Global stability and co-balancing numbers in a system of rational difference equations

Najmeddine Attia; Ahmed Ghezal; Najmeddine Attia; Ahmed Ghezal

doi:10.3934/era.2024097

Electronic Research Archive

2024, Volume 32, Issue 3: 2137-2159. doi: 10.3934/era.2024097

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Research article

Global stability and co-balancing numbers in a system of rational difference equations

Najmeddine Attia ^{1,2
,
,},
Ahmed Ghezal ³

1.
Department of Mathematics and Statistics, College of Science, King Faisal University, Al-Ahsa 31982, Saudi Arabia
2.
Department of Mathematics, Faculty of Sciences of Monastir, University of Monastir, Monastir-5000, Tunisia
3.
Department of Mathematics, Abdelhafid Boussouf University Center of Mila, Algeria

Received: 27 January 2024 Revised: 26 February 2024 Accepted: 04 March 2024 Published: 14 March 2024

This paper investigates both the local and global stability of a system of rational difference equations and its connection to co-balancing numbers. The study delves into the intricate dynamics of mathematical models and their stability properties, emphasizing the broader implications of global stability. Additionally, the investigation extends to the role of co-balancing numbers, elucidating their significance in achieving equilibrium within the solutions of the rational difference equations. The interplay between global stability and co-balancing numbers forms a foundational aspect of the analysis. The findings contribute to a deeper understanding of the mathematical structures underlying dynamic systems and offer insights into the factors influencing their stability and equilibrium. This article serves as a valuable resource for mathematicians, researchers, and scholars interested in the intersection of global stability and co-balancing sequences in the realm of rational difference equations. Moreover, the presented examples and figures consistently demonstrate the global asymptotic stability of the equilibrium point throughout the paper.
- system of difference equations,
- global stability,
- balancing sequence,
- co-balancing sequence,
- rational difference equations
Citation: Najmeddine Attia, Ahmed Ghezal. Global stability and co-balancing numbers in a system of rational difference equations[J]. Electronic Research Archive, 2024, 32(3): 2137-2159. doi: 10.3934/era.2024097

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Abstract

This paper investigates both the local and global stability of a system of rational difference equations and its connection to co-balancing numbers. The study delves into the intricate dynamics of mathematical models and their stability properties, emphasizing the broader implications of global stability. Additionally, the investigation extends to the role of co-balancing numbers, elucidating their significance in achieving equilibrium within the solutions of the rational difference equations. The interplay between global stability and co-balancing numbers forms a foundational aspect of the analysis. The findings contribute to a deeper understanding of the mathematical structures underlying dynamic systems and offer insights into the factors influencing their stability and equilibrium. This article serves as a valuable resource for mathematicians, researchers, and scholars interested in the intersection of global stability and co-balancing sequences in the realm of rational difference equations. Moreover, the presented examples and figures consistently demonstrate the global asymptotic stability of the equilibrium point throughout the paper.

