Introduction to the Geometric Theory of ODEs with Applications to Chemical Processes

  • Dietrich FlockerziEmail author
Part of the Modeling and Simulation in Science, Engineering and Technology book series (MSSET)


We give an introduction to the geometric theory of ordinary differential equations (ODEs) tailored to applications to biochemical reaction networks and chemical separation processes. Quite often, the ordinary differential equations under investigation are “reduced” partial differential equations (PDEs) as in the search of traveling wave solutions. So, we also address ODE topics that have their origin in the PDE context.

We present the mathematical theory of invariant and integral manifolds, in particular, of center and slow manifolds, which reflect the splitting of variables and/or processes into slow and fast ones. The invariance of a smooth manifold is characterized by a quasilinear partial differential equation, and the widely used approximations of invariant manifolds are derived from such PDEs. So we also offer, to some extent, an introduction to quasilinear PDEs. The basic ideas and crucial tools are illustrated with numerous examples and exercises. Concerning the proofs, we confine ourselves to outline the crucial steps and refer, especially in the first three sections, to the literature.

The final Sects. 1.4 and 1.5 on reaction–separation processes and on chromatographic separation present new results, including their proofs. They are the outcome of many fruitful discussions with my colleagues Malte Kaspereit and Achim Kienle.


Stability Integral manifolds and method of characteristics Center manifolds and asymptotic phases Reduction methods and bifurcations Quasi-stationary approximations and singular perturbations Slow invariant manifolds Reactive and chromatographic separation networks 


  1. 1.
    Aris, R.: Elementary Chemical Reactor Analysis. Dover, Mineola (1989) Google Scholar
  2. 2.
    Arrowsmith, D.K., Place, C.M.: An Introduction to Dynamical Systems. Cambridge University Press, Cambridge (1990) zbMATHGoogle Scholar
  3. 3.
    Arrowsmith, D.K., Place, C.M.: Dynamical Systems. Chapman and Hall Mathematics, London (1992) CrossRefzbMATHGoogle Scholar
  4. 4.
    Barbosa, D., Doherty, M.F.: A new set of composition variables for the representation of reactive phase diagrams. Proc. R. Soc. Lond. Ser. A, Math. Phys. Sci. 413, 459–464 (1987) CrossRefGoogle Scholar
  5. 5.
    Barbosa, D., Doherty, M.F.: Design and minimum-reflux calculations for single-feed multicomponent reactive distillation columns. Chem. Eng. Sci. 43, 1523–1537 (1988) CrossRefGoogle Scholar
  6. 6.
    Berman, A., Plemmons, R.J.: Nonnegative Matrices in the Mathematical Sciences. SIAM Classics in Applied Mathematics, vol. 9 (1994) CrossRefzbMATHGoogle Scholar
  7. 7.
    Betounes, D.: Partial Differential Equations for Computational Science: With Maple and Vector Analysis. Springer, New York (1998) CrossRefzbMATHGoogle Scholar
  8. 8.
    Bohmann, A.: Reaction invariants. Student’s Thesis, University of Magdeburg (2008) Google Scholar
  9. 9.
    Borghans, J.A.M., deBoer, R.J., Segel, L.A.: Extending the quasi-steady state approximation by changing variables. Bull. Math. Biol. 58, 43–63 (1996) CrossRefzbMATHGoogle Scholar
  10. 10.
    Brauer, F., Castillo-Chavez, C.: Mathematical Methods in Population Biology and Epidemiology. Texts in Applied Mathematics, vol. 40. Springer, New York (2001) CrossRefGoogle Scholar
  11. 11.
    Bressan, A., Serre, D., Williams, M., Zumbrun, K.: Hyperbolic Systems of Balance Laws. Cetraro, Italy, 2003. Lecture Notes in Mathematics, vol. 1911. Springer, Berlin (2007) CrossRefGoogle Scholar
  12. 12.
