Exploring dynamics and pattern formation of a fractional-order three-variable Oregonator model

Jia Yu Yang; Yu Lan Wang; Zhi Yuan Li; Jia Yu Yang; Yu Lan Wang; Zhi Yuan Li

doi:10.3934/nhm.2025052

Networks and Heterogeneous Media

2025, Volume 20, Issue 4: 1201-1229. doi: 10.3934/nhm.2025052

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Research article

Exploring dynamics and pattern formation of a fractional-order three-variable Oregonator model

1.
Department of Mathematics, Inner Mongolia University of Technology, Hohhot 010051, China
2.
College of Date Science and Application, Inner Mongolia University of Technology, Hohhot 010080, China

Received: 30 July 2025 Revised: 17 October 2025 Accepted: 03 November 2025 Published: 11 November 2025

In this paper, we investigated the nonlinear dynamics and pattern formation of a fractional-order three-variable Oregonator model. We first performed a linear stability analysis of the model without diffusion, deriving equilibrium points and Jacobian eigenvalues, and verified Matignon's stability conditions. A high-precision numerical scheme was developed, and simulations revealed that even tiny variations in fractional order produce significant changes in long-term trajectories. For the reaction-diffusion model, we analyzed Turing instability under integer-order diffusion and derived the critical wave-number conditions via Routh-Hurwitz criteria. Weakly nonlinear analysis near the Turing threshold yielded coupled amplitude equations whose coefficients predicted stripe, hexagon, and mixed patterns. Extensive two-dimensional numerical experiments confirmed the theoretical predictions: Depending on diffusion coefficients and other parameters, the model evolved into bullseye, spiral, labyrinthine, or spot-stripe mixtures.
- fractional Oregonator model,
- Grünwald–Letnikov derivative,
- turing instability,
- spatiotemporal patterns,
- weakly nonlinear analysis
Citation: Jia Yu Yang, Yu Lan Wang, Zhi Yuan Li. Exploring dynamics and pattern formation of a fractional-order three-variable Oregonator model[J]. Networks and Heterogeneous Media, 2025, 20(4): 1201-1229. doi: 10.3934/nhm.2025052

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Abstract

In this paper, we investigated the nonlinear dynamics and pattern formation of a fractional-order three-variable Oregonator model. We first performed a linear stability analysis of the model without diffusion, deriving equilibrium points and Jacobian eigenvalues, and verified Matignon's stability conditions. A high-precision numerical scheme was developed, and simulations revealed that even tiny variations in fractional order produce significant changes in long-term trajectories. For the reaction-diffusion model, we analyzed Turing instability under integer-order diffusion and derived the critical wave-number conditions via Routh-Hurwitz criteria. Weakly nonlinear analysis near the Turing threshold yielded coupled amplitude equations whose coefficients predicted stripe, hexagon, and mixed patterns. Extensive two-dimensional numerical experiments confirmed the theoretical predictions: Depending on diffusion coefficients and other parameters, the model evolved into bullseye, spiral, labyrinthine, or spot-stripe mixtures.

