Hopf bifurcation of uncertain nonlinear differential equations

Xiuying Guo; Caiyun Huang; Qiubao Wang; Zikun Han; Zeman Wang; Xiyuan Chen; Xiuying Guo; Caiyun Huang; Qiubao Wang; Zikun Han; Zeman Wang; Xiyuan Chen

doi:10.3934/math.20251242

AIMS Mathematics

2025, Volume 10, Issue 12: 28243-28263. doi: 10.3934/math.20251242

Previous Article Next Article

Research article

Hopf bifurcation of uncertain nonlinear differential equations

1.
Department of Mathematics and Physics, Shijiazhuang Tiedao University, Shijiazhuang, China
2.
Department of Engineering Mechanics, Shijiazhuang Tiedao University, Shijiazhuang, China

Received: 30 September 2025 Revised: 18 November 2025 Accepted: 21 November 2025 Published: 01 December 2025
MSC : 34A12, 47E05

It is reasonable to use uncertain nonlinear differential equations to model systems that are subject to noise disturbances of unstable frequency, such as uncertain financial systems, biological population systems, infectious disease systems, and pharmacokinetic systems. Due to the coupling effect of nonlinearity and uncertainty, it is challenging to directly solve the responses of these equations. This paper addressed the challenge of studying the responses of these systems, especially the changes in steady-state behavior caused by parameter variations, known as "bifurcation" phenomena. We defined the concept of Hopf bifurcation in uncertain differential equations using cross-entropy and investigated the bifurcation phenomena in a class of second-order uncertain nonlinear differential equations. An efficient algorithm was designed to verify uncertain Hopf bifurcation and quantify the bifurcation threshold, with the validity of our definition confirmed through numerical simulations. This paper extended the classical Hopf bifurcation of ordinary differential equations to uncertain differential equations via the $ \alpha $-path, thereby proposing a theoretical framework for uncertain bifurcation within uncertain dynamics.
- Hopf bifurcation,
- uncertain nonlinear differential equation,
- cross-entropy
Citation: Xiuying Guo, Caiyun Huang, Qiubao Wang, Zikun Han, Zeman Wang, Xiyuan Chen. Hopf bifurcation of uncertain nonlinear differential equations[J]. AIMS Mathematics, 2025, 10(12): 28243-28263. doi: 10.3934/math.20251242

Related Papers:

Abstract

It is reasonable to use uncertain nonlinear differential equations to model systems that are subject to noise disturbances of unstable frequency, such as uncertain financial systems, biological population systems, infectious disease systems, and pharmacokinetic systems. Due to the coupling effect of nonlinearity and uncertainty, it is challenging to directly solve the responses of these equations. This paper addressed the challenge of studying the responses of these systems, especially the changes in steady-state behavior caused by parameter variations, known as "bifurcation" phenomena. We defined the concept of Hopf bifurcation in uncertain differential equations using cross-entropy and investigated the bifurcation phenomena in a class of second-order uncertain nonlinear differential equations. An efficient algorithm was designed to verify uncertain Hopf bifurcation and quantify the bifurcation threshold, with the validity of our definition confirmed through numerical simulations. This paper extended the classical Hopf bifurcation of ordinary differential equations to uncertain differential equations via the $ \alpha $-path, thereby proposing a theoretical framework for uncertain bifurcation within uncertain dynamics.

