Global asymptotic stability and qualitative analysis of fuzzy difference equations with composite exponential saturation and parabolic fuzzy parameters

Xiong Xiao; Qianhong Zhang; Xiong Xiao; Qianhong Zhang

doi:10.3934/math.2026661

AIMS Mathematics

2026, Volume 11, Issue 6: 16063-16094. doi: 10.3934/math.2026661

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Research article

Global asymptotic stability and qualitative analysis of fuzzy difference equations with composite exponential saturation and parabolic fuzzy parameters

Xiong Xiao ^1,2,
Qianhong Zhang ^{1,2
,
,}

1.
School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang 550025, China
2.
School of Big Data Statistics, Guizhou University of Finance and Economics, Guiyang 550025, China

Received: 09 March 2026 Revised: 24 May 2026 Accepted: 27 May 2026 Published: 05 June 2026
MSC : 39A10

This paper investigates the dynamic behavior of a class of nonlinear fuzzy difference equations with compound exponential terms. By using g-division, Lyapunov stability analysis, algebraic inequalities, and mathematical induction, the existence and uniqueness of positive fuzzy solutions are proved. In addition, the boundedness and persistence of these solutions are established, along with the global asymptotic stability of the unique positive equilibrium. Finally, numerical simulations are performed to verify the theoretical results. These findings refine the analysis framework for fuzzy systems with compound nonlinear terms. They also have potential applications in biological population control, economic resource allocation, and engineering systems with uncertain parameters.
- fuzzy difference equations,
- composite exponential terms,
- global asymptotic stability,
- Lyapunov stability,
- g-division
Citation: Xiong Xiao, Qianhong Zhang. Global asymptotic stability and qualitative analysis of fuzzy difference equations with composite exponential saturation and parabolic fuzzy parameters[J]. AIMS Mathematics, 2026, 11(6): 16063-16094. doi: 10.3934/math.2026661

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Abstract

This paper investigates the dynamic behavior of a class of nonlinear fuzzy difference equations with compound exponential terms. By using g-division, Lyapunov stability analysis, algebraic inequalities, and mathematical induction, the existence and uniqueness of positive fuzzy solutions are proved. In addition, the boundedness and persistence of these solutions are established, along with the global asymptotic stability of the unique positive equilibrium. Finally, numerical simulations are performed to verify the theoretical results. These findings refine the analysis framework for fuzzy systems with compound nonlinear terms. They also have potential applications in biological population control, economic resource allocation, and engineering systems with uncertain parameters.

