Abstract
Understanding the complexity of reaction mechanisms is pivotal for advancing scientific and engineering solutions. This study addresses the challenge of distinguishing between slow and fast species in high-dimensional reaction mechanism, where direct analytical solutions are intractable. Through a hybrid approach combining analytical derivations and computational techniques including numerical simulations via MATLAB toolboxes we elucidate the kinetic behavior of species, quantify parameter sensitivities and calculating Surface Bifurcation. This paper provides a comprehensive examination of complex reaction mechanisms. The findings based on the analysis of the elementary steps involved in product formation show that, the hydrogen-assisted propagation step offers the dominant net contribution, while the direct radical recombination step stays close to equilibrium. The relative availability of molecular hydrogen, reactive radicals, and product-driven reverse reactions control step dominance. The integration of computational modeling with analytical insights not only clarifies reaction dynamics but also facilitates the optimization of reaction conditions and the design of efficient analytical frameworks. This work highlights the critical role of model reduction strategies in demystifying complex systems, offering a robust foundation for future innovations in sustainable technology and industrial applications.
Data availability
The data sets used during the current study available from the corresponding author on reasonable request.
References
F. Sultan, W. A. Khan, M. Ali, M. Shahzad, F. Khan, M. Waqas, Slow invariant manifolds and its approximation in a multi-route reaction mechanism: a case study of iodized H2/O mechanism, J. Mol. Liquids, vol. 288, pp. 111048.
Fernandez, A. M. & Carta, G. Characterization of protein adsorption by composite silica-polyacrylamide gel anion exchangers I. Equilibrium and mass transfer in agitated contactors. J. Chromatogr. A 746(2), 169–183 (1996).
Maas, U. & Pope, S. B. Simplifying chemical kinetics: intrinsic low-dimensional manifolds in composition space. Combust. Flame 88(3–4), 239–264 (1992).
Bykov, V., Goldfarb, I., Gol’Dshtein, V. & Maas, U. On a modified version of ILDM approach: asymptotic analysis based on integral manifolds. IMA J. Appl. Math. 71(3), 359–382 (2006).
Chiavazzo, E., Gorban, A. N. & Karlin, I. V. Comparison of invariant manifolds for model reduction in chemical kinetics. Commun. Comput. Phys 2, 964–992 (2007).
Rehman, A. U., Shahzad, M. & Aldila, D. Comparison of invariant surfaces in multi-route reaction mechanisms. React. Kinet. Mech. Catal. 135(4), 1719–1737 (2022).
F. Sultan, M. S. Ishaq, Y. Elmasry, S. Knani, I. N. Cangul Computational modeling and analysis on chemically reacting species in catalytic oxidation reaction mechanism: Novel model reduction and NSFD scheme. Results Chem., 102814, 2025.
Shahzad, M., Sultan, F., Haq, I. & Wahab, A. H. Computing the low dimension manifold in dissipative dynamical systems. Nucleus 53(2), 107–113 (2016).
M. Shahzad and F. Sultan, "Complex reactions and dynamics, In Adv Chem Kinet, 2018.
J. Alvarez-Ramirez, M. Meraz and Martinez-Martine, Analysis of fast and slow dynamics of chemical kinetics using singular value decomposition., Reaction Kinetics, Mechanisms and Catalysis,, vol. 136, no. 2, pp. 567–586, 2023.
Bodenstein, M. Eine theorie der photochemischen reaktionsgeschwindigkeiten. Z. Phys. Chem. 85(1), 329–397 (1913).
Briggs, G. E. & Haldane, J. B. A note on the kinetics of enzyme action. Biochem. J. 19(2), 338 (1925).
Gorban, A. N. & Shahzad, M. The michaelis-menten-stueckelberg theorem. Entropy 13(5), 966–1019 (2011).
G. V. Yablonskii, V. I. Bykov and V. I. Elokhin, Kinetic models of catalytic reactions, Elsevier., 1991.
Fenichel, N. Geometric singular perturbation theory for ordinary differential equations. J. Differential Equations 31(1), 53–98 (1979).
Ciliberto, A., Capuani, F. & Tyson, J. J. Modeling networks of coupled enzymatic reactions using the total quasi-steady state approximation. PLoS Comput. Biol. 3(3), e45 (2007).
Rabitz, H. Chemical sensitivity analysis theory with applications to molecular dynamics and kinetics. Comput. Chem. 5(4), 167–180 (1981).
Zi, Z. Sensitivity analysis approaches applied to systems biology models. IET Syst. Biol. 5(6), 336–346 (2011).
F. Sultan, M. Yaseen, R. Fatima and M. Shahzad, Model simplification and sensitivity analysis of chemical species for oxidation behavior in multi route complex chemical mechanism., Heliyon, vol. 10, no. 12, 2024.
Khoshnaw, S. H., Shahzad, M., Ali, M. & Sultan, F. A quantitative and qualitative analysis of the COVID–19 pandemic model. Chaos Solitons Fractals 138, 109932 (2020).
Aldila, D., Khoshnaw, S., Safitri, E. & Anwar, Y. A mathematical study on the spread of COVID-19 considering social distancing and rapid assessment: The case of Jakarta, Indonesia. Chaos Solitons Fractals 139, 110042 (2020).
Hamad, H. J., Khoshnaw, S. H. & Shahzad, M. Simplified modeling of butane dehydrogenation: deeper understanding of the system’s dynamics. Phys. Scr. 99(7), 075250 (2024).
X. Xin, F. Sultan, M. Yaseen and E. S. Sherif, Numerical and computational analysis on a dissipative dynamical system: Slow invariant manifold for complex chemical mechanism, Heliyon, vol. 10, no. 16, (2024).
Fatah, S. M., Khoshnaw, S. H. & Mustafa, A. N. Understanding Transmission Dynamics of HIV/AIDS with Sensitivity Analysis and Optimal Control Strategies. Eur. J. Pure Appl. Math. 18(2), 6142–6142 (2025).
Shahzad, M., Mustafa, S. & Khoshnaw, S. H. Inference of complex reaction mechanisms applying model reduction techniques. Phys. Scr. 99(4), 045242 (2024).
Farman, M., Xu, C., Abbas, P., Sambas, A. & Sultan, F. Stability and chemical modeling of quantifying disparities in atmospheric analysis with sustainable fractal fractional approach. Commun. Nonlinear Sci. Numer. Simul. 142, 108525 (2025).
Acknowledgements
The authors extend their appreciation to the Deanship of Research and Graduate Studies at King Khalid University for funding this work through Large Research Project under grant number RGP2/43/46.
Funding
The authors declare that there is no funding source available.
Author information
Authors and Affiliations
Contributions
All authors reviewed the manuscript.
Corresponding authors
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.
About this article
Cite this article
Khatoon, A., Shahzad, M., Elmasry, Y. et al. Exploring the dynamics of chemical species interactions in complex reaction mechanism: classification of fast and slow species and bifurcation analysis. Sci Rep (2026). https://doi.org/10.1038/s41598-026-37965-2
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s41598-026-37965-2
