Skip to main content
SpringerLink
Log in

Long-Time Behavior of Non-Autonomous FitzHugh–Nagumo Lattice Systems

  • Published:
Qualitative Theory of Dynamical Systems Aims and scope Submit manuscript

Abstract

In Abdallah (J Appl Math 3: 273–288, 2005), Boughoufala and Abdallah (Disc Cont Dyn Ays B), Li and Wang (J Math Anal Appl 325: 141–156, 2007), Vleck and Wang (Physica D 212: 317–336, 2005), Wang (Int J Bifurcation Chaos 17: 1673–1685, 2007) the existence of global attractors for autonomous and non-autonomous deterministic FitzHugh–Nagumo lattice dynamical systems (LDSs) with nonlinear parts of the form \(G_{i}\left( u_{i}\right) \) have been studied. Here the existence of the uniform global attractor for such non-autonomous systems with nonlinear parts of the form \(G_{i}\left( u_{k}\mid k\in I_{iq}\right) \) is carefully investigated, where such nonlinear parts present the main difficulty of this work.

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.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abdallah, A.Y.: Attractors for first order lattice systems with almost periodic nonlinear part. Disc. Cont. Dyn. Sys. B 25, 1241–1255 (2020)

    MATH  Google Scholar 

  2. Abdallah, A.Y.: Long-time behavior for second order lattice dynamical systems. Acta Appl. Math. 106, 47–59 (2009)

    Article  MathSciNet  Google Scholar 

  3. Abdallah, A.Y.: Upper semicontinuity of the attractor for lattice dynamical systems of partly dissipative reaction-diffusion systems. J. Appl. Math. 2005(3), 273–288 (2005)

    Article  MathSciNet  Google Scholar 

  4. Abdallah, A.Y.: Uniform global attractors for first order non-autonomous lattice dynamical systems. Proc. Amer. Math. Soc. 138, 3219–3228 (2010)

    Article  MathSciNet  Google Scholar 

  5. Abdallah, A.Y., Wannan, R.T.: Second order non-autonomous lattice systems and their uniform attractors. Comm. Pure Appl. Anal. 18, 1827–1846 (2019)

    Article  MathSciNet  Google Scholar 

  6. Bates, P.W., Lu, K., Wang, B.: Attractors for lattice dynamical systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 11, 143–153 (2001)

    Article  MathSciNet  Google Scholar 

  7. Boughoufala, A. M., Abdallah, A. Y.: Attractors for FitzHugh-Nagumo lattice systems with almost periodic nonlinear parts, Disc. Cont. Dyn. Ays. -B. https://doi.org/10.3934/dcdsb.2020172

  8. Caraballo, T., Morillas, F., Valero, J.: Random attractors for stochastic lattice systems with non-Lipschitz nonlinearity. J. Diff. Eqs. Appl. 17, 161–184 (2011)

    Article  MathSciNet  Google Scholar 

  9. Caraballo, T., Morillas, F., Valero, J.: Attractors of stochastic lattice dynamical systems with a multiplicative noise and non-Lipschitz nonlinearities. J. Diff. Eqs. 253, 667–693 (2012)

    Article  MathSciNet  Google Scholar 

  10. Chate, H., Courbage, M.: (Eds.), “Lattice Systems,” Phys. D 103 1-4 (1997), 1-612

  11. Chepyzhov, V.V., Vishik, M.I.: Attractors of non-autonomous dynamical systems and their dimension. J. Math. Pures Appl. 73, 279–333 (1994)

    MathSciNet  MATH  Google Scholar 

  12. Chow, S.N.: Lattice Dynamical Systems, Lecture Notes in Mathematics, Dynamical System, pp. 1–102. Springer, Berlin (2003)

    Google Scholar 

  13. Gu, A., Li, Y.: Singleton sets random attractor for stochastic FitzHugh-Nagumo lattice equations driven by fractional Brownian motions. Commun. Nonlinear Sci. Numer. Simul. 19, 3929–3937 (2014)

    Article  MathSciNet  Google Scholar 

  14. Gu, A., Li, Y., Li, J.: Random attractors on lattice of stochastic FitzHugh–Nagumo systems driven by \(\beta \)-stable Lévy noises, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 24, 9 (2014)

    MATH  Google Scholar 

  15. Han, X.: Asymptotic behavior of stochastic partly dissipative lattice systems in weighted spaces. Int. J. Diff. Eqs. 2011, 23 (2011)

    MATH  Google Scholar 

  16. Huang, J.: The random attractor of stochastic FitzHugh-Nagumo equations in an infinite lattice with white noises. Phys. D. 233, 83–94 (2007)

