Kybernetika 55 no. 5, 852-869, 2019

New results on stability of periodic solution for CNNs with proportional delays and $D$ operator

Bo DuDOI: 10.14736/kyb-2019-5-0852

Abstract:

The problems related to periodic solutions of cellular neural networks (CNNs) involving $D$ operator and proportional delays are considered. We shall present Topology degree theory and differential inequality technique for obtaining the existence of periodic solution to the considered neural networks. Furthermore, Laypunov functional method is used for studying global asymptotic stability of periodic solutions to the above system.

Keywords:

stability, periodic solution, $D$ operator, existence

Classification:

34D05, 34D20

paper.pdf

References:

  1. C. Aouiti and I. B. Gharbia at all.: Dynamics of impulsive neutral-type BAM neural networks. J. Franklin Inst. 356 (2019), 2294-2324.   DOI:10.1016/j.jfranklin.2019.01.028
  2. S. Arik: A modified Lyapunov functional with application to stability of neutral-type neural networks with time delays. J. Franklin Inst. 356 (2019), 276-291.   DOI:10.1016/j.jfranklin.2018.11.002
  3. E. Askari, S. Setarehdan, A. Sheikhani A. M. Mohammadi and H. Teshnehlab: Designing a model to detect the brain connections abnormalities in children with autism using 3D-cellular neural networks. J. Integr. Neurosci. 17 (2018), 391-411.   DOI:10.3233/jin-180075
  4. I. Barbalat: Systems d'equations differential d'oscillationsn onlinearities. Rev. Rounmaine Math. Pure Appl. 4 (1959), 267-270.   CrossRef
  5. Z. Cheng and F. Li: Positive periodic solutions for a kind of second-order neutral differential equations with variable coefficient and delay. Mediterr. J. Math. 15 (2018), 134-153.   DOI:10.1007/s00009-018-1184-y
  6. L. Chua and L. Yang: Cellular neural networks: application. IEEE. Trans. Circuits Syst. 35 (1988), 1273-1290.   DOI:10.1109/31.7601
  7. S. Dharani, R. Rakkiyappan and J. Cao: New delay-dependent stability criteria for switched hopfield neural networks of neutral type with additive time-varying delay components. Neurocomputing 151 (2015), 827-834.   DOI:10.1016/j.neucom.2014.10.014
  8. H. Ding, J. Liang and T. Xiao: Existence of almost periodic solutions for SICNNs with time-varying delays. Physics Lett. A 372 (2008), 5411-5416.   DOI:10.1016/j.physleta.2008.06.042
  9. R. Gaines and J. Mawhin: Coincidence Degree and Nonlinear Differential Equations. Springer, Berlin 1977.   DOI:10.1007/bfb0089538
  10. Y. Gang: New results on the stability of fuzzy cellular neural networks with time-varying leakage delays. Neural Computing Appl. 25 (2014), 1709-1715.   DOI:10.1007/s00521-014-1662-5
  11. K. Guan: Global power-rate synchronization of chaotic neural networks with proportional delay via impulsive control. Neurocomputing 283 (2018), 256-265.   DOI:10.1016/j.neucom.2018.01.027
  12. R. Guo, W. Ge and Z. Zhang at all.: Finite time state estimation of complex-valued BAM neutral-type neural networks with time-varying delays. Int. J. Control, Automat. Systems 17 (2019), 3, 801-809.   DOI:10.1007/s12555-018-0542-7
  13. Z. Huang: Almost periodic solutions for fuzzy cellular neural networks with multi-proportional delays. Int. J. Machine Learning Cybernet. 8 (2017), 1323-1331.   DOI:10.1007/s13042-016-0507-1
  14. Y. Li, B. Li, S. Yao and L. Xiong: The global exponential pseudo almost periodic synchronization of quaternion-valued cellular neural networks with time-varying delay. Neurocomputing 303 (2018), 75-87.   DOI:10.1016/j.neucom.2018.04.044
  15. X. Li, L. Huang and H. Zhou: Global stability of cellular neural networks with constant and variable delays. Nonlinear Anal. TMA 53 (2003), 319-333.   DOI:10.1016/s0362-546x(02)00176-1
  16. B. Liu: Finite-time stability of CNNs with neutral proportional delays and time-varying leakage delays. Math. Methods App. Sci. 40 (2017), 167-174.   DOI:10.1002/mma.3976
  17. R. Manivannan, R. Samidurai, J. Cao and A. Alsaedi: New delay-interval-dependent stability criteria for switched hopfield neural networks of neutral type with successive time-varying delay components. Cognit. Neurodyn. 10 (2016), 6, 543-562.   DOI:10.1007/s11571-016-9396-y
