Journal of Mathematical Chemistry

, Volume 52, Issue 1, pp 93–114 | Cite as

Quadratic first integrals of kinetic differential equations

  • Ilona NagyEmail author
  • János Tóth
Original Paper


Classes of kinetic differential equations are delineated which do have a quadratic first integral, and classes which can not have one. Example reactions corresponding to the obtained kinetic differential equations are shown, and a few figures showing the trajectories of the corresponding systems are also included. Connections to other areas are mentioned and unsolved problems collected. The new results are theoretical, although computational tools are heavily used. Applications from biology and combustion theory will come later.


First integral Kinetic differential equation Computational biology and chemistry 



The present work has partially been supported by the European Science Foundation Research Networking Programme: Functional Dynamics in Complex Chemical and Biological Systems, by the Hungarian National Scientific Foundation, No. 84060, and by the COST Action CM901: Detailed Chemical Kinetic Models for Cleaner Combustion. This work is connected to the scientific program of the “Development of quality-oriented and harmonized R+D+I strategy and functional model at BME” project. This project is supported by the New Széchenyi Plan (Project ID: TÁMOP-4.2.1/B-09/1/KMR-2010-0002).


  1. 1.
    E. Bertolazzi, Positive and conservative schemes for mass action kinetics. Comput. Math. 32, 29–43 (1996)Google Scholar
  2. 2.
    F.W. Carroll, A polynomial in each variable separately is a polynomial. Am. Math. Mon. 68, 42 (1961)CrossRefGoogle Scholar
  3. 3.
    V. Chellaboina, S.P. Bhat, W.M. Haddad, D.S. Bernstein, Modeling and analysis of mass-action kinetics. IEEE Control Syst. Mag. 29, 60–78 (2009)CrossRefGoogle Scholar
  4. 4.
    J. Deák, J. Tóth, B. Vizvári, Anyagmegmaradás összetett kémiai mechanizmusokban (Mass conservation in complex chemical mechanisms). Alk. Mat. Lapok 16(1–2), 73–97 (1992)Google Scholar
  5. 5.
    R. Dilao, Turing instabilities and patterns near a Hopf bifurcation. Appl. Math. Comput. 164(2), 391–414 (2005)CrossRefGoogle Scholar
  6. 6.
    P. Érdi, J. Tóth, Mathematical Models of Chemical Reactions. Theory and Applications of Deterministic and Stochastic Models (Princeton University Press, Princeton, 1989)Google Scholar
  7. 7.
    I. Faragó, Nonnegativity of the difference schemes. Pure Math. Appl. 6, 147–159 (1996)Google Scholar
  8. 8.
    Gy Farkas, Kinetic lumping schemes. Chem. Eng. Sci. 54, 3909–3915 (1999)CrossRefGoogle Scholar
  9. 9.
    M. Feinberg, F.J.M. Horn, Chemical mechanism structure and the coincidence of the stoichiomeric and kinetic subspaces. Arch. Ratl. Mech. Anal. 66(1), 83–97 (1977)CrossRefGoogle Scholar
  10. 10.
    F. Gonzalez-Gascon, D.P. Salas, On the first integrals of Lotka-Volterra systems. PLA 266(4–6), 336–340 (2000)CrossRefGoogle Scholar
  11. 11.
    A. Halmschlager, L. Szenthe, J. Tóth, Invariants of kinetic differential equations. Electron. J. Qual. Theory Differ. Equ. 14, 1–14 (2004)Google Scholar
  12. 12.
    P. Hanusse, De l’existence d’un cycle limite dans l’évolution des systémes chimiques ouverts. C. R. Acad. Sci. Ser. C 274, 1245–1247 (1972)Google Scholar
  13. 13.
    P. Hanusse, Étude des systémes dissipatifs chimiques á deux et trois espéces itermédiaires. C. R. Acad. Sci. Ser. C 263–266 (1973)Google Scholar
  14. 14.
    V. Hárs, J. Tóth, On the inverse problem of reaction kinetics. Colloquia Mathematica Societatis János Bolyai. Qualitative Theory of, Differential Equations (1979), pp. 363–379Google Scholar
  15. 15.
    F. Horn, R. Jackson, General mass action kinetics. Arch. Ratl. Mech. Anal. 47, 81–116 (1972)CrossRefGoogle Scholar
  16. 16.
    S. Karátson, J. Korotov, A discrete maximum principle in hilbert space with applications to nonlinear cooperative elliptic systems. Helsinki University of Technology Institue of Mathematics Research Reports, A 552 (2008)Google Scholar
  17. 17.
    G. Li, H. Rabitz, J. Tóth, A general analysis of exact nonlinear lumping in chemical kinetics. Chem. Eng. Sci. 49(3), 343–361 (1994)CrossRefGoogle Scholar
  18. 18.
    S. Marin, G. S. Yablonsky, Kinetics of Chemical Reactions. Decoding Complexity (Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, 2011)Google Scholar