References

[1]	I. M. Elbaz, M. A. Sohaly, H. El-Metwally, Random dynamics of an SIV epidemic model, Commun. Nonlinear Sci. Numer. Simul., 131 (2024), 107779. https://doi.org/10.1016/j.cnsns.2023.107779 doi: 10.1016/j.cnsns.2023.107779
[2]	R. Ceccon, G. Livieri, S. Marmi, The Yoccoz Birkeland livestock population model coupled with random price dynamics, Commun. Nonlinear Sci. Numer. Simul., 118 (2023), 106982. https://doi.org/10.1016/j.cnsns.2022.106982 doi: 10.1016/j.cnsns.2022.106982
[3]	C. Fan, J. Wu, T. Yao, H. Xiao, J. Xu, J. Leng, et al., Investigation of pump scheme on the dynamics of brightness-enhanced random Raman fiber lasers, Opt. Laser Technol., 172 (2024), 110507. https://doi.org/10.1016/j.optlastec.2023.110507 doi: 10.1016/j.optlastec.2023.110507
[4]	J. Lin, P. B. Kahn, Random effects in population models with hereditary effects, J. Math. Biol., 10 (1980), 101–112. https://doi.org/10.1007/BF00275836 doi: 10.1007/BF00275836
[5]	D. Kahneman, A. Tversky, Prospect theory: An analysis of decisions under risk, Econometrica, 47 (1979), 263–291. https://doi.org/10.1017/CBO9780511609220.014 doi: 10.1017/CBO9780511609220.014
[6]	B. Liu, Uncertainty theory, 4 Eds., Berlin, Heidelberg: Springer, 2015. https://doi.org/10.1007/978-3-662-44354-5
[7]	J. B. Walsh, An introduction to stochastic partial differential equations, In: École d'Été de Probabilités de Saint Flour XIV - 1984, 1180 (1986), 265–439. https://doi.org/10.1007/BFb0074920
[8]	P. L. Chow, Generalized solution of some parabolic equations with a random drift, Appl. Math. Optim., 20 (1989), 1–17. https://doi.org/10.1007/bf01447642 doi: 10.1007/bf01447642
[9]	K. Peter, Existence, uniqueness and smoothness for a class of function valued stochastic partial differential equations, Stoch. Stoch. Rep., 41 (1992), 177–199. https://doi.org/10.1080/17442509208833801 doi: 10.1080/17442509208833801
[10]	S. Peszat, J. Zabczyk, Stochastic evolution equations with a spatially homogeneous Wiener process, Stoch. Proc. Appl., 72 (1997), 187–204. https://doi.org/10.1016/s0304-4149(97)00089-6 doi: 10.1016/s0304-4149(97)00089-6
[11]	X. Yang, K. Yao, Uncertain partial differential equation with application to heat conduction, Fuzzy Optim. Decis. Making, 16 (2017), 379–403. https://doi.org/10.1007/s10700-016-9253-9 doi: 10.1007/s10700-016-9253-9
[12]	Z. Peng, K. Iwamura, A sufficient and necessary condition of uncertainty distribution, J. Interdiscip. Math., 13 (2010), 277–285. https://doi.org/10.1080/09720502.2010.10700701 doi: 10.1080/09720502.2010.10700701
[13]	B. Liu, Some research problems in uncertainty theory, J. Uncertain Syst., 3 (2009), 3–10.
[14]	B. Liu, Uncertain reliability analysis, In: Uncertainty theory: A Branch of mathematics for modeling human uncertainty, 300 (2010), 125–130. https://doi.org/10.1007/978-3-642-13959-8_4
[15]	B. Liu, Fuzzy process, hybrid process and uncertain process, J. Uncertain Syst., 2 (2008), 3–16.
[16]	X. Chen, B. Liu, Existence and uniqueness theorem for uncertain differential equations, Fuzzy Optim. Decis. Making, 9 (2010), 69–81. https://doi.org/10.1007/s10700-010-9073-2 doi: 10.1007/s10700-010-9073-2
[17]	K. Yao, J. Gao, Y. Gao, Some stability theorems of uncertain differential equation, Fuzzy Optim. Decis. Making, 12 (2013), 3–13. https://doi.org/10.1007/s10700-012-9139-4 doi: 10.1007/s10700-012-9139-4
[18]	K. Yao, A type of uncertain differential equations with analytic solution, J. Uncertain. Anal. Appl., 1 (2013), 8. https://doi.org/10.1186/2195-5468-1-8 doi: 10.1186/2195-5468-1-8
[19]	Y. Liu, An analytic method for solving uncertain differential equations, J. Uncertain Syst., 6 (2012), 244–249.
[20]	Y. Sheng, J. Gao, Exponential stability of uncertain differential equation, Soft Comput., 20 (2016), 3673–3678. https://doi.org/10.1007/s00500-015-1727-0 doi: 10.1007/s00500-015-1727-0
[21]	K. Yao, H. Ke, Y. Sheng, Stability in mean for uncertain differential equation, Fuzzy Optim. Decis. Making, 14 (2015), 365–379. https://doi.org/10.1007/s10700-014-9204-2 doi: 10.1007/s10700-014-9204-2