References

[1]	T. Zhang, H. Qu, J. Zhou, Asymptotically almost periodic synchronization in fuzzy competitive neural networks with Caputo-Fabrizio operator, Fuzzy Sets Syst., 471 (2023), 108676. https://doi.org/10.1016/j.fss.2023.108676 doi: 10.1016/j.fss.2023.108676
[2]	T. Zhang, Y. Li, Global exponential stability of discrete-time almost automorphic Caputo–Fabrizio BAM fuzzy neural networks via exponential Euler technique, Knowl.-Based Syst., 246 (2022), 108675. https://doi.org/10.1016/j.knosys.2022.108675 doi: 10.1016/j.knosys.2022.108675
[3]	T. Zhang, Y. Li, Exponential Euler scheme of multi-delay Caputo–Fabrizio fractional-order differential equations, Appl. Math. Lett., 124 (2022), 107709. https://doi.org/10.1016/j.aml.2021.107709 doi: 10.1016/j.aml.2021.107709
[4]	T. Zhang, Y. Liuc, H. Qub, Global mean-square exponential stability and random periodicity of discrete-time stochastic inertial neural networks with discrete spatial diffusions and Dirichlet boundary condition, Comput. Math. Appl., 141 (2023), 116–128. https://doi.org/10.1016/j.camwa.2023.04.011 doi: 10.1016/j.camwa.2023.04.011
[5]	A. Ghezal, M. Balegh, I. Zemmouri, Markov-switching threshold stochastic volatility models with regime changes, AIMS Math., 9 (2024), 3895–3910. https://doi.org/10.3934/math.2024192 doi: 10.3934/math.2024192
[6]	A. Ghezal, I. Zemmouri, On Markov-switching asymmetric logGARCH models: stationarity and estimation, Filomat, 37 (2023), 1–19. doi: 10.2298/FIL2301001G
[7]	A. Ghezal, I. Zemmouri, M-estimation in periodic Threshold GARCH models: Consistency and asymptotic normality, Miskolc Math. Notes, in Press.
[8]	A. Ghezal, QMLE for periodic time-varying asymmetric logGARCH models, Commun. Math. Stat., 9 (2021), 273–297. http://doi.org/10.1007/s40304-019-00193-4 doi: 10.1007/s40304-019-00193-4
[9]	A. Ghezal, I. Zemmouri, On the Markov-switching autoregressive stochastic volatility processes, SeMA J., 2023. http://doi.org/10.1007/s40324-023-00329-1
[10]	A. Ghezal, A doubly Markov switching AR model: Some probabilistic properties and strong consistency, J. Math. Sci., 271 (2023), 66–75. http://doi.org/10.1007/s10958-023-06262-y doi: 10.1007/s10958-023-06262-y
[11]	A. Bibi, A. Ghezal, QMLE of periodic time-varying bilinear-GARCH models, Commun. Stat.- Theory Methods, 48 (2019), 3291–3310. https://doi.org/10.1080/03610926.2018.1476703 doi: 10.1080/03610926.2018.1476703
[12]	A. Ghezal, Spectral representation of Markov-switching bilinear processes, Sao Palo J. Math. Sci., 2023. https://doi.org/10.1007/s40863-023-00380-w
[13]	A. Bibi, A. Ghezal, QMLE of periodic bilinear models and of PARMA models with periodic bilinear innovations, Kybernetika, 54 (2017), 375–399. http://doi.org/10.14736/kyb-2018-2-0375 doi: 10.14736/kyb-2018-2-0375
[14]	A. Bibi, A. Ghezal, Minimum distance estimation of Markov-switching bilinear processes, Statistics, 50 (2016), 1290–1309. https://doi.org/10.1080/02331888.2016.1229783 doi: 10.1080/02331888.2016.1229783
[15]	A. Bibi, A. Ghezal, On periodic time-varying bilinear processes: structure and asymptotic inference, Stat. Methods Appl., 25 (2016), 395–420. https://doi.org/10.1007/s10260-015-0344-5 doi: 10.1007/s10260-015-0344-5
[16]	A. Bibi, A. Ghezal, Consistency of quasi-maximum likelihood estimator for Markov-switching bilinear time series models, Stat. Probab. Lett., 100 (2015), 192–202. https://doi.org/10.1016/j.spl.2015.02.010 doi: 10.1016/j.spl.2015.02.010
