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Global dynamics of the system of difference equations

  • Merve Kara EMAIL logo and Yasin Yazlik
Published/Copyright: May 7, 2025
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Abstract

In the current paper, we investigate the global dynamics of the following difference equations system of exponential form:

Υn+1=Γ1+δ1eΥn1Θ1+Ψn,Ψn+1=Γ2+δ2eΨn1Θ2+Ωn,Ωn+1=Γ3+δ3eΩn1Θ3+Υn,nN0,

where the coefficients Γp, δp, Θp for p ∈ {1, 2, 3} and the initial conditions Υ−s, Ψ−s, Ω−s, for s ∈ {0, 1} are positive constants. Further, a numerical example is presented to confirm the obtained theoretical results.

MSC 2010: 39A10; 39A20; 39A23
  1. (Communicated by Jozef Džurina)

References

[1] Abo-Zeid, R.: Global behavior and oscillation of a third order difference equation, Quaest. Math. 44(9) (2021), 1261–1280.10.2989/16073606.2020.1787537Search in Google Scholar

[2] Elaydi, S.: An Introduction to Difference Equations, Springer, 2005.Search in Google Scholar

[3] El-Metwally, H.—Grove, E. A.—Ladas, G.—Levins, R.: On the difference equation xn+1 = α + βxn−1e−xn Nonlinear Anal. Theory Meth. Appl. 47(7)(2001), 4623–4634.10.1016/S0362-546X(01)00575-2Search in Google Scholar

[4] Elsayed, E. M.—Alzahrani, F.—Abbas, I.—Alotaibi, N. H.: Dynamical behavior and solution of nonlinear difference equation via Fibonacci sequence, J. Appl. Anal. Comput. 10(1) (2020), 282–296.10.11948/20190143Search in Google Scholar

[5] Ghezal, A.: Note on a rational system of (4k + 4)−order difference equations: periodic solution and convergence, J. Appl. Math. Comput. 69 (2022), 2207–2215.10.1007/s12190-022-01830-ySearch in Google Scholar

[6] Grove, E. A.—Ladas, G.: Advances in Discrete Mathematics and Applications, Chapman and hall/CRC, 2005.Search in Google Scholar

[7] Halim, Y.—Touafek, N.—Yazlik, Y.: Dynamic behavior of a second-order nonlinear rational difference equation, Turkish J. Math. 39(6) (2015), 1004–1018.10.3906/mat-1503-80Search in Google Scholar

[8] Ibrahim, T. F.: Periodicity and global attractivity of difference equation of higher order, J. Comput. Anal. Appl. 16(1) (2014), 552–564.Search in Google Scholar

[9] Kara, M.—Yazlik, Y.: Solvable three-dimensional system of higher-order nonlinear difference equations, Filomat 36(10) (2022), 3453–3473.10.2298/FIL2210449KSearch in Google Scholar

[10] Kara, M.—Yazlik, Y.: On a solvable system of rational difference equations of higher order, Turkish. J. Math. 46(2) (2022), 587–611.Search in Google Scholar

[11] Kara, M.—Yazlik, Y.: On the solutions of three-dimensional system of difference equations via recursive relations of order two and Applications, J. Appl. Anal. Comput. 12(2) (2022), 736–753.10.11948/20210305Search in Google Scholar

[12] Kara,M.: Investigation of the global dynamics of two exponential-form difference equations systems, Electron. Res. Arch. 31(11) (2023), 6697–6724.10.3934/era.2023338Search in Google Scholar

[13] Khan, A. Q.—Qureshi, M. N.: Behavior of an exponential system of difference equations, Discrete Dyn. Nat. Soc. 2014 (2014), 1–9.10.1155/2014/607281Search in Google Scholar

[14] Khan, A. Q.—Sharif, A.: Global dynamics of some 3 × 6 systems of exponential difference equations, Discrete Dyn. Nat. Soc. 2018 (2018), 1–35.10.1155/2018/8362837Search in Google Scholar

[15] Khan, A. Q.—Noorani, M. S. M.—Alayachi, H. S.: Global dynamics of higher-order exponential systems of difference equations, Discrete Dyn. Nat. Soc. 2019 (2019), 1–19.10.1155/2019/3825927Search in Google Scholar

[16] Kocic, V. L.—Ladas, G.: Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Springer Science & Business Media, 1993.10.1007/978-94-017-1703-8Search in Google Scholar

[17] Mylona, C.—Psarros, N.—Papaschinopoulos, G.—Schinas, C. J.: Stability of two 3 × 3 close-to-cyclic systems of exponential difference equations, Math. Meth. Appl. Sci. 41(17) (2018), 7936–7948.10.1002/mma.5256Search in Google Scholar

