Dynamics of the rational difference equations

Year 2024, Volume: 12 Issue: 2, 177 - 184, 27.12.2024

Burak Oğul

https://doi.org/10.51354/mjen.1420761

Abstract

Discrete-time systems are sometimes used to explain natural phenomena that happen in non-linear sciences. We study the periodicity, boundedness, oscillation, stability, and certain exact solutions of nonlinear difference equations of generalized order in this paper. Using the standard iteration method, exact solutions are obtained. Some well-known theorems are used to test the stability of the equilibrium points. Some numerical examples are also provided to confirm the theoretical work’s validity. The numerical component is implemented with Wolfram Mathematica. The method presented may be simply applied to other rational recursive issues. nIn this research, we examine the qualitative behavior of rational recursive sequences provided that the initial conditions are arbitrary real numbers. We examine the behavior of solutions on graphs according to the state of their initial valueb 𝑥𝑛+1 = 𝑥𝑛𝑥𝑛−8 ±𝑥𝑛−7 ± 𝑥𝑛𝑥𝑛−7𝑥𝑛−8 , 𝑛 ∈ N0.

Keywords

Equilibrium point, solution of difference equation, stability, boundedness, global asymptotic stability *Corresponding

References

• [1] Abdelrahman M.A.E, Moaaz O., ”On the New Class of The Nonlinear Rational Difference Equations,” Electronic Journal of Mathematical Analysis and Applications, 6 (1), 117-125, (2018).
• [2] Ahmed A.E.S., Iriˇcanin B., KosmalaW., Stevi´c S., Smarda Z, ”Note on constructing a family of solvable sine-type difference equations,” Advances in Difference Equations, 2021(1), 1-11, (2021).
• [3] Agarwal R.P., ”Difference Equations and Inequalities,” Marcel Dekker, New York, 1992, 2nd edition, 2000.
• [4] Agarwal R.P.and Elsayed E.M., ”On the solution of fourthorder rational recursive sequence,” Advanced Studies in Contemporary Mathematics, 20(4), 525-545 (2010).
• [5] H.S. Alayachi, M.S.M. Noorani, A.Q. Khan, M.B. Almatrafi, Analytic Solutions and Stability of Sixth Order Difference Equations, Mathematical Problems in Engineering, 2020, Article ID 1230979, 12-23 (2020).
• [6] Aloqeili M., ”Dynamics of a rational difference equation,” Applied Mathematics and Computation, 176(2), 768-774, (2006).
• [7] Almaslokh, A. and Qian, C., ”Global attractivity of a higher order nonlinear difference equation with unimodal terms,” Opuscula Mathematica, 43(2), 131-143 (2023).
• [8] M.B. Almatrafi, M.M. Alzubaidi, Analysis of the Qualitative Behaviour of an Eighth-Order Fractional Difference Equation, Open Journal of Discrete Applied Mathematics, 2, No. 1, 41-47 (2019).
• [9] Almatrafi, M.B. and Alzubaidi, M.M., Qualitative analysis for two fractional difference equations. Nonlinear Engineering, 9(1), 265-272 (2020).
• [10] Amleh A.M., Grove G.A., Ladas G., Georgiou, D.A., ”On the recursive sequence 𝑦𝑛+1 = 𝛼+ 𝑦𝑛−1 𝑦𝑛 ,” J. of ath. Anal. App. 233, 790-798 (1999).
• [11] DeVault R., Ladas G., Schultz S.W., ”On the recursive sequence 𝑥𝑛+1 = 𝐴 𝑥𝑛 + 1 𝑥𝑛−2 ,” Proc.Amer. Math. Soc.126 (11) 3257-3261 (1998).
• [12] S. Elaydi, An introduction to difference equations, 3rd Ed., Springer, USA, (2005).
• [13] Elsayed E.M., ”On The Difference Equation 𝑥𝑛+1 = 𝑥𝑛−5 −1+𝑥𝑛−2 𝑥𝑛−5 ,” Inter. J. Contemp. Math. Sci., 3 (33) 1657- 1664, (2008).
• [14] Gibbons C.H., Kulenovic M.R.S., Ladas G., ”On the recursive sequence 𝛼+𝛽𝑥𝑛−1 𝜒+𝛽𝑥𝑚−1 ,” Math. Sci. Res. Hot-Line, 4(2), 1-11 (2000).