[17]	S. Elaydi, An Introduction to Difference Equations, Springer, New York, 2005.
[18]	E. A. Grove, G. Ladas, Periodicities in Nonlinear Difference Equations, Chapman and Hall/CRC, New York, 2005. https://doi.org/10.1201/9781420037722
[19]	V. L. Kocic, G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Academic Publishers, Dordrecht, 1993.
[20]	M. R. S. Kulenovic, G. Ladas, Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures, Chapman Hall-CRC, New York, 2002. https://doi.org/10.1201/9781420035384
[21]	A. Ghezal, Note on a rational system of (4k+4)-order difference equations: periodic solution and convergence, J. Appl. Math. Comput., 69 (2023), 2207–2215. https://doi.org/10.1007/s12190-022-01830-y doi: 10.1007/s12190-022-01830-y
[22]	E. M. Elsayed, Solution and attractivity for a rational recursive sequence, Discrete Dyn. Nat. Soc., 2011 (2011), 982309. https://doi.org/10.1155/2011/982309 doi: 10.1155/2011/982309
[23]	A. Ghezal, M. Balegh, I. Zemmouri, Solutions and local stability of the Jacobsthal system of difference equations, AIMS Math., 9 (2024), 3576–3591. https://doi.org/10.3934/math.2024175 doi: 10.3934/math.2024175
[24]	D. Simșek, B. Oğul, C. Çınar, Solution of the rational difference equation $x_{n+1} = x_{n-17}/\left(1+x_{n-5}.x_{n-11}\right)$, Filomat, 33 (2019), 1353–1359. https://doi.org/10.2298/FIL1905353S doi: 10.2298/FIL1905353S
[25]	D. T. Tollu, Y. Yazlik, N. Taskara, On the solutions of two special types of Riccati difference equation via Fibonacci numbers, Adv. Differ. Equations, 2013 (2013). https://doi.org/10.1186/1687-1847-2013-174 doi: 10.1186/1687-1847-2013-174
[26]	Y. Zhang, X. Yang, G. M. Megson, D. J. Evans, On the system of rational difference equations, Appl. Math. Comput., 176 (2006), 403–408. https://doi.org/10.1016/j.amc.2005.09.039 doi: 10.1016/j.amc.2005.09.039
[27]	Q. Zhang, L Yang, J. Liu, Dynamics of a system of rational third order difference equation, Adv. Differ. Equations, 2012 (2012). https://doi.org/10.1186/1687-1847-2012-136 doi: 10.1186/1687-1847-2012-136
[28]	İ. Okumuș, Y. Soykan, On the solutions of systems of difference equations via Tribonacci numbers, preprint, arXiv: 1906.09987. https://doi.org/10.48550/arXiv.1906.09987
[29]	R. F. Zhang, M. C. Li, Bilinear residual network method for solving the exactly explicit solutions of nonlinear evolution equations, Nonlinear Dyn., 108 (2022), 521–531. https://doi.org/10.1007/s11071-022-07207-x doi: 10.1007/s11071-022-07207-x
[30]	R. F. Zhang., M. C. Li, H. M. Yin, Rogue wave solutions and the bright and dark solitons of the (3+1)-dimensional Jimbo-Miwa equation, Nonlinear Dyn., 103 (2021), 1071–1079. https://doi.org/10.1007/s11071-020-06112-5 doi: 10.1007/s11071-020-06112-5
[31]	R. F. Zhang, S. Bilige, T. Chaolu, Fractal solitons, arbitrary function solutions, exact periodic wave and breathers for a nonlinear partial differential equation by using bilinear neural network method, J. Syst. Sci. Complex, 34 (2021), 122–139. https://doi.org/10.1007/s11424-020-9392-5 doi: 10.1007/s11424-020-9392-5
[32]	R. F. Zhang, S. Bilige, Bilinear neural network method to obtain the exact analytical solutions of nonlinear partial differential equations and its application to p-gBKP equation, Nonlinear Dyn., 95 (2019), 3041–3048. https://doi.org/10.1007/s11071-018-04739-z doi: 10.1007/s11071-018-04739-z
