# A Review on Dynamical Nature of Systems of Nonlinear Difference Equations

## DOI:

https://doi.org/10.26713/jims.v11i2.1110## Keywords:

Difference equations, Equilibrium point, Boundedness, Stability, Periodicity## Abstract

The goal of this paper is to review about the dynamical behavior of the positive solutions of the systems of difference equations. The present study gives review of recent studies in systems of difference equations. We focus on papers dealing with two-dimensional, third-dimensional and multi-dimensional systems of nonlinear difference equations.

### Downloads

## References

H. Bao, Dynamical behavior of a system of second-order nonlinear difference equations, International Journal of Differential Equations, 2015 (2015), Article ID 679017, 7 pages, DOI: 10.1155/2015/679017.

E. Camouzis and G. Papaschinopoulos, Global asymptotic behavior of positive solutions on the system of rational difference equations (x_{n+1}=1+frac{ x_{n}}{y_{n-m}}), (y_{n+1}=1+frac{y_{n}}{x_{n-m}}), Applied Mathematics Letters 17(6) (2004), 733 -- 737, DOI: 10.1016/S0893-9659(04)90113-9.

D. Clark and M. R. S. Kulenovic, A coupled system of rational difference equations, Computers and Mathematics with Applications 43 (6-7) (2002), 849 -- 867, DOI: 10.1016/S0898-1221(01)00326-1.

D. Clark, M. R. S. Kulenovic and J. F. Selgrade, Global asymptotic behavior of a two-dimensional difference equation modelling competition, Nonlinear Analysis: Theory, Methods & Applications 52(7) (2003), 1765 -- 1776, DOI: 10.1016/S0362-546X(02)00294-8.

I. Dekkar, N. Touafek and Y. Yazlik, Global stability of a third-order nonlinear system of difference equations with period-two coefficients, Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas 111 (2017), 325 -- 347, DOI: 10.1007/s13398-016-0297-z.

Q. Din, Asymptotic behavior of an anti-competitive system of second-order difference equations, Journal of the Egyptian Mathematical Society 24(1) (2016), 37 -- 43, DOI: 10.1016/j.joems.2014.08.008.

M. F. Elettreby and H. El-Metwally, On a system of difference equations of an economic model, Discrete Dynamics in Nature and Society 2013 (2013), Article ID 405628, 6 pages, DOI: 10.1155/2013/405628.

E. M. Elsayed and H. El-Metwally, On the solutions of some nonlinear systems of difference equations, Advances in Difference Equations 2013 (2013):161, DOI: 10.1186/1687-1847-2013-161.

E. M. Elsayed, A. Alotaibi and H. A. Almaylabi, On a solutions of fourth order rational systems of difference equations, J. Comp. Anal. Appl. 22(7) (2017), 1298 -- 1308.

E. M. Elsayed, Solution for systems of difference equations of rational form of order two, Comp. Appl. Math. 33(3) (2014), 751 -- 765, DOI: 10.1007/s40314-013-0092-9.

N. Fotiades and G. Papaschinopoulos, On a system of difference equations with maximum, Applied Mathematics and Computation 221 (2013), 684 -- 690, DOI: 10.1016/j.amc.2013.07.014.

M. Gumus and Y. Soykan, Global character of a six-dimensional nonlinear system of difference equations, Discrete Dynamics in Nature and Society 2016 (2016), Article ID 6842521, 7 pages, DOI: 10.1155/2016/6842521.

E. A. Grove, D. Hadley, E. Lapierre and S. W. Schultz, On the global behavior of the rational system (x_{n+1}=frac{alpha _{1}}{x_{n}+y_{n}}), (y_{n+1}=frac{alpha _{2}+beta _{2}x_{n}+y_{n}}{y_{n}}), Sarajova J. Mathematics 8(21) (2016), 283 -- 292, DOI: 10.5644/SJM.08.2.09.

N. Haddad, N. Touafek and J. F. T. Rabago, Well-defined solutions of a systems of difference equations, J. Appl. Math. Comput. 2018 (56) (2018), 439 -- 458, DOI: 10.1007/s12190-017-1081-8.

A. Q. Khan, Global dynamics of two systems of exponential difference equations by Lyapunov function, Advances in Difference Equations 2014 (1) (2014), 1 -- 21, DOI: 10.1186/1687-1847-2014-297.

M. R. S. Kulenovic and M. Nurkanovic, Global asymptotic behavior of a two-dimensional system of difference equations modeling cooperation, Journal of Difference Equations and Applications 9 (1) (2003), 149 -- 159, DOI: 10.1080/10236100309487541.

M. R. S. Kulenovic and M. Nurkanovic, Global behavior of a three-dimensional linear fractional system of difference equations, J. Mathematical Analysis and Applications 310 (2) (2005), 673 -- 689, DOI: 10.1016/j.jmaa.2005.02.042.

