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Stability preserving NSFD scheme for a cooperative and supportive network

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Abstract

A continuous dynamical system of a Cooperative Supportive Neural Network is discretized using Non-Standard Finite Difference scheme. Results in the direction of the existence of equilibria, sufficient conditions for local and global stability of equilibrium are established for the discrete form of the network. Results are compared with those of the continuous model. Theoretical numerical examples with simulations are provided to understand the results. Our study establishes that the Non-Standard Finite Difference scheme chosen here preserves the properties of the continuous system for any step size. Also, the input-output relations of difference equation model are tested using a recently developed technique. Our study is the first of its kind in this area of neural networks.

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References

  1. Dang QA, Hoang MT (2020) Positivity and global stability preserving NSFD schemes for a mixing propagation model of computer viruses. J Comput Appl Math 374:112753. https://doi.org/10.1016/j.cam.2020.112753

    Article  MathSciNet  MATH  Google Scholar 

  2. Anguelov R, Lubuma JM-S, Shillor M (2009) Dynamically consistent non standard finite difference schemes for continuous dynamical systems. Discrete Contin Dyn Syst 2009(Supplement):34–43

  3. Meksianis N, Nursanti A, Supriatna Asep K (2018) Comparison of the differential transformation method and non standard finite difference scheme for solving plant disease mathematical model. Commun Biomath Sci 1(2):110. https://doi.org/10.5614/cbms.2018.1.2.4

    Article  Google Scholar 

  4. Mickens RE (2002) Nonstandard finite difference schemes for differential equations. J Differ Equ Appl 8(9):823–847. https://doi.org/10.1080/1023619021000000807

    Article  MathSciNet  MATH  Google Scholar 

  5. Sree Hari Rao V, Raja Sekhara Rao P (2018) Time varying stimulations in simple neural networks and convergence to desired outputs. Differ Equ Dyn Syst 26:81–104. https://doi.org/10.1007/s12591-016-0312-z

    Article  MathSciNet  MATH  Google Scholar 

  6. Dawit B ParchaKalyani (2016) Application of non standard finite difference method on logistic differential equation and comparison with standard difference methods. IOSR J Math 12(3:104–112

    Google Scholar 

  7. Dang Quang A, Tuan HM (2020) Numerical dynamics of nonstandard finite difference schemes for a computer virus propagation model. Int J Dyn Control 8:772–778. https://doi.org/10.1007/s40435-019-00604-y

    Article  MathSciNet  Google Scholar 

  8. Raja Sekhara Rao P, Venkata Ratnam K, Lalitha P (2015) Delay independent stability of cooperative and supportive neural network. Nonlinear Dyn Syst Theory 15(2):184–197

    MathSciNet  MATH  Google Scholar 

  9. Raja Sekhara Rao P, Venkata Ratnam K, Lalitha P, Kumar SD (2017) Global dynamics of a cooperative and supportive network with subnetwork deactivation. Nonlinear Dyn Syst Theory 17(2):205–216

    MathSciNet  MATH  Google Scholar 

  10. Sree Hari Rao V, Raja Sekhara Rao P (2007) Cooperative and supportive neural network. Phys Lett A 371(1):101–110. https://doi.org/10.1016/j.physleta.2007.06.049

    Article  MATH  Google Scholar 

  11. Shabbir MS, Din Q, Safeer M et al (2019) A dynamically consistent nonstandard finite difference scheme for a predator-prey model. Adv Differ Equ. https://doi.org/10.1186/s13662-019-2319-6

    Article  MathSciNet  MATH  Google Scholar 

  12. Xiaolan Z, Qi W, Jiechang W (2018) Numerical dynamics of nonstandard finite difference method for nonlinear delay differential equation. Int J Bifurc Chaos 28(11):1850133. https://doi.org/10.1142/S021812741850133X

    Article  MathSciNet  MATH  Google Scholar 

  13. Anguelov R, Lubuma JM-S, Shillor M (2009) Dynamically consistent non standard finite difference schemes for continuous dynamical systems. Discrete Contin Dyn Syst 2009(Supplement):34–43

    MATH  Google Scholar 

  14. Hamza Alaa E, El-Sayed MA (1998) Stability problem of some nonlinear difference equations. Int J Math Math Sci 21(2):331–340. https://doi.org/10.1155/S0161171298000453

    Article  MathSciNet  MATH  Google Scholar 

  15. Danumjaya P, Merina D (2019) Stability preserving non-standard finite difference schemes for diabetes with tuberculosis infectious model. Lett Biomath 6(2):1–18. https://doi.org/10.1080/23737867.2019.1618743

