Solution of the inverse problem for chemical reactions with chaotic dynamics

Objectives. To develop and test a method for solving the inverse problem of chemical kinetics for estimating the frequencies of elementary stages of complex chemical reactions occurring in a chaotic regime.

Methods. The method is based on the representation of nonstationary experimental data on reagent concentrations and the rates of their change in the form of a matrix of a particular structure.

Results. The effectiveness of the method is demonstrated by examples of reactions proceeding according to stage schemes similar to the Willamowski–Rossler mechanism, characterized by undamped aperiodic oscillations.

Conclusions. The method allows the frequencies of stages for reactions proceeding according to mechanisms with non-monotonic dynamics of any complexity to be determined with high accuracy.

1. Ismagilova A.S., Spivak S.I. Obratnye zadachi khimicheskoi kinetiki (Inverse Problems of Chemical Kinetics). Saarbrucken: Lap Lambert Academic Publ.; 2013, 125 p. (In Russ.). ISBN 978-36593118474. https://www.elibrary.ru/glykiv

2. Yagola A.G., Yanfei V., Stepanova I.EH., Titarenko V.N. Obratnye zadachi i metody ikh resheniya. Prilozheniya k geofizike (Inverse Problems and Methods of their Solution. Applications to Geophysics). Moscow: Laboratoriya znanii; 2021, 217 p. (In Russ.). ISBN 978-5-93208-555-4

3. Leonov A.S. Reshenie nekorrektno postavlennykh obratnykh zadach: ocherk teorii, prakticheskie metody i demonstratsii v MATLAB (Solution of ill-Posed Inverse Problems: Outline of theory, Practical Methods and Demonstrations in MATLAB). Moscow: Librokom; 2024, 368 p. (In Russ.). ISBN 978-5-9710-9965-9

4. Pisarenko E.V., Pisarenko V.N. Analysis and simulation of the nonlinear kinetics of reacting chemical systems. Theor. Found. Chem. Eng. 2013;47(2):128–135. https://doi.org/10.1134/S004057951302005X [Original Russian Text: PisarenkoE.V., PisarenkoV.N. Analysis and simulation of the nonlinear kinetics of reacting chemical systems. Teoreticheskie Osnovy Khimicheskoi Tekhnologii. 2013;47(2): 173–181 (in Russ.). https://doi.org/10.7868/S0040357113020061 ]

5. Shatalov M.Y., Fedotov S.I., Shatalov Y.M. New methods of determination of kinetic parameters of the theoretical models from experimental data. Theor. Found. Chem. Eng. 2013;47(3): 207–216. https://doi.org/10.1134/S0040579513020097 [Original Russian Text: Shatalov M.Yu., Fedotov S.I., Shatalov Yu.M. New methods of determination of kinetic parameters of the theoretical models from experimental data. Teoreticheskie Osnovy Khimicheskoi Tekhnologii. 2013;47(3): 260–270 (in Russ.). https://doi.org/10.7868/S0040357113020103 ]

6. Gorsky V.G. A prior parameter identifiability analysis of fixed structure models. In: Letzky E.K. (Ed.). Design of Experiments and Data Analysis: New Trends and Results. Moscow: Antal; 1993. P. 92–131.

7. Fedotov V.Kh., Kol’tsov N.I. Method of solving the inverse problem of chemical kinetics for catalytic reactions in which each step involves main reactants. Rus. J. Phys. Chem. B. 2016;10(5): 753–759. https://doi.org/10.1134/S1990793116050195 [Original Russian Text: Fedotov V.Kh., Kol’tsov N.I. Method of solving the inverse problem of chemical kinetics for catalytic reactions in which each step involves main reactants. Khimicheskaya Fizika. 2016;35(10):9–15 (in Russ.). https://doi.org/10.7868/S0207401X1610006X ]

8. Kol’tsov N.I. Method for Solving the Inverse Problem of the Chemical Kinetics of Multistage Reactions. Kinet. Catal. 2020;61(6):833–838. https://doi.org/10.1134/S0023158420040096 [Original Russian Text: Kol’tsov N.I. Method for Solving the Inverse Problem of the Chemical Kinetics of Multistage Reactions. Kinetika i Kataliz. 2020;61(6):783–788 (in Russ.). https://doi.org/10.31857/S0453881120040127 ]

9. Kol’tsov N.I. Method for Determining the Rate Constants of Chemical Reaction Stages in an Enclosed Gradientless Reactor. Russ. J. Applied Chem. 2020;93(10):1544–1552. https://doi.org/10.1134/S1070427220100092 [Original Russian Text: Kol’tsov N.I. Method for Determining the Rate Constants of Chemical Reaction Stages in an Enclosed Gradientless Reactor. Zhurnal Prikladnoi Khimii. 2020;93(10):1474–1481 (in Russ.). https://doi.org/10.31857/S0044461820100096 ]

