The forbidden set, solvability and stability of a circular system of complex Riccati type difference equations

George L. Karakostas; George L. Karakostas

doi:10.3934/math.20231434

AIMS Mathematics

2023, Volume 8, Issue 11: 28033-28050. doi: 10.3934/math.20231434

Previous Article Next Article

Research article Special Issues

The forbidden set, solvability and stability of a circular system of complex Riccati type difference equations

George L. Karakostas ^,

Department of Mathematics, University of Ioannina, Ioannina 45110, Greece

Received: 13 July 2023 Revised: 04 October 2023 Accepted: 05 October 2023 Published: 12 October 2023
MSC : 39A45, 39A05, 39A23, 39A30, 39A99

In this paper, the circular system of Riccati type complex difference equations of the form

$ u_{n+1}^{(j)} = \frac{a_ju_n^{(j-1)}+b_j}{c_ju_n^{(j-1)}+d_j}, \; n = 0, 1, 2, \cdots, \; j = 1, 2, \cdots, k, $

where $ u_n^{(0)}: = u_n^{(k)} $ for all $ n $, is investigated. First, the forbidden set of the equation is given. Then the solvability of the system is examined and the expression of the solutions, given in terms of their initial values. Next, the asymptotic behaviour of the solutions is studied. Finally, in case of negative Riccati real numbers

$ R_j: = \frac{a_jd_j-b_jc_j}{[a_j+d_j]^2}, \; j\in\overline{1, k}, $

it is shown that there exists a unique positive fixed point which attracts all solutions starting from positive states.
- difference equations,
- solvability,
- asymptotic behaviour
Citation: George L. Karakostas. The forbidden set, solvability and stability of a circular system of complex Riccati type difference equations[J]. AIMS Mathematics, 2023, 8(11): 28033-28050. doi: 10.3934/math.20231434

Related Papers:

Abstract

In this paper, the circular system of Riccati type complex difference equations of the form

$ u_{n+1}^{(j)} = \frac{a_ju_n^{(j-1)}+b_j}{c_ju_n^{(j-1)}+d_j}, \; n = 0, 1, 2, \cdots, \; j = 1, 2, \cdots, k, $

where $ u_n^{(0)}: = u_n^{(k)} $ for all $ n $, is investigated. First, the forbidden set of the equation is given. Then the solvability of the system is examined and the expression of the solutions, given in terms of their initial values. Next, the asymptotic behaviour of the solutions is studied. Finally, in case of negative Riccati real numbers

$ R_j: = \frac{a_jd_j-b_jc_j}{[a_j+d_j]^2}, \; j\in\overline{1, k}, $

it is shown that there exists a unique positive fixed point which attracts all solutions starting from positive states.