References

[1]	K. Sriram, S. Bernard, Complex dynamics in the Oregonator model with linear delayed feedback, Chaos, 18 (2008), 023126. https://doi.org/10.1063/1.2937015 doi: 10.1063/1.2937015
[2]	H. Mahara, T. Yamaguchi, Y. Morikawa, T. Amemiya, T. Yamamoto, H. Miike, et al., Forced excitations and excitable chaos in the photosensitive Oregonator under periodic sinusoidal perturbations, Physica D, 205 (2005), 275–282. https://doi.org/10.1016/j.physd.2005.01.014 doi: 10.1016/j.physd.2005.01.014
[3]	R. J. Field, R. M. Mazo, N. Manz, Science, serendipity, coincidence, and the Oregonator at the University of Oregon, 1969–1974, Chaos, 32 (2022), 052101. https://doi.org/10.1063/5.0087455 doi: 10.1063/5.0087455
[4]	K. Imaeda, T. Tei, Bifurcation and chaos caused by an external periodic force in the Oregonator of BZ reaction, J. Phys. Soc. Jpn., 55 (1986), 743–752. https://doi.org/10.1143/JPSJ.55.743 doi: 10.1143/JPSJ.55.743
[5]	C. P. Li, Z. Wang, Non-uniform L1/discontinuous Galerkin approximation for the time-fractional convection equation with weak regular solution, Math. Comput. Simul., 182 (2021), 838–857. https://doi.org/10.1016/j.matcom.2020.12.007 doi: 10.1016/j.matcom.2020.12.007
[6]	Z. Y. Li, Y. L. Wang, F. G. Tan, X. H. Wan, H. Yu, J. S. Duan, Solving a class of linear nonlocal boundary value problems using the reproducing kernel, Appl. Math. Comput., 265 (2015), 1098–1105. https://doi.org/10.1016/j.amc.2015.05.117 doi: 10.1016/j.amc.2015.05.117
[7]	I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
[8]	H. Che, Y. L. Wang, Z. Y. Li, Novel patterns in a class of fractional reaction-diffusion models with the Riesz fractional derivative, Math. Comput. Simul., 202 (2022), 149–163. https://doi.org/10.1016/j.matcom.2022.05.037 doi: 10.1016/j.matcom.2022.05.037
[9]	C. Han, Y. L. Wang, Numerical solutions of variable-coefficient fractional-in-space KdV equation with the Caputo fractional derivative, Fractal Fract., 6 (2022), 207. https://doi.org/10.3390/fractalfract6040207 doi: 10.3390/fractalfract6040207
[10]	X. L. Gao, Z. Y. Li, Y. L. Wang, Chaotic dynamic behavior of a fractional-order financial system with constant inelastic demand, Int. J. Bifurcation Chaos, 34 (2024), 2450111. https://doi.org/10.1142/S0218127424501116 doi: 10.1142/S0218127424501116
[11]	K. Diethelm, The analysis of fractional differential equations: An application-oriented exposition using differential operators of Caputo type, Lect. Notes Math., 2004.
[12]	K. Diethelm, N. J. Ford, Analysis of fractional differential equations, J. Math. Anal. Appl., 265 (2002), 229–248. https://doi.org/10.1006/jmaa.2000.7194 doi: 10.1006/jmaa.2000.7194
[13]	K. Diethelm, N. J. Ford, A. D. Freed, A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dyn., 29 (2002), 3–22. https://doi.org/10.1023/A:1016592219341 doi: 10.1023/A:1016592219341
[14]	C. P. Li, F. H. Zeng, Numerical Methods for Fractional Calculus, Chapman and Hall/CRC, Boca Raton, 2015.
[15]	C. P. Li, M. Cai, Theory and Numerical Approximations of Fractional Integrals and Derivatives, Society for Industrial and Applied Mathematics, Philadelphia, 2019.
[16]	H. L. Zhang, Y. L. Wang, J. X. Bi, S. H. Bao, Novel pattern dynamics in a vegetation-water reaction-diffusion model, Math. Comput. Simul., 241 (2026), 97–116. https://doi.org/10.1016/j.matcom.2025.09.020 doi: 10.1016/j.matcom.2025.09.020
[17]	S. Zhang, H. L. Zhang, Y. L. Wang, Z. Y. Li, Dynamic properties and numerical simulations of a fractional phytoplankton-zooplankton ecological model, Networks Heterogen. Media, 20 (2025), 648–669. https://doi.org/10.3934/nhm.2025028 doi: 10.3934/nhm.2025028