    Britton, N.F.: Reaction–Diffusion Equations and Their Applications in Biology. Academic Press, Orlando (1986) Google Scholar
  13. 13.
    Brunovský, P.: Tracking invariant manifolds without differential forms. Acta Math. Univ. Comen. 65(1), 23–32 (1996) zbMATHGoogle Scholar
  14. 14.
    Brunovský, P.: C r-inclination theorems for singularly perturbed equations. J. Differ. Equ. 155, 133–152 (1999) CrossRefzbMATHGoogle Scholar
  15. 15.
    Canon, E., James, F.: Resolution of the Cauchy problem for several hyperbolic systems arising in chemical engineering. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 9(2), 219–238 (1992) MathSciNetzbMATHGoogle Scholar
  16. 16.
    Chicone, C.: Ordinary Differential Equations with Applications. Texts in Applied Mathematics, vol. 34. Springer, New York (1999) zbMATHGoogle Scholar
  17. 17.
    Chow, S.N., Hale, J.K.: Methods of Bifurcation Theory, 2nd edn. Grundlehren, vol. 251. Springer, New York (1996) Google Scholar
  18. 18.
    Conradi, C., Flockerzi, D.: Multistationarity in mass action networks with applications to ERK activation. J. Math. Biol. 65(1), 107–156 (2012) MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Conradi, C., Flockerzi, D.: Switching in mass action networks based on linear inequalities. SIAM J. Appl. Dyn. Syst. 11(1), 110–134 (2012) MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Conradi, C., Flockerzi, D., Stelling, J., Raisch, J.: Subnetwork analysis reveals dynamic features of complex (bio)chemical networks. Proc. Natl. Acad. Sci. USA 104(49), 19175–19180 (2007) CrossRefGoogle Scholar
  21. 21.
    Conradi, C., Flockerzi, D., Raisch, J.: Multistationarity in the activation of a MAPK: parametrizing the relevant region in parameter space. Math. Biosci. 211, 105–131 (2008) MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Doherty, M.F., Malone, M.: Conceptual Design of Distillation Systems. McGraw-Hill, Boston (2001) Google Scholar
  23. 23.
    Edelstein-Keshet, L.: Mathematical Models in Biology. SIAM Classics in Applied Mathematics, vol. 46 (2005) CrossRefzbMATHGoogle Scholar
  24. 24.
    Elnashaie, S.S.E.H., Elshishini, S.S.: Modelling, Simulation and Optimization of Industrial Fixed Bed Catalytic Reactors. Gordon and Beach Science, Amsterdam (1993) Google Scholar
  25. 25.
    Evans, L.C.: Partial Differential Equations, 2nd edn. AMS Graduate Studies in Mathematics, vol. 19. AMS, Providence (2010) zbMATHGoogle Scholar
  26. 26.
    Fall, C.P., Maland, E.S., Wagner, J.M., Tyson, J.J.: Computational Cell Biology. Interdisciplinary Applied Mathematics, vol. 20. Springer, New York (2002) zbMATHGoogle Scholar
  27. 27.
    Fenichel, N.: Geometric singular perturbation theory for ordinary differential equations. J. Differ. Equ. 31, 53–98 (1979) MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Fife, P.C.: Mathematical Aspects of Reacting and Diffusing Systems. Lecture Notes in Biomathematics, vol. 28. Springer, Berlin (1979) zbMATHGoogle Scholar
  29. 29.
    Flockerzi, D.: Existence of small periodic solutions of ordinary differential equations in \(\mathbb{R}^{2}\). Arch. Math. 33(3), 263–278 (1979) MathSciNetCrossRefGoogle Scholar
  30. 30.
    Flockerzi, D.: Bifurcation formulas for ordinary differential equations in \(\mathbb{R}^{n}\). Nonlinear Anal. 5(3), 249–263 (1981) MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Flockerzi, D.: Weakly nonlinear systems and the bifurcation of higher dimensional tori. In: Equadiff 82. Lecture Notes in Mathematics, vol. 1017, pp. 185–193. Springer, Berlin (1983) CrossRefGoogle Scholar
  32. 32.