[18] Mylona, C.—Psarros, N.—Papaschinopoulos, G.—Schinas, C. J.: Stability of the non-hyperbolic zero equilibrium of two closeto-symmetric systems of difference equations with exponential terms, Symmetry 10(6) (2018), 188.10.3390/sym10060188Search in Google Scholar

[19] Ozturk, I.—Bozkurt, F.—Ozen, S.: On the difference equation yn+1=α+βeynγ+yn1, Appl. Math. Comput. 181(2) (2006), 1387–1393.10.1016/j.amc.2006.03.007Search in Google Scholar

[20] Papaschinopoluos, G.—Radin, M. A.—Schinas, C. J.: On the system of two difference equations of exponential form: xn+1=a+bxn1eyn,yn+1=c+dyn1exn, Math. Comput. Model. 54(11–12) (2011), 2969–2977.Search in Google Scholar

[21] Papaschinopoluos, G.—Radin, M.—Schinas, C. J.: Study of the asymptotic behavior of the solutions of three systems of difference equations of exponential form, Appl. Math. Comput. 218(9) (2012), 5310–5318.10.1016/j.amc.2011.11.014Search in Google Scholar

[22] Papaschinopoluos, G.—Schinas, C. J.: On the dynamics of two exponential type systems of difference equations, Comput. Math. Appl. 64(7) (2012), 2326–2334.10.1016/j.camwa.2012.04.002Search in Google Scholar

[23] Papaschinopoulos, G.—Ellina, G.—Papadopoulos, K. B.: Asymptotic behavior of the positive solutions of an exponential type system of difference equations, Appl. Math. Comput. 245 (2014), 181–190.10.1016/j.amc.2014.07.074Search in Google Scholar

[24] Pituk, M.: More on Poincares and Perrons theorems for difference equations, J. Difference Equ. Appl. 8(3) (2002), 201–216.10.1080/10236190211954Search in Google Scholar

[25] Psarros, N.—Papaschinopoulos, G.—Schinas, C. J.: On the stability of some systems of exponential difference equations, Opuscula Math. 38(1) (2018), 95–115.10.7494/OpMath.2018.38.1.95Search in Google Scholar

[26] Sedaghat, H.: Nonlinear Difference Equations: Theory with Applications to Social Science Models, Kluwer Academic Publishers, 2003.10.1007/978-94-017-0417-5Search in Google Scholar

[27] Taskara, N.—Tollu, D. T.—Touafek, N.—Yazlik, Y.: A solvable system of difference equations, Commun. Korean. Math. Soc. 35(1) (2020), 301–319.Search in Google Scholar

[28] Thai, T. H.—Dai, N. A.—Anh, P. T.: Global dynamics of some system of second-order difference equations, Electron. Res. Arch. 29(6) (2021), 4159–4175.10.3934/era.2021077Search in Google Scholar

[29] Tollu, D. T.—Yazlik, Y.—Taskara, N.: Behavior of positive solutions of a difference equation, J. Appl. Math. Inform. 35(3) (2017), 217–230.10.14317/jami.2017.217Search in Google Scholar

[30] Tollu, D. T.—Yazlik, Y.—Taskara, N.: On a solvable nonlinear difference equation of higher order, Turkish J. Math. 42(4) (2018), 1765–1778.10.3906/mat-1705-33Search in Google Scholar

[31] Touafek, N.: On a general system of difference equations defined by homogeneous functions, Math. Slovaca 71(3) (2021), 697–720.10.1515/ms-2021-0014Search in Google Scholar

[32] Wang, W.—Feng, H.: On the dynamics of positive solutions for the difference equation in a new population model, J. Nonlinear Sci. Appl. 9 (2016), 1748–1754.10.22436/jnsa.009.04.30Search in Google Scholar

[33] Yalcinkaya, I.—Tollu, D. T.: Global behavior of a second order system of difference equations, Adv. Stud. Contemp. Math. 26(4) (2016), 653–667.Search in Google Scholar

[34] Yazlik, Y.—Tollu, D. T.—Taskara, N.: On the solutions of difference equation systems with Padovan numbers, Appl. Math. 4(12A) (2013), 1–15.10.1186/1687-1847-2013-174Search in Google Scholar

[35] Yazlik, Y.—Kara, M.: On a solvable system of difference equations of higher-order with period two coefficients, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 68(2) (2019), 1675–1693.10.31801/cfsuasmas.548262Search in Google Scholar

Received: 2023-11-21
Accepted: 2024-12-05
Published Online: 2025-05-07
Published in Print: 2025-04-28

© 2025 Mathematical Institute Slovak Academy of Sciences

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