[18]	I. Petr'aš, Fractional-order Nonlinear Systems: Modeling, Analysis and Simulation, Springer Science & Business Media, Heidelberg, 2011.
[19]	C. Xu, M. Liao, M. Farman, A. Shehzad, Hydrogenolysis of glycerol by heterogeneous catalysis: A fractional order kinetic model with analysis, MATCH Commun. Math. Comput. Chem., 91 (2024), 635–664. https://doi.org/10.46793/match.91-3.635X doi: 10.46793/match.91-3.635X
[20]	C. Xu, M. Farman, Y. Pang, Z. Liu, M. Liao, L. Yao, et al., Mathematical analysis and dynamical transmission of SEIrIsR model with different infection stages by using fractional operator, Int. J. Biomath., 2025 (2025), 2450151. https://doi.org/10.1142/S1793524524501511 doi: 10.1142/S1793524524501511
[21]	C. G. Liu, J. L. Wang, Passivity of fractional-order coupled neural networks with multiple state/derivative couplings, Neurocomputing, 455 (2021), 379–389. https://doi.org/10.1016/j.neucom.2021.05.050 doi: 10.1016/j.neucom.2021.05.050
[22]	C. Jiang, A. Zada, M. T. Şenel, T. Li, Synchronization of bidirectional N-coupled fractional-order chaotic systems with ring connection based on antisymmetric structure, Adv. Differ. Equ., 2019 (2019), 1–16. https://doi.org/10.1186/s13662-019-2380-1 doi: 10.1186/s13662-019-2380-1
[23]	T. Jia, X. Chen, L. He, F. Zhao, J. Qiu, Finite-time synchronization of uncertain fractional-order delayed memristive neural networks via adaptive sliding mode control and its application, Fractal Fract., 6 (2022), 502. https://doi.org/10.3390/fractalfract6090502 doi: 10.3390/fractalfract6090502
[24]	Y. Zhao, Y. Sun, Z. Liu, Y. Wang, Solvability for boundary value problems of nonlinear fractional differential equations with mixed perturbations of the second type, AIMS Math., 5 (2020), 557–567. https://doi.org/10.3934/math.2020037 doi: 10.3934/math.2020037
[25]	Z. Liu, Y. Ding, C. Liu, C. Zhao, Existence and uniqueness of solutions for singular fractional differential equation boundary value problem with p-Laplacian, Adv. Differ. Equ., 2020 (2020), 83. https://doi.org/10.1186/s13662-019-2482-9 doi: 10.1186/s13662-019-2482-9
[26]	R. Xing, M. Xiao, Y. Zhang, J. Qiu, Stability and Hopf bifurcation analysis of an (n+ m)-neuron double-ring neural network model with multiple time delays, J. Syst. Sci. Complexity, 35 (2022), 159–178. https://doi.org/10.1007/s11424-021-0108-2 doi: 10.1007/s11424-021-0108-2
[27]	X. Du, M. Xiao, J. Qiu, Y. Lu, J. Cao, Stability and dynamics analysis of time-delay fractional-order large-scale dual-loop neural network model with cross-coupling structure, IEEE Trans. Neural Networks Learn. Syst., 36 (2024), 7873–7887. https://doi.org/10.1109/TNNLS.2024.3413366 doi: 10.1109/TNNLS.2024.3413366
[28]	H. Wang, M. Xiao, B. Tao, F. Xu, Z. Wang, C. Huang, et al., Improving dynamics of integer-order small-world network models under fractional-order PD control, Sci. China Inf. Sci., 63 (2020), 112206. https://doi.org/10.1007/s11432-018-9933-6 doi: 10.1007/s11432-018-9933-6
[29]	P. Li, R. Gao, C. Xu, Y. Li, A. Akgül, D. Baleanu, Dynamics exploration for a fractional-order delayed zooplankton–phytoplankton system, Chaos, Solitons Fractals, 166 (2023), 112975. https://doi.org/10.1016/j.chaos.2022.112975 doi: 10.1016/j.chaos.2022.112975
[30]	L. Yang, M. Dolnik, A. M. Zhabotinsky, I. R. Epstein, Pattern formation arising from interactions between Turing and wave instabilities, J. Chem. Phys., 117 (2002), 7259–7265. https://doi.org/10.1063/1.1507110 doi: 10.1063/1.1507110