    Flockerzi, D.: Generalized bifurcation of higher-dimensional tori. J. Differ. Equ. 55(3), 346–367 (1984) MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Flockerzi, D., Conradi, C.: Subnetwork analysis for multistationarity in mass action kinetics. J. Phys. Conf. Ser. 138, 012006 (2008) CrossRefGoogle Scholar
  34. 34.
    Flockerzi, D., Sundmacher, K.: On coupled Lane–Emden equations arising in dusty fluid models. J. Phys. Conf. Ser. 268, 012006 (2011) CrossRefGoogle Scholar
  35. 35.
    Flockerzi, D., Bohmann, A., Kienle, A.: On the existence and computation of reaction invariants. Chem. Eng. Sci. 62(17), 4811–4816 (2007) CrossRefGoogle Scholar
  36. 36.
    Flockerzi, D., Kaspereit, M., Kienle, A.: Spectral properties of bi-Langmuir isotherms. Chem. Eng. Sci. 104, 957–959 (2013) CrossRefGoogle Scholar
  37. 37.
    Flockerzi, D., Holstein, K., Conradi, C.: N-site phosphorylation systems with 2N−1 steady states. Bull. Math. Biol. 76, 1892–1916 (2014) MathSciNetCrossRefGoogle Scholar
  38. 38.
    Forssén, P., Arnell, R., Kaspereit, M., Seidel-Morgenstern, A., Fornstedt, T.: Effects of a strongly adsorbed additive on process performance in chiral preparative chromatography. J. Chromatogr. A 1212, 89–97 (2008) CrossRefGoogle Scholar
  39. 39.
    Gadewar, S.R., Schembecker, G., Doherty, M.F.: Selection of reference components in reaction invariants. Chem. Eng. Sci. 60, 7168–7171 (2005) CrossRefGoogle Scholar
  40. 40.
    Granas, A., Dugundji, J.: Fixed Point Theory. Monographs in Mathematics. Springer, New York (2003) CrossRefzbMATHGoogle Scholar
  41. 41.
    Gray, P., ScottS, K.: Chemical Oscillations and Instabilities: Nonlinear Chemical Kinetics. Oxford University Press, Oxford (1990) Google Scholar
  42. 42.
    Grüner, S., Kienle, A.: Equilibrium theory and nonlinear waves for reactive distillation columns and chromatographic reactors. Chem. Eng. Sci. 59, 901–918 (2004) CrossRefGoogle Scholar
  43. 43.
    Grüner, S., Mangold, M., Kienle, A.: Dynamics of reaction separation processes in the limit of chemical equilibrium. AIChE J. 52(3), 1010–1026 (2006) CrossRefGoogle Scholar
  44. 44.
    Guiochon, G., Felinger, A., Shirazi, D.G., Katti, A.K.: Fundamentals of Preparative and Nonlinear Chromatography, 2nd edn. Elsevier, San Diego (2006) Google Scholar
  45. 45.
    Hale, J.K.: Ordinary Differential Equations. Wiley, New York (1969) zbMATHGoogle Scholar
  46. 46.
    Harrison, G.W.: Global stability of predator–prey interactions. J. Math. Biol. 8, 159–171 (1979) MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Hassard, B.D., Kazarinoff, N.D., Wan, Y.-H.: Theory and Applications of the Hopf Bifurcation. London Mathematical Society Lecture Notes Series, vol. 41. Cambridge University Press, Cambridge (1980) Google Scholar
  48. 48.
    Helfferich, F., Klein, G.: Multicomponent Chromatography: Theory of Interference. Marcel Dekker, New York (1970) Google Scholar
  49. 49.
    Hofbauer, J., Sigmund, K.: Theory of Evolution and Dynamical Systems. Cambridge University Press, Cambridge (1988) zbMATHGoogle Scholar
  50. 50.