[33]	R. Abo-Zeid, Global behavior of two third order rational difference equations with quadratic terms, Math. Slovaca, 69 (2019), 147–158. https://doi.org/10.1515/ms-2017-0210 doi: 10.1515/ms-2017-0210
[34]	E. M. Elsayed, A. Alshareef, F. Alzahrani, Qualitative behavior and solution of a system of three-dimensional rational difference equations, Math. Methods Appl. Sci., 45 (2022), 5456–5470. https://doi.org/10.1002/mma.8120 doi: 10.1002/mma.8120
[35]	E. M. Elsayed, B. S. Aloufi, O. Moaaz, The behavior and structures of solution of fifth-order rational recursive sequence, Symmetry, 14 (2022), 641. https://doi.org/10.3390/sym14040641 doi: 10.3390/sym14040641
[36]	E. M. Elsayed., F. Alzahrani, H. S. Alayachi, Global attractivity and the periodic nature of third order rational difference equation, J. Comput. Anal. Appl., 23 (2017), 1230–1241.
[37]	M. Kara, Investigation of the global dynamics of two exponential-form difference equations systems, Electron. Res. Arch., 31 (2023), 6697–6724. http://doi.org/10.3934/era.2023338 doi: 10.3934/era.2023338
[38]	M. Berkal, R. Abo-Zeid, On a rational (p+1)th order difference equation with quadratic term, Univ. J. Math. Appl., 5 (2022), 136–144. https://doi.org/10.32323/ujma.198471 doi: 10.32323/ujma.198471
[39]	A. Ghezal, I. Zemmouri, Solution forms for generalized hyperbolic cotangent type systems of p-difference equations, Bol. Soc. Paran. Mat., in Press.
[40]	A. Ghezal, I. Zemmouri, Global stability of a multi-dimensional system of rational difference equations of higher-order with Pell-coeffcients, Bol. Soc. Paran. Mat., in Press.
[41]	A. Ghezal, I. Zemmouri, Higher-order system of p-nonlinear difference equations solvable in closed-form with variable coefficients, Bol. Soc. Paran. Mat., 41 (2022). https://doi.org/10.5269/bspm.63529 doi: 10.5269/bspm.63529
[42]	G. K. Panda, Sequence balancing and cobalancing numbers, Fibonacci Q., 45 (2007), 265–271.
[43]	G. K. Panda, P. K. Ray, Cobalancing numbers and cobalancers, Int. J. Math. Math. Sci., 2005 (2005), 1189–1200. http://doi.org/10.1155/S0161171205405116 doi: 10.1155/S0161171205405116
[44]	A. Behera, G. Panda, On the square roots of triangular numbers, Fibonacci Q., 37 (1999), 98–105.
[45]	R. F. Zhang, M. C. Li, A. Cherraf, S. R. Vadyala, The interference wave and the bright and dark soliton for two integro-differential equation by using BNNM, Nonlinear Dyn., 111 (2023), 8637–8646. https://doi.org/10.1007/s11071-023-08257-5 doi: 10.1007/s11071-023-08257-5
[46]	R. F. Zhang, M. C. Li, J. Y. Gan, Q. Li, Z. Z. Lan, Novel trial functions and rogue waves of generalized breaking soliton equation via bilinear neural network method, Chaos, Solitons Fractals, 154 (2022), 111692. https://doi.org/10.1016/j.chaos.2021.111692 doi: 10.1016/j.chaos.2021.111692
[47]	R. F. Zhang, M. C. Li, M. Albishari, F. C. Zheng, Z. Z. Lan, Generalized lump solutions, classical lump solutions and rogue waves of the (2+1)-dimensional Caudrey-Dodd-Gibbon-Kotera-Sawada-like equation, Appl. Math. Comput., 403 (2021), 126201. https://doi.org/10.1016/j.amc.2021.126201 doi: 10.1016/j.amc.2021.126201
[48]	R. F. Zhang, S. Bilige, J. G. Liu, M. Li, Bright-dark solitons and interaction phenomenon for p-gBKP equation by using bilinear neural network method, Phys. Scr., 96 (2021), 025224. https://doi.org/10.1088/1402-4896/abd3c3 doi: 10.1088/1402-4896/abd3c3

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