References

[1]	Q. Zhang, L. Yang, J. Liu, Dynamics of a system of rational third-order difference equation, Adv. Differ. Equ., 2012 (2012), 136. https://doi.org/10.1186/1687-1847-2012-136 doi: 10.1186/1687-1847-2012-136
[2]	Q. Din, Global behavior of a plant-herbivore model, Adv. Differ. Equ., 2015 (2015), 119. https://doi.org/10.1186/s13662-015-0458-y doi: 10.1186/s13662-015-0458-y
[3]	M. I. Chaudhry, M. J. Miranda, Complex price dynamics in vertically linked cobweb markets, Econ. Model., 72 (2018), 363–378. https://doi.org/10.1016/j.econmod.2018.02.012 doi: 10.1016/j.econmod.2018.02.012
[4]	S. H. Streipert, G. S. K. Wolkowicz, An alternative delayed population growth difference equation model, J. Math. Biol., 83 (2021), 25. https://doi.org/10.1007/s00285-021-01652-9 doi: 10.1007/s00285-021-01652-9
[5]	Z. Y. He, A. Abbes, H. Jahanshahi, N. D. Alotaibi, Y. Wang, Fractional-order discrete-time SIR epidemic model with vaccination: Chaos and complexity, Mathematics, 10 (2022), 165. https://doi.org/10.3390/math10020165 doi: 10.3390/math10020165
[6]	T. Liu, H. Zhang, S. Wang, A new high-order compact CN-ADI scheme on graded meshes for three-dimensional nonlinear PIDEs with multiple weakly singular kernels, Appl. Math. Lett., 171 (2025), 109697. https://doi.org/10.1016/j.aml.2025.109697 doi: 10.1016/j.aml.2025.109697
[7]	T. N. Kaye, D. A. Pyke, The effect of stochastic technique on estimates of population viability from transition matrix models, Ecology, 84 (2003), 1464–1476. https://doi.org/10.1890/0012-9658(2003)084[1464:TEOSTO]2.0.CO;2 doi: 10.1890/0012-9658(2003)084[1464:TEOSTO]2.0.CO;2
[8]	G. J. Klir, B. Yuan, Fuzzy sets and fuzzy logic: Theory and applications, Pearson College Div, 1995.
[9]	L. A. Zadeh, Fuzzy sets, Inf. Control, 8 (1965), 338–353. https://doi.org/10.1016/S0019-9958(65)90241-X
[10]	G. Papaschinopoulos, B. K. Papadopoulos, On the fuzzy difference equation $ x_{n+1} = A+x_n/x_{n-m} $, Fuzzy Sets Syst., 129 (2002), 73–81. https://doi.org/10.1016/S0165-0114(01)00198-1 doi: 10.1016/S0165-0114(01)00198-1
[11]	E. Y. Deeba, A. D. Korvin, Analysis by fuzzy difference equations of a model of $ CO_2 $ level in the blood, Appl. Math. Lett., 12 (1999), 33–40. https://doi.org/10.1016/S0893-9659(98)00168-2 doi: 10.1016/S0893-9659(98)00168-2
[12]	G. Papaschinopoulos, B. K. Papadopoulos, On the fuzzy difference equation $ x_{n+1} = A+B/x_n $, Soft Comput., 6 (2002), 456–461. http://dx.doi.org/10.1007/s00500-001-0161-7 doi: 10.1007/s00500-001-0161-7
[13]	J. Zhang, B. Zhou, H. Zhang, D. Yang, Fully distributed Event-Triggered secondary control for islanded microgrid restoration with communication link faults, IEEE Trans. Autom. Sci. Eng., 23 (2026), 2465–2474. https://doi.org/10.1109/tase.2025.3650604 doi: 10.1109/tase.2025.3650604
[14]	A. Calpbinici, E. Irmak, E. Kabalci, Design and implementation of an energy management system with Event-Triggered distributed secondary control in DC microgrids, Energies, 17 (2024), 662. https://doi.org/10.3390/en17030662 doi: 10.3390/en17030662
[15]	Y. Chen, D. Qi, Z. Li, Z. Wang, X. Yang, J. Zhang, Distributed event-triggered control for frequency restoration in islanded microgrids with reduced trigger condition checking, CSEE J. Power Energy Syst., 10 (2024), 2156–2165. https://doi.org/10.17775/cseejpes.2020.05730 doi: 10.17775/cseejpes.2020.05730
[16]	H. El-Metwally, E. A. Grove, G. Ladas, R. Levins, M. Radin, On the difference equation $ x_{n+1} = \alpha+\beta x_{n-1}e^{-x_n} $, Nonlinear Anal., 47 (2001), 4623–4634. https://doi.org/10.1016/S0362-546X(01)00575-2 doi: 10.1016/S0362-546X(01)00575-2
[17]	I. Ozturk, F. Bozkurt, S. Ozen, On the difference equation $ y_{n+1} = \frac{\alpha+\beta e^{-y_n}}{\gamma+y_{n-1}} $, Appl. Math. Comput., 181 (2006), 1387–1393. https://doi.org/10.1016/j.amc.2006.03.007 doi: 10.1016/j.amc.2006.03.007
