Skip to main content
SpringerLink
Log in

Uniform Approximation of Impulsive Hopfield Cellular Neural Networks by Piecewise Constant Arguments on \([\tau , \infty )\)

  • Published:
Acta Applicandae Mathematicae Aims and scope Submit manuscript

Abstract

In this paper we give a uniform approximation of a CNN-Hopfield type impulsive system by means of an IDEPCA approximating system. As a consequence of the uniform approximation, certain properties like boundedness are inherited. We also consider the analysis of a constant coefficients case. These results are novel in the impulsive differential equations frame. Examples are simulated, illustrating the effectiveness of our results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  1. Abbas, S., Pinto, M., Sepúlveda, D., Tyagi, S.: Approximation of solutions of fractional-order delayed cellular neural network on \([0,\infty )\). Mediterr. J. Math. 12(1), 23 (2017). https://doi.org/10.1007/s00009-016-0826-1. Springer International Publishing

    Article  MathSciNet  MATH  Google Scholar 

  2. Abouagwa, M., Khalaf, A.D., Mustafa, A., Wang, X.: Stochastic Volterra integral equations with jumps and the strong superconvergence of the Euler–Maruyama approximation. J. Comput. Appl. Math. 382, 113071 (2021). https://doi.org/10.1016/j.cam.2020.113071

    Article  MathSciNet  MATH  Google Scholar 

  3. Akhmet, M.: Principles of Discontinuous Dynamical Systems. Springer, New York (2010)

    Book  Google Scholar 

  4. Akhmet, M.: Nonlinear Hybrid Continuous/Discrete-Time Models. Atlantis Press, Amsterdam (2011)

    Book  Google Scholar 

  5. Akhmet, M., Yilmaz, E.: Impulsive Hopfield-type neural network system with piecewise constant argument. Nonlinear Anal., Real World Appl. 11, 2584–2593 (2010). https://doi.org/10.1016/j.nonrwa.2009.09.003

    Article  MathSciNet  MATH  Google Scholar 

  6. Bohner, M., Erhan, I., Georgiev, S.: The Euler method for dynamic equations on time scales. Nonlinear Stud. 27(2), 415–431 (2020)

    MathSciNet  MATH  Google Scholar 

  7. Bozkurt, F.: Modeling a tumor growth with piecewise constant arguments. Discrete Dyn. Nat. Soc. 2013, 841764 (2013). https://doi.org/10.1155/2013/841764

    Article  MathSciNet  MATH  Google Scholar 

  8. Busenberg, S., Cooke, K.: Models of vertically transmitted diseases with sequential-continuous dynamics, pp. 179–187 (1982). https://doi.org/10.1016/B978-0-12-434170-8.50028-5

    Book  MATH  Google Scholar 

  9. Castillo, S., Pinto, M., Torres, R.: Asymptotic formulae for impulsive differential equations with piecewise constant argument of generalized type. Electron. J. Differ. Equ. 2019(40), 40 (2019). https://ejde.math.txstate.edu/Volumes/2019/40/castillo.pdf

    MathSciNet  MATH  Google Scholar 

  10. Chávez, A., Castillo, S., Pinto, M.: Discontinuous almost periodic type functions, almost automorphy of solutions of differential equations with discontinuous delay and applications. Electron. J. Qual. Theory Differ. Equ. 2014(75), 75 (2015). https://doi.org/10.14232/ejqtde.2014.1.75

    Article  MathSciNet  MATH  Google Scholar 

  11. Chiu, K.S.: Existence and global exponential stability of equilibrium for impulsive cellular neural network models with piecewise alternately advanced and retarded argument. Abstr. Appl. Anal. 2013 (2013). https://doi.org/10.1155/2013/196139

  12. Chiu, K.S., Pinto, M.: Periodic solutions of differential equations with a general piecewise constant argument and applications. Electron. J. Qual. Theory Differ. Equ. 2010(46), 46 (2010). https://doi.org/10.14232/ejqtde.2010.1.46

    Article  MathSciNet  MATH  Google Scholar 

  13. Chiu, K.S., Pinto, M., Jeng, J.: Existence and global convergence of periodic solutions in recurrent neural network models with a general piecewise alternately advanced and retarded argument. Acta Appl. Math. 133(1), 133–152 (2014). https://doi.org/10.1007/s10440-013-9863-y

    Article  MathSciNet  MATH  Google Scholar 

  14. Cooke, K., Győri, I.: Numerical approximation of the solutions of delay differential equations on an infinite interval using piecewise constant arguments. Comput. Math. Appl. 28(1–3), 81–92 (1994). https://doi.org/10.1016/0898-1221(94)00095-6

    Article  MathSciNet  MATH  Google Scholar 

  15. Coronel, A., Maulén, C., Pinto, M., Sepúlveda, D.: Dichotomies and asymptotic equivalence in alternately advanced and delayed differential systems. J. Math. Anal. Appl. 450(2), 1434–1458 (2017). https://doi.org/10.1016/j.jmaa.2017.01.087

    Article  MathSciNet  MATH  Google Scholar 

  16. Dai, L.: Nonlinear Dynamics of Piecewise Constant Systems and Implementation of Piecewise Constant Arguments. World Scientific, Singapore (2008)