  18. N. Ozcan: Stability analysis of Cohen-Grossberg neural networks of neutral-type: Multiple delays case. Neural Networks 113 (2019), 20-27.   DOI:10.1016/j.neunet.2019.01.017
  19. R. Rakkiyappan and P. Balasubramaniam: New global exponential stability results for neutral type neural networks with distributed time delays. Neurocomputing 71 (2008), 1039-1045.   DOI:10.1016/j.neucom.2007.11.002
  20. R. Samidurai, S. Rajavel, R. Sriraman, J. Cao, A. Alsaedi and F. E. Alsaadi: Novel results on stability analysis of neutral-type neural networks with additive time-varying delay components and leakage delay. Int. J. Control Automat. Syst. 15 (2017), 4, 1888-1900.   DOI:10.1007/s12555-016-9483-1
  21. R. Saml et all.: Some generalized global stability criteria for delayed Cohen-Grossberg neural networks of neutral-type. Neural Networks 116 (2019), 198-207.   DOI:10.1016/j.neunet.2019.04.023
  22. K. Shi, H. Zhu, S. Zhong, Y. Zeng and Y. Zhang: New stability analysis for neutral type neural networks with discrete and distributed delays using a multiple integral approach. J. Frankl. Inst. 352 (2015), 1, 155-176.   DOI:10.1016/j.jfranklin.2014.10.005
  23. V. Singh: Improved global robust stability criterion for delayed neural networks. Chao. Solit. Fract. 31 (2007), 224-229.   DOI:10.1016/j.chaos.2005.09.050
  24. V. Singh: On global robust stability of interval Hopfield neural networks with delay. Chao. Solit. Fract. 33 (2007), 1183-1188.   DOI:10.1016/j.chaos.2006.01.121
  25. S. Xiao: Global exponential convergence of HCNNs with neutral type proportional delays and D operator. Neural Process. Lett. 49 (2019), 347-356.   DOI:10.1007/s11063-018-9817-5
  26. Y. Xin and Z. B. Cheng: Neutral operator with variable parameter and third-order neutral differential equation. Adv. Diff. Equ. 273 (2014), 1-18.   DOI:10.1186/1687-1847-2014-273
  27. L. Yao: Global convergence of CNNs with neutral type delays and D operator. Neural Comput. Appl. 29 (2018), 105-109.   DOI:10.1007/s00521-016-2403-8
  28. Y. Yu: Global exponential convergence for a class of neutral functional differential equations with proportional delays. Math. Methods Appl. Sci. 39 (2016), 4520-4525.   DOI:10.1002/mma.3880
  29. Y. Yu: Global exponential convergence for a class of HCNNs with neutral time-proportional delays. Appl. Math. Comput. 285 (2016), 1-7.   DOI:10.1016/j.amc.2016.03.018
  30. X. Zhang and Q. Han: Global asymptotic stability analysis for delayed neural networks using a matrix-based quadratic convex approach. Neural Networks 54 (2014), 57-69.   DOI:10.1016/j.neunet.2014.02.012
  31. X. Zhang and Q. Han: Neuronal state estimation for neural networks with two additive time-varying delay components. IEEE Trans. Cybernetics 47 (2017), 3184-3194.   DOI:10.1109/tcyb.2017.2690676
  32. X. Zhang, Q. Han and L. Wang: Admissible delay upper bounds for global asymptotic stability of neural networks with time-varying delays. IEEE Trans. Neural Networks Learning Systems 29 (2018), 5319-5329.   DOI:10.1109/tnnls.2018.2797279
  33. X. Zhang, Q. Han and Z. Zeng: Hierarchical type stability criteria for delayed neural networks via canonical Bessel-Legendre inequalities. IEEE Trans. Cybernetics 48 (2018), 1660-1671.   DOI:10.1109/tcyb.2017.2776283
  34. H. Zhang and T. Ma et all.: Robust global exponential synchronization of uncertain chaotic delayed neural networks via dual-stage impulsive control. IEEE Trans. Systems Man Cybernet. 40 (2010), 831-844.   DOI:10.1109/tsmcb.2009.2030506
  35. H. Zhang, Z. Qiu and L. Xiong: Stochastic stability criterion of neutral-type neural networks with additive time-varying delay and uncertain semi-Markov jump. Neurocomputing 333 (2019), 395-406.   DOI:10.1016/j.neucom.2018.12.028
  36. M. Zheng and L. Li et all.: Finite-time stability and synchronization of memristor-based fractional-order fuzzy cellular neural networks. Comm. Nonlinear Sci. Numer. Simul. 59 (2018), 272-291.   DOI:10.1016/j.cnsns.2017.11.025
  37. Q. Zhou: Weighted pseudo anti-periodic solutions for cellular neural networks with mixed delays. Asian J. Control 19 (2017), 1557-1563.   DOI:10.1002/asjc.1468
  38. Q. Zhou and J. Shao: Weighted pseudo-anti-periodic SICNNs with mixed delays. Neural Computing Appl. 29 (2018), 272-291.   DOI:10.1007/s00521-016-2582-3