• [15] Ibrahim T.F., Khan A.Q., Ibrahim, A., Qualitative behavior of a nonlinear generalized recursive sequence with delay, Mathematical Problems in Engineering, (2021).
• [16] Ibrahim, T. F., S¸ims¸ek, D., O˘gul, B. The Solution and Dynamic Behaviour of Difference Equations of Twenty-First Order, Manas Journal of Engineering, 11(1), 159-165, (2023).
• [17] O. Karpenko, O. Stanzhytskyi, The relation between the existence of bounded solutions of differential equations and the corresponding difference equations, Journal of Difference Equations and Applications, 19, No. 12, 1967-1982 (2013). https://doi.org/10.1080/10236198.2013.794795
• [18] M. Bohner, O. Karpenko, O. Stanzhytskyi, Oscillation of solutions of second-order linear differential equations and corresponding difference equations, Journal of Difference Equations and Applications, 20, No. 7, 1112-1126 (2014). https://doi.org/10.1080/10236198.2014.893297
• [19] Khan A.Q., El-Metwally H., Global dynamics, boundedness, and semicycle analysis of a difference equation, Discrete Dynamics in Nature and Society, (2021).
• [20] Khyat, T., Kulenovic, M.R. and Pilav, E. The invariant curve caused byNeimark-Sacker bifurcation of a perturbed Beverton-Holt difference equation, International Journal of Difference Equations, 12(2), 267-280 (2017).
• [21] Kocic V.L., Ladas G., ”Global behavior of nonlinear difference equations of higher order with applications,” volume 256 of Mathematics and its Applications, Kluwer Academic Publishers Group, Dordrecht, 1993.
• [22] Kostrov, Y. and Kudlak, Z. On a second-order rational difference equation with a quadratic term. International Journal of Difference Equations, 11(2), 179-202 (2016).
• [23] Kulenovic M.R.S., Ladas G., Sizer W.S., ”On the recursive sequence 𝛼𝑥𝑛+𝛽𝑥𝑛−1 𝜒𝑥𝑛+𝛽𝑥𝑛−1 ,” Math. Sci. Res.Hot-Line, 2(5), 1-16 (1998).
• [24] Kulenovic M.R.S., Ladas G., ”Dynamics of second order rational difference equations” Chapman & Hall/CRC, Boca Raton, FL, 2002. With open problems and conjectures.
• [25] O˘gul, B., S¸ims¸ek, D., ¨O ˘g¨unmez, H., and Kurbanlı, A.S. Dynamical behavior of rational difference equation xn+ 1= xn-17±1±xn-2 xn-5 xn-8 xn-11 xn-14 xn-17, Bolet´ın de la Sociedad Matem´atica Mexicana, 27(2), 49 (2021).
• [26] O˘gul, B., S¸ims¸ek, D., Kurbanlı, A. S., ¨O ˘g¨unmez, H., Dynamical Behavior of Rational Difference Equation xn+ 1= xn-15±1±xn-3 xn-7 xn-11 xn-15, Differential Equations and Dynamical Systems, 1-16 (2021).
• [27] Ogul, B., and Simsek, D. Dynamical behavior of one rational fifth-order difference equation. Carpathian Mathematical Publications, 15(1), 43-51 (2023).
• [28] Sanbo, A., Elsayed, E.M., Some properties of the solutions of the difference equation 𝑥𝑛+1 = 𝛼𝑥𝑛 + MANAS Journal of Engineering, Volume 12 (Issue 2) © (2024) www.journals.manas.edu.kg B. O˘gul / MANAS Journal of Engineering, 12 (2) (2024) 184 (𝑏𝑥𝑛𝑥𝑛−4)/(𝑐𝑥𝑛−3 + 𝑑𝑥𝑛−4), Open Journal of Discrete Applied Mathematics, 2, No. 2, 31–47 (2019).
• [29] Simsek, D., O˘gul, B., Abdullayev, F., Dynamical behavior of solution of fifteenth-order rational difference equation, Filomat, 24(3), 997-1008 (2024)
• [30] Yalcinkaya ˙I., C¸ alıs¸kan V., Tollu D.T., ”On a nonlinear fuzzy difference equation,” Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 71(1), 68-78, (2022).