    Book  Google Scholar 

  17. González, L.: Aproximación de soluciones casi periódicas de ecuaciones diferenciales mediante argumento constante a trozos. Master’s thesis, Facultad de Ciencias, Universidad de Chile (2013)

  18. Hopfield, J.: Neural networks and physical systems with emergent collective computational abilities. Proc. Natl. Acad. Sci. 79(8), 2554–2558 (1982). https://doi.org/10.1073/pnas.79.8.2554

    Article  MathSciNet  MATH  Google Scholar 

  19. Kartal, S.: Mathematical modeling and analysis of tumor-immune system interaction by using Lotka-Volterra predator-prey like model with piecewise constant arguments. Period. Eng. Nat. Sci. 2(1), 7–12 (2014). https://doi.org/10.21533/pen.v2i1.36

    Article  Google Scholar 

  20. Myshkis, A.: On certain problems in the theory of differential equations with deviating arguments. Usp. Mat. Nauk 32(2), 173–202 (1977). https://doi.org/10.1070/RM1977v032n02ABEH001623

    Article  Google Scholar 

  21. Pinto, M.: Asymptotic equivalence of nonlinear and quasilinear differential equations with piecewise constant argument. Math. Comput. Model. 49, 1750–1758 (2009). https://doi.org/10.1016/j.mcm.2008.10.001

    Article  MATH  Google Scholar 

  22. Pinto, M.: Cauchy and green matrices type and stability in alternately advanced and delayed differential systems. J. Differ. Equ. Appl. 17(2), 235–254 (2011). https://doi.org/10.1080/10236198.2010.549003

    Article  MathSciNet  MATH  Google Scholar 

  23. Pinto, M., Robledo, G.: Existence and stability of almost periodic solutions in impulsive neural network models. Appl. Comput. Math. 217(8), 4167–4177 (2010). https://doi.org/10.1016/j.amc.2010.10.033

    Article  MathSciNet  MATH  Google Scholar 

  24. Pinto, M., Sepúlveda, D., Torres, R.: Exponential periodic attractor of an impulsive Hopfield-type neural network system with piecewise constant argument of generalized type. Electron. J. Qual. Theory Differ. Equ. 2018(34), 34 (2018). https://doi.org/10.14232/ejqtde.2018.1.34

    Article  MATH  Google Scholar 

  25. Rojas, R.: Neural Networks – A Systematic Introduction. Springer, Berlin (1996)

    MATH  Google Scholar 

  26. Samoilenko, A., Perestyuk, N.: Impulsive Differential Equations. World Scientific, Singapore (1995)

    Book  Google Scholar 

  27. Shah, S., Wiener, J.: Advanced differential equations with piecewise constant argument deviations. Int. J. Math. Math. Sci. 6, 671–703 (1983). https://doi.org/10.1155/S0161171283000599

    Article  MathSciNet  MATH  Google Scholar 

  28. Torres, R.: Differential equations with piecewise constant argument of generalized type with impulses. Master’s thesis, Facultad de Ciencias, Universidad de Chile (2015)

  29. Veloz, T., Pinto, M.: Existence, computability and stability for solutions of the diffusion equation with general piecewise constant argument. J. Appl. Math. Anal. Appl. 426(1), 330–339 (2014). https://doi.org/10.1016/j.jmaa.2014.10.045

    Article  MathSciNet  MATH  Google Scholar 

  30. Wiener, J.: Generalized Solutions of Functional Differential Equations. World Scientific, Singapore (1993)

    Book  Google Scholar 

  31. Wiener, J., Lakshmikantham, V.: Differential equations with piecewise constant arguments and impulsive equations. Nonlinear Stud. 7(1), 60–69 (2000)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

Samuel Castillo thanks for the support of DIUBB 164408 3/R. Marko Kostić thanks for the support of MNTR project no 451-03-68/2020/14/200156. Manuel Pinto thanks for the support of Fondecyt project 1170466. Ricardo Torres thanks for the support of Fondecyt project 1120709 and sincerely thanks to his colleague, Prof. Bastián Viscarra of Universidad Austral de Chile, for his support providing the plots used in this paper.

Author information

Authors and Affiliations

  1. Instituto de Ciencias Físicas y Matemáticas, Facultad de Ciencias, Universidad Austral de Chile, Campus Isla Teja, Valdivia, Chile

    R. Torres

  2. Departamento de Matemáticas, Facultad de Ciencias, Universidad de Chile, Santiago, Chile

    M. Pinto

  3. Grupo de Investigación en Sistemas Dinámicos y Aplicaciones (GISDA), Departamento de Matemática, Facultad de Ciencias, Universidad del Bío-Bío, Casilla 5-C, Concepción, Chile

    S. Castillo

  4. Faculty of Technical Sciences, University of Novi Sad, Trg D. Obradovića 6, Novi Sad, 21125, Serbia

    M. Kostić

Authors
  1. R. Torres
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Pinto
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. S. Castillo
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. M. Kostić
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to R. Torres.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Torres, R., Pinto, M., Castillo, S. et al. Uniform Approximation of Impulsive Hopfield Cellular Neural Networks by Piecewise Constant Arguments on \([\tau , \infty )\). Acta Appl Math 171, 8 (2021). https://doi.org/10.1007/s10440-020-00373-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10440-020-00373-3

Keywords

Mathematics Subject Classification (2010)

Search

Navigation