Global Dynamics of Solutions of A New Class of Rational Difference Equations

The purpose of this paper is to investigate the global dynamics of solutions of the following delay nonlinear difference equation $$ x_{n+1}=a+\frac{bx_{n-k}}{x_{n-l}}+\frac{cx_{n-l}}{x_{n-k}}\text{, }n=0,1,... $$ where the parameters $a,b,c$ are non-zero real numbers, $k,l\in \mathbb{Z}% ^{+}$ and the initial values $x_{-\max \{k,l\}},...,x_{-1},x_{0}\in \mathbb{R} -\{0\}$. The results obtained here improve and generalize some known ones in the literature. Moreover, several numerical simulations are provided to support obtained results.

Keywords:

Dynamics, two periodic solution boundedness,

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[1] A. M. Amleh, E. A. Grove, G. Ladas and D. A. Georgiou, On the recursive sequence $x_{n+1}=\alpha +(x_{n-1}/x_{n})$, Journal of Mathematical Analysis and Applications, 233 (1999), 790-798.
[2] R. Abo Zeid, Global attractivity of a higher-order difference equation, Discrete Dynamics in Nature and Society, Volume 2012, Article ID 930410, 11 pages.
[3] E. M. Elabbasy, M. Y. Barsoum, H. S. Alshawee, Behavior of solutions of a class of nonlinear rational difference equation $% x_{n+1}=\alpha x_{n-k}+(\beta x_{n-l}^{\delta }/\gamma x_{n-s}^{\delta })$, Electronic Journal of Mathematical Analysis and Applications, 4(2) (2016), 78-87.
[4] O. Moaaz and M. A. E. Abdelrahman, Behaviour of the new class of the rational difference equations, Electronic Journal of Mathematical Analysis and Applications, 4(2) (2016), 129-138.
[5] O. Moaaz, Comment on ”New method to obtain periodic solutions of period two and three of a rational difference equation” [Nonlinear Dyn 79:241–250], Nonlinear Dyn., 88 (2017), 1043-1049.
[6] O. Moaaz, Dynamics of difference equation $% x_{n+1}=f(x_{n-l},x_{n-k})$, Advances in Difference Equations, 2018(1), 447.
[7] E. M. Elsayed, New method to obtain periodic solutions of period two and three of a rational difference equation, Nonlinear Dynamics, 79(1) (2014), 241-250.
[8] E. M. Elsayed, Solution and attractivity for a rational recursive sequence, Discrete Dynamics in Nature and Society, Volume 2011, Article ID 982309, 18 pages.
[9] M. Gumus, The periodicity of positive solutions of the non-linear difference equation $x_{n+1}=\alpha +(x_{n-k}^{p}/x_{n}^{q})$, Discrete Dynamics in Nature and Society, vol.2013, Article ID 742912, 3 pages.
[10] V. Kocic. and G. Ladas, Global behavior of nonlinear difference equations of higher order with applications, Kluwer Academic Publishers, Dordrecht, (1993).
[11] M. R. S. Kulenovi´c and G. Ladas, Dynamics of second order rational difference equations, Chapman & Hall/CRC, (2001).
[12] O. Ocalan, Global dynamics of a non-autonomous rational difference equation, J. Appl. Math. & Informatics, 32(5-6) (2014), 843-848.
[13] I. Yalcinkaya and C. Cinar, On the dynamics of the difference equation $x_{n+1}=(ax_{n-k})/(b+cx_{n}^{p})$, Fasciculi Mathematici, 42 (2009), 141-148.

Yayın Aralığı: Yılda 2 Sayı
Başlangıç: 2013
Yayıncı: Mehmet Zeki SARIKAYA

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