[18]	I. Ozturk, F. Bozkurt, S. Ozen, Global asymptotic behavior of the difference equation $ y_{n+1} = \frac{\alpha e^{- (ny_n + (n-k)y_{n-k})}}{\beta + ny_n + (n-k)y_{n-k}} $, Appl. Math. Lett., 22 (2009), 595–599. https://doi.org/10.1016/j.aml.2008.06.037 doi: 10.1016/j.aml.2008.06.037
[19]	F. Bozkurt, Stability analysis of a nonlinear difference equation, Int. J. Mod. Nonlinear Theory Appl., 2 (2013), 1–6. https://doi.org/10.4236/ijmnta.2013.21001 doi: 10.4236/ijmnta.2013.21001
[20]	A. Q. Khan, M. S. M. Noorani, H. S. Alayachi, Global dynamics of higher-order exponential systems of difference equations, Discrete Dyn. Nat. Soc., 2019 (2019), 3825927. https://doi.org/10.1155/2019/3825927 doi: 10.1155/2019/3825927
[21]	S. C. Babu, D. S. Dilip, S. M. Mathew, Behavior of solutions of a discrete population model with mutualistic interaction, Comput. Math. Biophys., 12 (2024), 20230121. https://doi.org/10.1515/cmb-2023-0121 doi: 10.1515/cmb-2023-0121
[22]	G. Stefanidou, G. Papaschinopoulos, C. J. Schinas, On an exponential-type fuzzy difference equation, Adv. Differ. Equ., 2010 (2010), 196920. https://doi.org/10.1155/2010/196920 doi: 10.1155/2010/196920
[23]	Q. Zhang, L. Yang, D. Liao, On first order fuzzy Ricatti difference equation, Inf. Sci., 270 (2014), 226–236. https://doi.org/10.1016/j.ins.2014.02.086 doi: 10.1016/j.ins.2014.02.086
[24]	Q. Zhang, W. Zhang, F. Lin, D. Li, On dynamic behavior of second-order exponential-type fuzzy difference equation, Fuzzy Sets Syst., 419 (2021), 169–187. https://doi.org/10.1016/j.fss.2020.07.021 doi: 10.1016/j.fss.2020.07.021
[25]	K. Lin, Q. Zhang, Dynamical behavior of a second-Order exponential-type fuzzy difference equation with quadratic term, J. Math. Interdiscip. Appl., 1 (2025), 29–50. http://dx.doi.org/10.62762/JMIA.2025.999827 doi: 10.62762/JMIA.2025.999827
[26]	K. Lin, Q. Zhang, Dynamical analysis of a nonlinear fuzzy difference equation with exponential terms, J. Nonlinear Math. Phys., 32 (2025), 63. https://doi.org/10.1007/s44198-025-00318-0 doi: 10.1007/s44198-025-00318-0
[27]	S. Sand, C. Cattani, M. Tanveer, M. Usman, A. Tassaddiq, A study of global dynamics and oscillatory behavior of rational-type nonlinear fuzzy difference equations with exponential decay, Axioms, 14 (2025), 637. https://doi.org/10.3390/axioms14080637 doi: 10.3390/axioms14080637
[28]	S. Atpinar, Y. Yazlik, Qualitative behavior of exponential type of fuzzy difference equations system, J. Appl. Math. Comput., 69 (2023), 4135–4162. https://doi.org/10.1007/s12190-023-01919-y doi: 10.1007/s12190-023-01919-y
[29]	S. Atpinar, Y. Yazlik, Q. Zhang, The dynamics of a second-order symmetric system of exponential fuzzy difference equations, Comput. Appl. Math., 45 (2026), 78. https://doi.org/10.1007/s40314-025-03417-2 doi: 10.1007/s40314-025-03417-2
[30]	H. Garg, Ansha, Arithmetic operations on generalized parabolic fuzzy numbers and its application, Proc. Natl. Acad. Sci. India Sect. A Phys. Sci., 88 (2018), 15–26. https://doi.org/10.1007/s40010-016-0278-9 doi: 10.1007/s40010-016-0278-9
[31]	M. Jagadeeswari, V. L. GomathiNayagam, Approximation of parabolic fuzzy numbers, Front. Artif. Intell. Appl., 299 (2017), 107–124. https://doi.org/10.3233/978-1-61499-828-0-107 doi: 10.3233/978-1-61499-828-0-107
[32]	D. Dubois, H. Prade, Possibility theory: An approach to computerized processing of uncertainty, New York: Plenum Publishing Corporation, 1988.
[33]	C. Wu, B. Zhang, Embedding problem of noncompact fuzzy number space $ E^\sim $ I, Fuzzy Sets Syst., 105 (1999), 165–169. https://doi.org/10.1016/S0165-0114(97)00218-2 doi: 10.1016/S0165-0114(97)00218-2
[34]	L. Stefanini, A generalization of Hukuhara difference and division for interval and fuzzy arithmetic, Fuzzy Sets Syst., 161 (2010), 1564–1584. https://doi.org/10.1016/j.fss.2009.06.009 doi: 10.1016/j.fss.2009.06.009
[35]	V. L. Kocic, G. Ladas, Global behavior of nonlinear difference equations of higher order with applications, Dordrecht: Springer, 1993. https://doi.org/10.1007/978-94-017-1703-8

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