    Article  MathSciNet  Google Scholar 

  17. Huang, J., Han, X., Zhou, S.: Uniform attractors for non-autonomous Klein–Gordon–Schrödinger lattice systems. Appl. Math. Mech. Engl. Ed. 30, 1597–1607 (2009)

    Article  Google Scholar 

  18. Jia, X., Zhao, C., Yang, X.: Global attractor and Kolmogorov entropy of three component reversible Gray-Scott model on infinite lattices. Appl. Math. Comp. 218, 9781–9789 (2012)

    Article  MathSciNet  Google Scholar 

  19. Levitan, B.M., Zhikov, V.V.: Almost Periodic Functions and Differential Equations. Cambridge Univ. Press, Cambridge (1982)

    MATH  Google Scholar 

  20. Li, H., Sun, L.: Upper semicontinuity of attractors for small perturbations of Klein–Gordon–Schrödinger lattice system. Adv. Diff. Equ. 2014(300), 16 (2014)

    MATH  Google Scholar 

  21. Li, X., Wang, D.: Attractors for partly dissipative lattice dynamic systems in weighted spaces. J. Math. Anal. Appl. 325, 141–156 (2007)

    Article  MathSciNet  Google Scholar 

  22. Oliveira, J., Pereira, J., Perla, M.: Attractors for second order periodic lattices with nonlinear damping. J. Diff. Eqs. Appl. 14, 899–921 (2008)

    Article  MathSciNet  Google Scholar 

  23. Pazy, A.: ‘Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences, vol. 44. Springer, New York (1983)

    MATH  Google Scholar 

  24. Temam, R.: Infinite-dimensional Dynamical Systems in Mechanics and Physics. Applied Mathematical Sciences, vol. 68, 2nd edn. Springer, New York (1997)

    Book  Google Scholar 

  25. Van Vleck, E., Wang, B.: Attractors for lattice FitzHugh–Nagumo systems. Phys. D 212, 317–336 (2005)

    Article  MathSciNet  Google Scholar 

  26. Wang, B.: Asymptotic behavior of non-autonomous lattice systems. J. Math. Anal. Appl. 331, 121–136 (2007)

    Article  MathSciNet  Google Scholar 

  27. Wang, B.: Dynamical behavior of the almost-periodic discrete FitzHugh–Nagumo systems. Int. J. Bifurc. Chaos 17, 1673–1685 (2007)

    Article  MathSciNet  Google Scholar 

  28. Wang, Y., Liu, Y., Wang, Z.: Random attractors for partly dissipative stochastic lattice dynamical systems. J. Diff. Eqs. Appl. 14, 799–817 (2008)

    Article  MathSciNet  Google Scholar 

  29. Wang, Z., Zhou, S.: Random attractors for non-autonomous stochastic lattice FitzHugh–Nagumo systems with random coupled coefficients. Taiwanese J. Math. 20, 589–616 (2016)

    Article  MathSciNet  Google Scholar 

  30. Yang, X., Zhao, C., Cao, J.: Dynamics of the discrete coupled nonlinear Schrödinger–Boussinesq equations. Appl. Math. Comp. 219, 8508–8524 (2013)

    Article  Google Scholar 

  31. Zhao, C., Zhou, S.: Compact uniform attractors for dissipative lattice dynamical systems with delays. Disc. Cont. Dyn. Sys 21, 643–663 (2008)

    Article  MathSciNet  Google Scholar 

  32. Zhou, S.: Attractors for first order dissipative lattice dynamical systems. Phys. D 178, 51–61 (2003)

    Article  MathSciNet  Google Scholar 

  33. Zhou, S.: Attractors and approximations for lattice dynamical systems. J. Diff. Eqs. 200, 342–368 (2004)

    Article  MathSciNet  Google Scholar 

  34. Zhou, S., Zhao, M.: Uniform exponential attractor for second order lattice system with quasi-periodic external forces in weighted space, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 24(1), 9 (2014)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, Palestine Technical University - Kadoorie, Tulkarem, Palestine

    Rania T. Wannan

  2. Department of Mathematics, The University of Jordan, Amman, 11942, Jordan

    Ahmed Y. Abdallah

Authors
  1. Rania T. Wannan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ahmed Y. Abdallah
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ahmed Y. Abdallah.

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

Wannan, R.T., Abdallah, A.Y. Long-Time Behavior of Non-Autonomous FitzHugh–Nagumo Lattice Systems. Qual. Theory Dyn. Syst. 19, 78 (2020). https://doi.org/10.1007/s12346-020-00414-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12346-020-00414-0

Keywords

Mathematics Subject Classification

Search

Navigation