[22]	L. Yang, Y. Liu, Solution method and parameter estimation of uncertain partial diferential equation with application to China's population, Fuzzy Optim. Decis. Making, 23 (2024), 155–177. https://doi.org/10.1007/s10700-023-09415-5 doi: 10.1007/s10700-023-09415-5
[23]	Y. Liu, B. Liu, Estimating unknown parameters in uncertain differential equation by maximum likelihood estimation, Soft Comput., 26 (2022), 2773–2780. https://doi.org/10.1007/s00500-022-06766-w doi: 10.1007/s00500-022-06766-w
[24]	Z. M. Wang, Z. Liu, Z. K. Han, X. Y. Guo, Q. B. Wang, The inverse uncertainty distribution of the solutions to a class of higher-order uncertain differential equations, AIMS Mathematics, 9 (2024), 33023–33061. https://doi.org/10.3934/math.20241579 doi: 10.3934/math.20241579
[25]	K. Yao, X. Chen, A numerical method for solving uncertain differential equations, J. Intell. Fuzzy Syst., 25 (2013), 825–832. https://doi.org/10.3233/ifs-120688 doi: 10.3233/ifs-120688
[26]	K. Yao, Extreme values and integral of solution of uncertain differential equation, J. Uncertain. Anal. Appl., 1 (2013), 2. https://doi.org/10.1186/2195-5468-1-2 doi: 10.1186/2195-5468-1-2
[27]	X. Yang, Y. Shen, Runge-Kutta method for solving uncertain differential equations, J. Uncertain. Anal. Appl., 3 (2015), 17. https://doi.org/10.1186/s40467-015-0038-4 doi: 10.1186/s40467-015-0038-4
[28]	X. Yang, D. A. Ralescu, Adams method for solving uncertain differential equations, Appl. Math. Comput., 270 (2015), 993–1003. https://doi.org/10.1016/j.amc.2015.08.109 doi: 10.1016/j.amc.2015.08.109
[29]	X. Wang, Y. Ning, T. A. Moughal, X. Chen, Adams Simpson method for solving uncertain differential equation, Appl. Math. Comput., 271 (2015), 209–219. https://doi.org/10.1016/j.amc.2015.09.009 doi: 10.1016/j.amc.2015.09.009
[30]	R. Gao, Milne method for solving uncertain differential equations, Appl. Math. Comput., 274 (2016), 774–785. https://doi.org/10.1016/j.amc.2015.11.043 doi: 10.1016/j.amc.2015.11.043
[31]	K. Yao, Uncertain differential equations, Berlin, Heidelberg: Springer, 2016. https://doi.org/10.1007/978-3-662-52729-0
[32]	Y. Hou, L. Wu, Y. Sheng, Solving high-order uncertain differential equations via Adams–Simpson method, Comput. Appl. Math., 40 (2021), 252. https://doi.org/10.1007/s40314-020-01408-z doi: 10.1007/s40314-020-01408-z
[33]	X. Ji, J. Zhou, Solving high-order uncertain differential equations via Runge–Kutta method, IEEE Trans. Fuzzy Syst., 26 (2017), 1379–1386. https://doi.org/10.1109/tfuzz.2017.2723350 doi: 10.1109/tfuzz.2017.2723350
[34]	J. Kuang, M. Wang, J. Han, Y. Sheng, Improved Milne-Hamming method for resolving high-order uncertain differential equations, Appl. Math. Comput., 457 (2023), 128199. https://doi.org/10.1016/j.amc.2023.128199 doi: 10.1016/j.amc.2023.128199
[35]	J. E. Marsden, M. McCracken, The Hopf bifurcation and its applications, New York: Springer, 1976. https://doi.org/10.1007/978-1-4612-6374-6
[36]	G. Iooss, Bifurcation of maps and applications, Elsevier, 1979.
[37]	S. Wiggins, Introduction to applied nonlinear dynamical systems and chaos, New York: Springer, 2003. https://doi.org/10.1007/b97481
[38]	X. F. Liao, G. R. Chen, Hopf bifurcation and chaos analysis of Chen's system with distributed delays, Chaos Soliton Fract., 25 (2005), 197–220. https://doi.org/10.1016/j.chaos.2004.11.007 doi: 10.1016/j.chaos.2004.11.007
[39]	Q. B. Wang, Z. K. Han, X. Zhang, Y. J. Yang, Dynamics of the delay-coupled bubble system combined with the stochastic term, Chaos Soliton Fract., 148 (2021), 111053. https://doi.org/10.1016/j.chaos.2021.111053 doi: 10.1016/j.chaos.2021.111053
[40]	W. Q. Zhu, Z. L. Huang, Stochastic Hopf bifurcation of quasi-nonintegrable-Hamiltonian systems, Int. J. Non-Linear Mech., 34 (1999), 437–447. https://doi.org/10.1016/S0020-7462(98)00026-2 doi: 10.1016/S0020-7462(98)00026-2
[41]	X. Gao, L. F. Jia, S. Kar, A new definition of cross-entropy for uncertain variables, Soft Comput., 22 (2018), 5617–5623. https://doi.org/10.1007/s00500-017-2534-6 doi: 10.1007/s00500-017-2534-6

Reader Comments

Your name:*

Email:*
© 2025 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)