    Article  MathSciNet  Google Scholar 

  16. Dang Quang A, Tuan HM (2020) Numerical dynamics of nonstandard finite difference schemes for a computer virus propagation model. Int J Dyn Control 8:772–778. https://doi.org/10.1007/s40435-019-00604-y

  17. Mickens Ronald E (2005) Advances in the applications of nonstandard finite difference schemes. World Sci Singap. https://doi.org/10.1142/5884

    Article  MATH  Google Scholar 

  18. Raja Sekhara Rao P, Venkata Ratnam K, Lalitha P, Kumar SD (2017) Global dynamics of a cooperative and supportive network with subnetwork deactivation. Nonlinear Dyn Syst Theory 17(2):205–216

  19. Celik T (2016) Nonstandard finite difference method for ODEs for initial-value problems. New Trends Math Sci 4(4):27–32

    Article  Google Scholar 

  20. Raja Sekhara Rao P, Venkata Ratnam K, Sita Rama Murthy M (2018) Stability preserving non standard finite difference schemes for certain biological models. Int J Dyn Control 6:1496–1504. https://doi.org/10.1007/s40435-018-0410-6

  21. Koroglu C (2020) Exact and nonstandard finite difference schemes for the generalized KdV-Burgers equation. Adv Differ Equ 2020:134. https://doi.org/10.1186/s13662-020-02584-2

    Article  MathSciNet  Google Scholar 

  22. Obayomi A, Adetolaju S (2013) Construction of new non-standard finite difference schemes for the solution of free un-damped harmonic oscillator equation. J Math Theory Model 3:91–97

    Google Scholar 

  23. Mickens Ronald E (1993) Nonstandard finite difference models of differential equations. World Sci Singap. https://doi.org/10.1142/2081

    Article  MATH  Google Scholar 

  24. Mickens RE (2003) A nonstandard finite-difference scheme for the lotka-volterra system. Appl Numer Math, 45:309–314. ISSN 0168–9274. https://doi.org/10.1016/S0168-9274(02)00223-4

  25. Momani S, Rqayiq AA, Baleanu D (2012) A nonstandard finite difference scheme for two-sided space-fractional partial differential equations. Int J Bifurc Chaos 22(04):1250079

    Article  MathSciNet  Google Scholar 

  26. Platonov AV (2019) On the stability of non-stationary nonlinear difference systems with switching. Int J Dyn Control 7:1242–1251. https://doi.org/10.1007/s40435-019-00581-2

    Article  MathSciNet  Google Scholar 

  27. Raja Sekhara Rao P, Venkata Ratnam K, Sita Rama Murthy M (2019) Predictive dynamics of infectious diseases-a new technique. World J Model Simul 15(2):128–139

    Google Scholar 

  28. Letellier C, Elaydi S, Aguirre LA, Alaoui A (2004) Difference equations versus differential equations, a possible equivalence for the Rossler system? Phys D Nonlinear Phenom 195:29–49. https://doi.org/10.1016/j.physd.2004.02.007

    Article  MATH  Google Scholar 

  29. Mehdizadeh Khalsaraei M, Khodadosti F (2014) Nonstandard finite difference schemes for differential equations. Sahand Commun Math Anal 01(2):47–54

    MATH  Google Scholar 

  30. Raja Sekhara Rao P, Venkata Ratnam K, Sita Rama Murthy M (2018) Stability preserving non standard finite difference schemes for certain biological models. Int J Dyn Control 6:1496–1504. https://doi.org/10.1007/s40435-018-0410-6

    Article  MathSciNet  Google Scholar 

  31. Zhang L, Wang L, Xiaohua D (2014) Exact finite difference scheme and nonstandard finite difference scheme for burgers and burgers-fisher equations. J Appl Math 2:1–12. https://doi.org/10.1155/2014/597926

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors wish to thank the anonymous referees for their useful comments which led to a better presentation of the work and made it more self-contained.

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Authors and Affiliations

  1. Department of Mathematics, Birla Institute of Technology and Science-Pilani-Hyderabad Campus, Jawahar Nagar, Hyderabad, 500078, India

    K. Venkata Ratnam & G. Shirisha

  2. Govt.Polytechnic, Kalidindi, 521344, Krishna District, A.P., India

    P. Raja Sekhara Rao

Authors
  1. K. Venkata Ratnam
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  2. P. Raja Sekhara Rao
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  3. G. Shirisha
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Correspondence to G. Shirisha.

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Ratnam, K.V., Rao, P.R.S. & Shirisha, G. Stability preserving NSFD scheme for a cooperative and supportive network. Int. J. Dynam. Control 9, 1576–1588 (2021). https://doi.org/10.1007/s40435-021-00777-5

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  • DOI: https://doi.org/10.1007/s40435-021-00777-5

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