References

[1]	R. Abo-Zeid, Global asymptotic stability of a second order rational difference equation, J. Appl. Math. Inform., 2 (2010), 797–804.
[2]	R. P. Agarwal, Differences equations and inequalities, theory, methods and applications, Boca Raton: CRC Press, 2000. https://doi.org/10.1201/9781420027020
[3]	K. S. Al-Basyouni, E. M. Elsayed, On some solvable systems of some rational difference equations of third order, Mathematics, 11 (2023), 1047. https://doi.org/10.3390/math11041047 doi: 10.3390/math11041047
[4]	R. J. H. Beverton, S. J. Holt, On the dynamics of exploited fish populations, In: Fishery investigations, London: H. M. Stationery off., 1957.
[5]	L. Brand, A sequence defined by a difference equation, Am. Math. Mon., 62 (1955), 489–492. https://doi.org/10.2307/2307362 doi: 10.2307/2307362
[6]	D. Clark, M. R. S. Kulenović, A coupled system of rational difference equations, Comput. Math. Appl., 43 (2002), 849–867. https://doi.org/10.1016/S0898-1221(01)00326-1 doi: 10.1016/S0898-1221(01)00326-1
[7]	D. Cheraghi, T. Kuna, Dynamical systems, Maths of Planet Earth CDT Text Book, 2016. Available from: https://www.ma.imperial.ac.uk/ dcheragh/Teaching/2016-F-DS-MPE.pdf.
[8]	J. M. Cushing, An evolutionary Beverton-Holt model, In: Z. AlSharawi, J. Cushing, S. Elaydi, Theory and applications of difference equations and discrete dynamical dystems, Springer Proceedings in Mathematics & Statistics, 2014. https://doi.org/10.1007/978-3-662-44140-4_7
[9]	J. M. Cushing, S. M. Henson, Global dynamics of some periodically forced monotone difference equations, J. Differ. Equ. Appl., 7 (2001), 859–872. https://doi.org/10.1080/10236190108808308 doi: 10.1080/10236190108808308
[10]	S. Elaydi, R. J. Sacker, Global stability of periodic orbits of non-autonomous difference equations and population biology, J. Differ. Equ., 208 (2005), 258–273, https://doi.org/10.1016/j.jde.2003.10.024 doi: 10.1016/j.jde.2003.10.024
[11]	E. M. Elsayed, K. N. Alharbi, The expressions and behavior of solutions for nonlinear systems of rational difference equations, J. Innov. Appl. Math. Comput. Sci., 2 (2022), 78–91.
[12]	L. Edelstein-Keshet, Mathematical models in biology, New York: SIAM, 2005.
[13]	E. A. Grove, G. Ladas, Periodicity in nonlinear difference equations, Advances in Discrete Mathematics and Applications, Chapman and Hall/CRC, 2005.
[14]	M. Kara, Y. Yazlik, On a solvable three-dimensional system of difference equations, Filomat, 34 (2020), 1167–1186. https://doi.org/10.2298/FIL2004167K doi: 10.2298/FIL2004167K
[15]	V. L. Kocić, G. Ladas, I. W. Rodrigues, On rational recursive sequences, J. Math. Anal. Appl., 173 (1993), 127–157.
[16]	M. R. S. Kulenović, G. Ladas, Dynamics of second order rational difference equations, New York: Chapman and Hall/LRC, 2001. https://doi.org/10.1201/9781420035384
[17]	G. Ladas, G. Lugo, F. J. Palladino, Open problems and conjectures on rational systems in three dimensions, Sarajevo J. Math., 8 (2012), 311–321. https://doi.org/10.5644/SJM.08.2.11 doi: 10.5644/SJM.08.2.11
[18]	P. S. Laplace, Recherches sur l' intégration des équations différentielles aux différences finies et sur leur usage dans la théorie des hasards (in French), Mém. Acad. R. Sci. Paris, 7 (1776), 69–197.
[19]	S. Stević, On some solvable difference equations and systems of difference equations, Abstr. Appl. Anal., 2012 (2012), 1–11. https://doi.org/10.1155/2012/541761 doi: 10.1155/2012/541761
[20]	S. Stević, Representations of solutions to linear and bilinear difference equations and systems of bilinear difference equations, Adv. Differ. Equ., 2018 (2018), 474. https://doi.org/10.1186/s13662-018-1930-2 doi: 10.1186/s13662-018-1930-2
[21]	S. Stević, B. Iri$ \breve{{\rm{c}}} $anin, W. Kosmala, Z. $ \breve{{\rm{S}}}$marda, On some classes of solvable systems of difference equations, Adv. Differ. Equ., 2019 (2019), 39. https://doi.org/10.1186/s13662-019-1959-x doi: 10.1186/s13662-019-1959-x
[22]	S. Stević, Some representations of the general solution to a difference equation of additive type, Adv. Differ. Equ., 2019 (2019), 431. https://doi.org/10.1186/s13662-019-2365-0 doi: 10.1186/s13662-019-2365-0
[23]	S. Stević, General solutions to four classes of nonlinear difference equations and some of their representations, Electron. J. Qual. Theory Differ. Equ., 75 (2019), 1–19. https://doi.org/10.14232/ejqtde.2019.1.75 doi: 10.14232/ejqtde.2019.1.75
[24]	S. Stević, Solvability of a general class of two-dimensional hyperbolic-cotangent-type systems of difference equations, Adv. Differ. Equ., 2019 (2019), 294. https://doi.org/10.1186/s13662-019-2233-y doi: 10.1186/s13662-019-2233-y
[25]	S. Stević, A short proof of the Cushing-Henson conjecture, Discrete Dyn. Nat. Soc., 2006 (2006), 1–5, https://doi.org/10.1155/DDNS/2006/37264 doi: 10.1155/DDNS/2006/37264
[26]	S. Stević, B. Iri$ \breve{{\rm{c}}} $anin, W. Kosmala, Z. $ \breve{{\rm{S}}}$marda, On a solvable class of nonlinear difference equations of fourth order, Electron. J. Qual. Theory Differ. Equ., 37 (2022), 1–17. https://doi.org/10.14232/ejqtde.2022.1.37 doi: 10.14232/ejqtde.2022.1.37

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)