology, 2(3), 217-26. http://doi:10.2174/157488407781668811.
  3. [3]  Kosko, B. (1992), Neural Networks and Fuzzy systems : A Dynamical Systems Approach to Machine Intelligence, Prentice- Hall.
  4. [4]  Palm, G. (1987), Computing with neural networks, Science, 235(4793), 1227b-1228b, DOI:10.1126/sci- ence.235.4793.1227b.
  5. [5]  Zhang, Y., Guo, Q., and Wang, J. (2017), Big data analysis using neural networks, Advanced Engineering Sciences, 49(1), 9-18, htpp://doi:10.15961/j.jsuese.2017.01.002.
  6. [6]  Hopfield, J.J. (1982), Neural networks and physical systems with emergent collective computational abilities, Proc. Natl. Acad. Sci. USA, 79, 2554-2558. http://doi:10.1073/pnas.79.8.2554.
  7. [7]  Gopalswamy, K. and He, X.Z. (1994), Delay independent stability in BAM networks, IEEE Trans on Neural Networks, 5, 998-1002, http://doi:10.1109/72.329700.
  8. [8]  Huang, Z., Mohamad, S., and Xia, Y. (2009), Exponential periodic attractor of discrete-time BAM neural networks with transmission delays, Computational Mathematics and Modeling, 20(3), 258-277, DOI:10.1007/s10598-009-9035-0.
  9. [9]  Kosko, B. (1998), Bidirectional associative memories, IEEE Trans, Syst.Man Cybern. SMC, 18, 49-60, http://doi:10.1109/21.87054.
  10. [10]  Simpson, P.K. (1990), Artificial Neural Systems Foundations, Paradigms, Applications and Implementations, Pergamon Press, New York.
  11. [11]  Cai, G., Yao, Q., and Shao, H. (2012), Global Synchronization of weighted cellular neural network with time-varying coupling delays, Communications in Nonlinear Science and Numerical Simulation, 17(10), 3843-3847, http://doi:10.1016/j.cnsns.2012.02.010.
  12. [12]  Chua, L.O. and Yang, L. (1998), Cellular neural networks, IEEE Transactions on Circuits and Systems, 35, 1257-1272. http://doi:10.1109/31.7600.
  13. [13]  Shen, Y., Yu, H., and Jian, J. (2009), Delay-dependent global asymptotic stability for delayed cellular neural networks, Communications in Nonlinear Science and Numerical Simulation, 14(4), 1057-1063,
  14. [14] Salehinejad, H., Sankar, S., Barfett, J., Colak, E., and Valaee, S. (2018), Recent advances in recurrent neural networks, arXiv:1801.01078v3 [cs.NE].
  15. [15]  HZhang, H., Wang, Z., and Liu, D. (2008), Global asymptotic stability of recurrent neural networks with multiple time-varying delays, IEEE Transactions On Neural NetworKS, 19(5), 855-873, http://10.1109/TNN.2007.912319.
  16. [16] Qin, S., Fan, D., Su, P., and Liu, Q. (2014), A simplified recurrent neural network for pseudoconvex optimization subject to linear equality constraints, Communications in Nonlinear Science and Numerical Simulation, 19(4), 789-798, http://10.1016/j.cnsns.2013.08.034.
  17. [17]  Tavanaei, A., Ghodrati, M., Kheradpisheh, S.R., Masquelier, T., and Maida, A. (2019), Deep learning in spiking neural networks, Neural Networks, 111, 47-63,
  18. [18]  Vyacheslav, D. and Dmitry, N. (2018), Recurrent spiking neural network learning based on a competitive maximization of neuronal activity, Frontiers in Neuroinformatics, 12, DOI:10.3389/fninf.2018.00079.
  19. [19]  Ponulak, F. and Kasinski, A. (2011), Introduction to spiking neural networks: information processing, learning and applications, Acta Neurobiol Exp, 71(4), 409-33.
  20. [20]  Kulkarni, S.R. and Rajendran, B. (2018), Spiking neural networks for handwritten digit recognition-supervised learning and network optimization, Neural Networks, 103, 118-127, j.neunet.2018.03.019.
  21. [21]  Rao, V.S.H. and Rao, P.R.S. (2007), Cooperative and supportive neural network, Physics Letters A, 371(1), 101-110, 2007.06.049.
  22. [22]  Belair, J. (1993), Stability in a model of delayed neural networks, Journal of Dynamics and Differential Equations, 5, 607-623,
  23. [23]  Wang, W., Wang, M., Luo, X., Li, L., and Zhao, W. (2018), Passivity of memristive BAM neural networks with probabilistic and mixed time-varying delays, Mathematical Problems in Engineering, 10.1155/2018/5830160.
  24. [24]  Lin, W.J., He, Y., Zhang, C.K., and Wu, M. (2018), Stability analysis of neural networks with time-varying delay: enhanced stability criteria and conservatism comparisons, Communications in Nonlinear Science and Numerical Simulation, 54, 118-135, http://doi:10.1016/j.cnsns.2017.05.021.
  25. [25] Tan, Y. and Tan, M. (2009), Global asymptotical stability of continuous-time delayed neural networks without global lipschitz activation functions, Communications in Nonlinear Science and Numerical Simulation, 14(11), 3715-3722,
  26. [26]  Rao, P.R.S., Ratnam, K.V., and Lalitha, P., (2015), Delay independent stability of cooperative and supportive neural network, Nonlinear Dynamics and System Theory, 15(2), 184-197, http://e-ndst.kiev.ua184.
  27. [27]  Rao, P.R.S., Ratnam, K.V., Lalitha, P., and Satpathi, D.K. (2017), Global dynamics of a cooperative and supportive network with subnetwork deactivation, Nonlinear Dynamics and System Theory, 17(2), 205-216,
  28. [28]  Rao, P.R.S., Ratnam, K.V., and Lalitha, P. (2014), Estimation of inputs for desired output of a cooperative and supportive neural network, IJETCAS, 14-536, Issue 9 Volume 1, ISSN (Online), 2279-0055.
  29. [29]  Lv, C., Xing, Y., Zhang, J., Na, X., Li, Y., Liu, T., Cao, D., and Wang, F.Y. (2018), Levenberg marquardt backpropagation training of multilayer neural networks for state estimation of a safety-critical cyber-physical system, IEEE Transactions on Industrial Informatics, 14(8), 3436-3446. DOI:10.1109/TII.2017.2777460.
  30. [30]  Hagan, M.T. and Menhaj, M.B. (1994), Training feed forward network with the marquardt algorithm, IEEE Trans. on Neural Net, 5(6), 989-993.
  31. [31]  Yuan, Y.X. (2011), Recent advances in numerical methods for nonlinear equations and nonlinear least squares, Numerical Algebra, Control $\&$ Optimization, 1(1), 15-34, DOI:10.3934/naco.2011.1.15.