A. S. Kurbanli, C. Cinar and M. E. Erdogan, On the behavior of solutions of the system of rational difference equations: (x_{n+1}=frac{x_{n-1}}{ y_{n}x_{n-1}-1}), (y_{n+1}=frac{y_{n-1}}{x_{n}y_{n-1}-1}), (z_{n+1}= frac{x_{n}}{y_{n}z_{n-1}}), Applied Mathematics 2(8) (2011), 1031 -- 1038, DOI: 10.4236/AM.2011.28143.

A. S. Kurbanli, C. Cinar and I. Yalcinkaya, On the behavior of positive solutions of the system of rational difference equations (x_{n+1}=frac{x_{n-1}}{ y_{n}x_{n-1}+1}), (y_{n+1}=frac{y_{n-1}}{x_{n}y_{n-1}+1}), Mathematical and Computer Modelling 53(5) (2011), 1261 -- 1267, DOI: 10.1016/j.mcm.2010.12.009.

A. S. Kurbanli, C. Cinar and D. Simsek, On the periodicity of solutions of the system of rational difference equations (x_{n+1}=frac{x_{n-1}+y_{n}}{y_{n}x_{n-1}-1}), (y_{n+1}=frac{y_{n-1}+x_{n}}{x_{n}y_{n-1}-1}), Applied Mathematics 2 (2011), 410 -- 413, DOI: 10.4236/AM.2011.24050.

A. S. Kurbanli, On the behavior of solutions of the system of rational difference equations ( x_{n+1}=frac{x_{n-1}}{y_{n}x_{n-1}-1}), (y_{n+1}=frac{y_{n-1}}{ x_{n}y_{n-1}-1}), (z_{n+1}=frac{1}{y_{n}z_{n}}), Advances in Difference Equations 2011 (2011) (1), 1, DOI: 10.1186/1687-1847-2011-40.

A. S. Kurbanli, On the behavior of solutions of the system of rational difference equations: (x_{n+1}=frac{x_{n-1}}{y_{n}x_{n-1}-1}), (y_{n+1}=frac{y_{n-1}}{ x_{n}y_{n-1}-1}), (z_{n+1}=frac{z_{n-1}}{y_{n}z_{n-1}-1}), Discrete Dynamics in Nature and Society 2011 (2011), Article ID 932362, 12 pages, DOI: 10.1155/2011/932362.

M. Mansour, M. M. El-Dessoky and E. M. Elsayed, The form of the solutions and periodicity of some systems of difference equations, Discrete Dynamics in Nature and Society 2012 (2012), Article ID 406821, 17 pages, DOI: 10.1155/2012/406821.

O. Ozkan and A. S. Kurbanli, On a system of difference equations, Discrete Dynamics in Nature and Society 2013 (2013), Article ID 970316, 7 pages, DOI: 10.1155/2013/970316.

G. Papaschinopoulos and C. J. Schinas, On the system of two nonlinear difference equations (x_{n+1}=A+frac{x_{n-1}}{y_{n}}), (y_{n+1}=A+frac{y_{n-1}}{x_{n}}), International Journal Mathematics and Mathematical Sciences 23(12) (2000), 839 -- 848, DOI: 10.1155/S0161171200003227.

G. Papaschinopoulos and C. J. Schinas, On a system of two difference equations, J. Mathematical Analysis and Applications 219(2) (1998), 415 -- 426, DOI: 10.1006/jmaa.1997.5829.

G. Papaschinopoulos and C. J. Schinas, On the dynamics of two exponential type systems of difference equations, Computers and Mathematics with Application 64(7) (2012), 2326 -- 2334, DOI: 10.1016/j.camwa.2012.04.002.

G. Papaschinopoulos, M. A. Radin and C. J. Schinas, On the system of two difference equations of exponential form: (x_{n+1}=a+bx_{n-1}e^{-y_{n}}), ( y_{n+1}=c+dy_{n-1}e^{-x_{n}}): (x_{n+1}=a+bx_{n-1}e^{-y_{n}}), (y_{n+1}=c+dy_{n-1}e^{-x_{n}}), Mathematical and Computer Modelling 54 (11) (2011), 2969 -- 2977, DOI: 10.1016/j.mcm.2011.07.019.

G. Papaschinopoulos, M. A. Radin and C. J. Schinas, Study of the asymptotic behavior of the solutions of three systems of difference equations of exponential form, Applied Mathematics and Computation 218(9) (2012), 5310 -- 5318, DOI: 10.1016/j.amc.2011.11.014.

G. Papaschinopoulos, N. Fotiades and C. J. Schinas, On a system of difference equations including negative exponential terms, Journal of Difference Equations and Applications 20(5-6) (2014), 717 -- 732, DOI: 10.1080/10236198.2013.814647.