[31]	R. Peng, F. Sun, Turing pattern of the Oregonator model, Nonlinear Anal., 72 (2010), 2337–2345. https://doi.org/10.1016/j.na.2009.10.034 doi: 10.1016/j.na.2009.10.034
[32]	Y. Jia, F. Feng, Y. Zhang, Y. He, Pattern formation induced by subdiffusion in Oregonator model, J. Hebei Univ. (Nat. Sci. Ed.), 37 (2017), 19–25.
[33]	C. Xu, C. Aouiti, Z. Liu, P. Li, L. Yao, Bifurcation caused by delay in a fractional-order coupled Oregonator model in chemistry, MATCH Commun. Math. Comput. Chem., 88 (2022), 371–396. https://doi.org/10.46793/match.88-2.371X doi: 10.46793/match.88-2.371X
[34]	C. Xu, W. Zhang, C. Aouiti, Z. X. Liu, L. Yao, Bifurcation dynamics in a fractional-order Oregonator model including time delay, MATCH Commun. Math. Comput. Chem., 87 (2022), 397–414. https://doi.org/10.46793/match.87-2.397X doi: 10.46793/match.87-2.397X
[35]	H. Li, Y. Yao, M. Xiao, Z. Wang, L. Rutkowski, Turing pattern dynamics in a fractional-diffusion Oregonator model under PD control, Nonlinear Anal. Model. Control, 30 (2025), 1–21. https://doi.org/10.15388/namc.2025.30.38967 doi: 10.15388/namc.2025.30.38967
[36]	F. C. Liu, X. F. Wang, Multi-armed spiral waves in three-component reaction diffusion systems, J. Hebei Univ. (Nat. Sci. Ed.), 29 (2009), 376–380.
[37]	C. P. Li, F. H. Zeng, Numerical Methods for Fractional Calculus, Chapman and Hall/CRC, Boca Raton, 2015.
[38]	D. Y. Xue, C. N. Zhao, Y. Q. Chen, A modified approximation method of fractional order system, in Proceedings of IEEE International Conference on Mechatronics and Automation, (2006), 1043–1048. https://doi.org/10.1109/ICMA.2006.257769
[39]	D. Y. Xue, L. Bai, Numerical algorithms for Caputo fractional-order differential equations, Int. J. Control, 90 (2017), 1201–1211. https://doi.org/10.1080/00207179.2016.1158419 doi: 10.1080/00207179.2016.1158419
[40]	D. Y. Xue, Fractional Calculus and Fractional-Order Control, Science Press, Beijing, 2018.
[41]	X. H. Wang, H. L. Zhang, Y. L. Wang, Z. Y. Li, Dynamic properties and numerical simulations of the fractional Hastings-Powell model with the Grünwald-Letnikov differential derivative, Int. J. Bifurcation Chaos, 35 (2025), 2550145. https://doi.org/10.1142/S0218127425501457 doi: 10.1142/S0218127425501457
[42]	M. Bodson, Explaining the Routh–Hurwitz criterion: A tutorial presentation, IEEE Control Syst. Mag., 40 (2020), 45–51. https://doi.org/10.1109/MCS.2019.2949974 doi: 10.1109/MCS.2019.2949974
[43]	Z. Bao, Q. Zhou, Z. Yang, Q. Yang, L. Xu, T. Wu, A multi time-scale and multi energy-type coordinated microgrid scheduling solution—Part I: Model and methodology, IEEE Trans. Power Syst., 30 (2014), 2257–2266. https://doi.org/10.1109/TPWRS.2014.2367127 doi: 10.1109/TPWRS.2014.2367127
[44]	S. Ghorai, S. Poria, Emergent impacts of quadratic mortality on pattern formation in a predator–prey system, Nonlinear Dyn., 87 (2017), 2715–2734. https://doi.org/10.1007/s11071-016-3222-2 doi: 10.1007/s11071-016-3222-2
[45]	S. Kumari, R. K. Upadhyay, P. Kumar, V. Rai, Dynamics and patterns of species abundance in ocean: A mathematical modeling study, Nonlinear Anal. Real World Appl., 60 (2021), 103303. https://doi.org/10.1016/j.nonrwa.2021.103303 doi: 10.1016/j.nonrwa.2021.103303
[46]	X. L. Gao, H. L. Zhang, X. Y. Li, Research on pattern dynamics of a class of predator-prey model with interval biological coefficients for capture, AIMS Math., 9 (2024), 18506–18527. https://doi.org/10.3934/math.2024901 doi: 10.3934/math.2024901

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