  19. 19.
    M.F. Morales, On a possible mechanism for biological periodicity. Bull. Math. Biophys. 6, 65–70 (1944)CrossRefGoogle Scholar
  20. 20.
    A.L. Nagy, D. Papp, J. Tóth, ReactionKinetics: a mathematica package with applications. Chem. Eng. Sci. (2012, in press)Google Scholar
  21. 21.
    G. Póta, Two-component bimolecular systems cannot have limit cycles: a complete proof. J. Chem. Phys. 78, 1621–1622 (1983)CrossRefGoogle Scholar
  22. 22.
    J.S. Rosenbaum, Conservation properties for numerical integration methods for systems of differential equations. 2. J. Phys. Chem. 81(25), 2362–2365 (1977)CrossRefGoogle Scholar
  23. 23.
    B. Schuman, J. Tóth, No limit cycle in two species second order kinetics. Bull. Sci. Math. 127, 222–230 (2003)CrossRefGoogle Scholar
  24. 24.
    S. Schuster, T. Höfer, Determining all extreme semi-positive conservation relations in chemical reaction systems: a test criterion for conservativity. J. Chem. Soc. Faraday Trans. 87(16), 2561–2566 (1991)CrossRefGoogle Scholar
  25. 25.
    G. Szederkényi, Computing sparse and dense realizations of reaction kinetic systems. J. Math. Chem. 47, 551–568 (2010)CrossRefGoogle Scholar
  26. 26.
    G. Szederkényi, K.M. Hangos, Finding complex balanced and detailed balanced realizations of chemical reaction networks. J. Math. Chem. 49, 1163–1179 (2011)CrossRefGoogle Scholar
  27. 27.
    L. Szili, J. Tóth, Necessary condition of the Turing instability. Phys. Rev. E 48(1), 183–186 (1993)CrossRefGoogle Scholar
  28. 28.
    L. Szili, J. Tóth, On the origin of Turing instability. J. Math. Chem. 22(1), 39–53 (1997)CrossRefGoogle Scholar
  29. 29.
    J. Tóth, Gradient systems are cross-catalytic. React. Kinet. Catal. Lett. 12(3), 253–257 (1979)CrossRefGoogle Scholar
  30. 30.
    J. Tóth, On the global deterministic and stochastic models of formal reaction kinetics with applications. MTA SZTAKI Tanulmányok 129, 1–166 (1981)Google Scholar
  31. 31.
    J. Tóth, Bendixson-type theorems with applications. Zeitschrift für Angewandte Mathematik und Mechanik 67(1), 31–35 (1987)CrossRefGoogle Scholar
  32. 32.
    J. Tóth, P. Érdi, Kinetic Symmetries: Some Hints, eds. by M. Moreau, P. Turq, Chemical Reactivity in Liquids. Fundamental Aspects (Plenum Press, New York 1988), pp. 517–522. Paris, Sept. 7–11, 1987Google Scholar
  33. 33.
    J. Tóth, V. Hárs, Orthogonal transforms of the Lorenz- and Rössler-equations. Physica 19D, 135–144 (1986)Google Scholar
  34. 34.
    J. Tóth, V. Hárs, Specification of oscillating chemical models starting form a given linearized form. Theor. Chim. Acta 70, 143–150 (1986)CrossRefGoogle Scholar
  35. 35.
    J. Tóth, A.L. Nagy, D. Papp, Reaction Kinetics: Exercises, Programs and Theorems (in preparation)Google Scholar
  36. 36.
    J. Tóth, A.L. Nagy, I. Zsély, Structural analysis of combustion models. CES (2013)Google Scholar
  37. 37.
    J.J. Tyson, J.C. Light, Properties of two-component bimolecular and trimolecular chemical reaction systems. J. Chem. Phys. 59(8), 4164–4273 (1973)CrossRefGoogle Scholar
  38. 38.
    A.I. Volpert, Differential equations on graphs. Mat. Sb. 88(130), 578–588 (1972)Google Scholar
  39. 39.
    A.I. Volpert, S.I. Hudyaev, Analyses in Classes of Discontinuous Functions and Equations of Mathematical Physics (Martinus Nijhoff Publishers, Dordrecht, 1985)Google Scholar
  40. 40.
    A. Weber, T. Sturm, W. Seiler, E.O. Abdel-Rahman, Parametric qualitative analysis of ordinary differential equations: computer algebra methods for excluding oscillations. LNCS 6244, 267–279 (2010). Extended Abstract of an Invited Talk, CASC 2010Google Scholar
  41. 41.
    J. Zador, R.X. Fernandes, Y. Georgievskii, G. Meloni, C.A. Taatjes, J.A. Miller, The reaction of hydroxyethyl radicals with \(\text{ O }_2\): A theoretical analysis of experimental product study. Proc. Combust. Inst. 32, 271–277 (2009)CrossRefGoogle Scholar

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© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Mathematical AnalysisBudapest University of Technology and EconomicsBudapestHungary
  2. 2.Laboratory for Chemical KineticsEötvös UniversityBudapest Hungary

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