Qualitative Analysis of Discrete Dynamical Systems Involving k-Mersenne Numbers

• Amira Khelifa ^[1] ; Yacine Halim ^[3] ; Turki D. Alharbi ^[4] ; J. G. Al-Juaid ^[2]
  1. [1] University Dr Yahia Fares Medea

     University Dr Yahia Fares Medea

     Argelia

     
  2. [2] Taif University

     Taif University

     Arabia Saudí

     
  3. [3] Mohamed Seddik ben Yahia University, Abdelhafid Boussouf University Center
  4. [4] Al-Leith University College

  Mostrar afiliaciones +
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 24, Nº 6, 2025
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • This paper studies the dynamics of a nonlinear difference system related to the k-Mersenne sequence, a generalization of classical Mersenne numbers. By transforming the nonlinear system into a linear form, explicit solutions are obtained in terms of k-Mersenne numbers. The analysis establishes the global asymptotic stability of the system’s positive equilibrium and characterizes the convergence rate using the Jacobian’s spectral radius. Numerical examples confirm the theoretical findings and illustrate the system’s stability and behavior.

• Referencias bibliográficas
  • 1. Elsayed, E.M.: On a system of two nonlinear difference equations of order two. Proc. Jangjeon Math. Soc. 18(3), 353–368 (2015)
  • 2. Elsayed, E.M., Ibrahim, T.F.: Periodicity and solutions for some systems of nonlinear rational difference equations. Hacet. J. Math. Stat....
  • 3. Elsayed, E.M.: Solution for systems of difference equations of rational form of order two. Comput. Appl. Math. 33, 751–765 (2014)
  • 4. Elsayed, E.M.: On the solutions and periodic nature of some systems of difference equations. Int. J. Biomath. 7, 1450067 (2014). (ID)
  • 5. Elsayed, E.M.: Solutions of rational difference systems of order two. Math. Comput. Modelling 55, 378–384 (2012)
  • 6. Gümü¸s, M., Soykan, Y.: Global character of a six-dimensional nonlinear system of difference equations. Discrete Dyn. Nat. Soc. Article...
  • 7. Gümü¸s, M.: The global asymptotic stability of a system of difference equations. J. Differ. Equ. Appl. 24(6), 976–991 (2018)
  • 8. Gümü¸s, M.: Analysis of periodicity for a new class of non-linear difference equations by using a new method. Electron. J. Math. Anal....
  • 9. Gümü¸s, M.: Global asymptotic behavior of a discrete system of difference equations with delays. Filomat 37(1), 251–264 (2023)
  • 10. Gümü¸s, M., Yalçinkaya, I., Tollu, D.T.: Dynamic analysis of high-order fuzzy difference equation. J. Appl. Math. Comput. 71, 1285–1308...
  • 11. Halim, Y., Khelifa, A., Berkal, M., Bouchair, A.: On a solvable system of p difference equations of higher order. Period. Math. Hung....
  • 12. Halim, Y., Rabago, J.F.T.: On the solutions of a second-order difference equations in terms of generalized Padovan sequences. Math. Slovaca...
  • 13. Halim, Y., Berkal, M., Khelifa, A.: On a three-dimensional solvable system of difference equations. Turk. J. Math. 44(4), 1263–1288 (2020)
  • 14. Halim, Y.: A system of difference equations with solutions associated to Fibonacci numbers. Int. J. Difference Equ. 11(1), 65–77 (2016)
  • 15. Hamioud, H., Dekkar, I., Touafek, N.: Solvability of a third-order system of nonlinear difference equations via a generalized Fibonacci...
  • 16. Kara, M., Yazlik, Y.: Solvability of a system of nonlinear difference equations of higher order. Turk. J. Math. 43(3), 1533–1565 (2019)
  • 17. Kara, M., Yazlik, Y., Touafek, N., Akrour, Y.: On a three-dimensional system of difference equations with variable coefficients. J. Appl....
  • 18. Kara, M., Yazlik, Y.: On eight solvable systems of difference equations in terms of generalized Padovan sequences. Miskolc Math. Notes...
  • 19. Kara, M., Yazlik, Y.: Representation of solutions of eight systems of difference equations via generalized Padovan sequences. Int. J....
  • 20. Kaouache, S., Feˇckan, M., Halim, Y., Khelifa, A.: Theoretical analysis of higher-order system of difference equations with generalized...
  • 21. Khelifa, A., Halim, Y., Berkal, M.: Solutions of a system of two higher-order difference equations in terms of Lucas sequence. Univers....
  • 22. Pituk, M.: More on Poincares and Perrons theorems for difference equations. J. Difference Equ. Appl. 8, 201–216 (2002)
  • 23. Uslu, K., Deniz, V.: Some identities of k-Mersenne numbers. Adv. Appl. Discrete Math. 18(4), 413–423 (2017)
  • 24. Touafek, N.: On a second order rational difference equation. Hacet. J. Math. Stat. 41(6), 867–874 (2012)
  • 25. Touafek, N.: On a general system of difference equations defined by homogeneous functions. Math. Slovaca 71(3), 697–720 (2021)
  • 26. Ta¸skara, N., Büyük, H.: On the solutions of three-dimensional difference equation systems via pell numbers. EJOSAT 34, 433–440 (2022)
  • 27. Tollu, D.T., Yazlik, Y.: Taskara, N: The solutions of four Riccati difference equations associated with Fibonacci numbers. BALKANJM 2(1),...
  • 28. Yazlik, Y., Tollu, D.T., Taskara, N.: Behaviour of solutions for a system of two higher-order difference equations. J. Sci. Arts. 45(4),...
  • 29. Yazlik, Y., Kara, M.: On a solvable system of difference equations of higher-order with period two coefficients. Commun. Fac. Sci. Univ....