D. Simsek, B. Demir and C. Cinar, On the solutions of the system of difference equation (x_{n+1}=max left { frac{A}{x_{n}},frac{y_{n}}{x_{n}} right }), (y_{n+1}=max left { frac{A}{y_{n}},frac{x_{n}}{y_{n}} right }), Discrete Dynamics in Nature and Society 2009 (2009), Article ID 325296, 11 pages, DOI: 10.1155/2009/325296.

T. Stevic and B. Iricanin, Long-term behavior of a cyclic max-type system of difference equations, Electronic Journal of Differential Equations 234 (2015), 1 -- 12.

S. Stevic, M. A. Alghamdi, A. Alotaibi and N. Shahzad, Boundedness character of a max-type system of difference equations second order, Electronic J. Qualitative Theory Differential Equ. 2015 (2014), (45), 1 -- 12, DOI: 10.14232/ejqtde.2014.1.45.

S. Stevic, M. A. Alghamdi, A. Alotaibi and N. Shahzad, On a nonlinear second order system of difference equations, Applied Mathematics and Computation 219(24) (2013), 11388 -- 11394, DOI: 10.1016/j.amc.2013.05.015.

S. Stevic, On a symmetric system of max-type difference equations, Applied Mathematics and Computation 219(15) (2013), 8407 -- 8412, DOI: 10.1016/j.amc.2013.02.008.

S. Stevic, On a third-order system of difference equations, Applied Mathematics and Computation 218(14) (2012), 7649 -- 7654, DOI: 10.1016/j.amc.2012.01.034.

S. Stevic, On some periodic systems of max-type difference equations, Applied Mathematics and Computation 218(23) (2012), 11483 -- 11487, DOI: 10.1016/j.amc.2012.04.077.

S. Stevic, Solutions of a max-type system of difference equations, Applied Mathematics and Computation 218(19) (2012), 9825 -- 9830, DOI: 10.1016/j.amc.2012.03.057.

Q. Wang, G. Zhang and L. Fu, On the behavior of the positive solutions of the system of the two higher-order rational difference equations, Applied Mathematics 4(08) (2013), 1220 -- 1225, DOI: 10.4236/am.2013.48164.

X. Yang, On the system of rational difference equations (x_{n+1}=A+frac{y_{n-1}}{x_{n-p}y_{n-q}}), (y_{n+1}=A+frac{x_{n-1}}{x_{n-r}y_{n-s}}), J. Math. Anal. Appl. 307(1) (2005), 305 -- 311, DOI: 10.1016/j.jmaa.2004.10.045.

Q. Zhang and W. Zhang, On a system of two high order nonlinear difference equations, Advances in Mathematical Physics 2014 (2014), Article ID 729273, 8 pages, DOI: 10.1155/2014/729273.

Q. Zhang, J. Liu and Z. Luo, Dynamical behavior of a system of third-order rational difference equation, Discrete Dynamic in Nature and Society 2015 (2015), Article ID 530453, 6 pages, DOI: 10.1155/2015/530453.

Q. Zhang, L. Yang and J. Liu, On the recursive system (x_{n+1}=A+frac{x_{n-m}}{y_{n}}), (y_{n+1}=B+frac{y_{n-m}}{x_{n}}), Acta Math. Univ. Comenianae 2 (2013), 201 -- 208.

D. Zhang, W. Ji, L. Wang and X. Li, On the symmetrical system of rational difference equation (x_{n+1}=A+frac{y_{n-k}}{y_{n}}), (y_{n+1}=A+frac{x_{n-k}}{x_{n} }), Applied Mathematics 4(05) (2013), 834 -- 837, DOI: 10.4236/am.2013.45114.

Q. Zhang, W. Zhang, Y. Shao and J. Liu, On the system of high order rational difference equations, International Scholarly Research Notices 2014 (2014), Article ID 760502, 5 pages, DOI: 10.1155/2014/760502.

Y. Zhang, X. Yang, D. J. Evans and C. Zhu, On the nonlinear difference equation system (x_{n+1}=A+frac{y_{n-m}}{x_{n}}), (y_{n+1}=A+frac{x_{n-m}}{y_{n}}), Computers and Mathematics with Applications 53(10) (2007), 1561 -- 1566, DOI: 10.1016/j.camwa.2006.04.030.

Y. Zhang, X. Yang, G. M. Megson and D. J. Evans, On the system of rational difference equations (x_{n+1}=A+frac{1}{y_{n-p}}), (y_{n+1}=A+frac{1}{x_{n-r}y_{n-s}}), Applied Mathematics and Computation 176(2) (2006), 403 -- 408, DOI: 10.1016%2Fj.amc.2005.09.039.

## Downloads

## Published

## How to Cite

*Journal of Informatics and Mathematical Sciences*,

*11*(2), 235–251. https://doi.org/10.26713/jims.v11i2.1110

## Issue

## Section

## License

Authors who publish with this journal agree to the following terms:- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a CCAL that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work.