10. Kol’tsov N.I. Solving the Inverse Problem for Chemical Reactions Occurring in a Plug-Flow Reactor. Theor. Found. Chem. Eng. 2021;55(6):1238–1245. https://doi.org/10.1134/S0040579521050237 [Original Russian Text: Kol’tsov N.I. Solving the Inverse Problem for Chemical Reactions Occurring in a Plug-Flow Reactor. Teoreticheskie Osnovy Khimicheskoi Tekhnologii. 2021;55(6):772–779 (in Russ.). https://doi.org/10.31857/S0040357121050043 ]

11. Kol’tsov N.I., Fedotov V.Kh. Invarianty i obratnye zadachi khimicheskoi kinetiki (Invariants and Inverse Problems of Chemical Kinetics). Cheboksary: Chuvash University Publ.; 2022, 240 p. (In Russ.). ISBN 978-5-7677-3431-3

12. Katsman E.A., Sokolova I.V., Temkin O.N. Solution of reverse kinetic problem for oscillatory reactions. Theor. Found. Chem. Eng. 2014;48(2):175–179. https://doi.org/10.1134/S0040579514020067 [Original Russian Text: KatsmanE.A., Sokolova I.V., Temkin O.N. Solution of reverse kinetic problem for oscillatory reactions. Teoreticheskie Osnovy Khimicheskoi Tekhnologii. 2014;48(2): 190–195 (in Russ.). https://doi.org/10.7868/S0040357114020067 ]

13. Gray P., Scott S.K. Chemical Oscillations and Instabilities. Non-linear Chemical Kinetics. Oxford: Clarendon Press; 1994, 470 р.

14. Alekseev B.V., Fedotov V.Kh., Kol’tsov N.I. Solution of the inverse problem of chemical kinetics for oscillatory reactions. Theor. Found. Chem. Eng. 2025. (In press).

15. Rössler O.E. Chaotic behavior in simple reaction system. Z. Naturfosch. 1976;A(31):259–264. https://doi.org/10.1515/zna-1976-3-408

16. Willamowski K.D., Rössler O.E. Irregular oscillations in a realistic abstract quadratic mass action system. Z. Naturforsch. 1980;35a:317–318. https://doi.org/10.1515/zna-1980-0308

17. Gaspard P. Rössler systems. In: Encyclopedia of Nonlinear Science. New York: Routledge; 2005. P. 808–811.

18. Stucki J., UrbanczikR. Entropy Production of the WillamowskiRossler Oscillator. Z. Naturforsch. 2005;60a:599–607. https://doi.org/10.1515/zna-2005-8-907

19. Shuster G. Determinirovannyi khaos: Vvedenie (Deterministic Chaos: An Introduction): transl. from Engl. Moscow: Mir; 1988, 248 p. (In Russ.). ISBN 5-03-001373-3

20. Kuznetsov S.P. Dinamicheskii khaos (Dynamic Chaos). Moscow: Fizmatlit; 2006, 356 p. (InRuss.). ISBN 5-94052-100-2

21. Kiperman S.L. Osnovy khimicheskoi kinetiki v geterogennom katalize (Fundamentals of Chemical Kinetics in Heterogeneous Catalysis). Moscow: Khimiya; 1979, 352 p. (In Russ.).

22. Zhabotinskii A.M. Kontsentratsionnye avtokolebaniya (Concentration Self-Oscillations). Moscow: Nauka; 1974, 179 p. (In Russ.).

23. Vygodskii M.Ya. Spravochnik po vysshei matematike (Handbook of Higher Mathematics). Moscow: AST; 2019, 703 p. (In Russ.). ISBN 978-5-17-117741-6

24. Alad’ev V.Z. Sistemy komp’yuternoi algebry: Maple: Iskusstvo programmirovaniya (Computer Algebra Systems: Maple: The Art of Programming). Moscow: Laboratoriya Bazovykh Znanii; 2006, 792 p. (In Russ.). ISBN 5-93208-189-9

25. Kol’tsov N.I. Chaotic oscillations in the simplest chemical reaction. ChemChemTech = Izv. Vyssh. Uchebn. Zaved. Khim. Khim. Tekhnol. 2018;61(4-5):133–135 (in Russ.). https://doi.org/10.6060/tcct.20186104-05.5654

26. Kol’tsov N.I., Fedotov V.Kh. Chaotic oscillations in a simple heterogeneous catalytic reaction. Butlerovskie soobshcheniya = Butlerov Communications. 2017;50(6):30–33 (in Russ.). https://elibrary.ru/zefnpr