    Holstein, K., Flockerzi, D., Conradi, C.: Multistationarity in sequential distributed multisite phosphorylation networks. Bull. Math. Biol. 75, 2028–2058 (2013) MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Huang, Y.-S., Sundmacher, K., Qi, Z., Schlünder, E.-U.: Residue curve maps of reactive membrane separation. Chem. Eng. Sci. 59, 2863–2879 (2004) CrossRefGoogle Scholar
  52. 52.
    Huang, Y.-S., Schlünder, E.-U., Sundmacher, K.: Feasibility analysis of membrane reactors—discovery of reactive arheotropes. Catal. Today 104, 360–371 (2005) CrossRefGoogle Scholar
  53. 53.
    Izhikevich, E.M.: Dynamical Systems in Neuroscience. MIT Press, Cambridge (2010), paperback edition Google Scholar
  54. 54.
    Jones, C.K.R.T.: Geometric singular perturbation theory. In: Dynamical Systems. Lecture Notes in Mathematics, vol. 1609. Springer, Berlin (1995) CrossRefGoogle Scholar
  55. 55.
    Jones, C.K.R.T., Kaper, T., Kopell, N.: Tracking invariant manifolds up to exponentially small errors. SIAM J. Math. Anal. 27(2), 558–577 (1996) MathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    Kaspereit, M.: Optimal synthesis and design of advanced chromatographic process concepts. Habilitationsschrift, University of Magdeburg (2011) Google Scholar
  57. 57.
    Kaspereit, M., Seidel-Morgenstern, A., Kienle, A.: Design of simulated moving bed processes under reduced purity requirements. J. Chromatogr. A 1162, 2–13 (2007) CrossRefGoogle Scholar
  58. 58.
    Kirchgraber, U., Palmer, K.J.: Geometry in the Neighborhood of Invariant Manifolds of Maps and Flows in Linearization. Pitman Research Notes in Mathematics, vol. 233. Longman Scientific & Technical, Harlow (1990) zbMATHGoogle Scholar
  59. 59.
    Kirsch, S., Hanke-Rauschenbach, R., El-Sibai, A., Flockerzi, D., Krischer, K., Sundmacher, K.: The S-shaped negative differential resistance during the electrooxidation of H2/CO in polymer electrolyte membrane fuel cells: modeling and experimental proof. J. Phys. Chem. C 115, 25315–25329 (2011) CrossRefGoogle Scholar
  60. 60.
    Klamt, S., Hädicke, O., van Kamp, A.: Stoichiometric and constraint-based analysis of biochemical reaction networks. In: Benner, P., et al. (eds.) Large-Scale Networks in Engineering and Life Sciences. Springer, Heidelberg (2014). Chap. 5 Google Scholar
  61. 61.
    Knobloch, H.W., Kappel, F.: Gewöhnliche Differentialgleichungen. B.G. Teubner, Stuttgart (1974) CrossRefzbMATHGoogle Scholar
  62. 62.
    Kumar, A., Josic, K.: Reduced models of networks of coupled enzymatic reactions. J. Theor. Biol. 278, 87–106 (2011) MathSciNetCrossRefGoogle Scholar
  63. 63.
    Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory, 2nd edn. Applied Mathematical Sciences, vol. 112. Springer, New York (1998) zbMATHGoogle Scholar
  64. 64.
    Kvaalen, E., Neel, L., Tondeur, D.: Directions of quasi-static mass and energy transfer between phases in multicomponent open systems. Chem. Eng. Sci. 40, 1191–1204 (1985) CrossRefGoogle Scholar
  65. 65.
    Lee, C.H., Othmer, H.: A multi-time scale analysis of chemical reaction networks: I. Deterministic systems. J. Math. Biol. 60(3), 387–450 (2010) MathSciNetCrossRefGoogle Scholar
  66. 66.
    Lindström, T.: Global stability of a model for competing predators. Nonlinear Anal. 39, 793–805 (2000) MathSciNetCrossRefzbMATHGoogle Scholar
  67. 67.
    Meiss, J.D.: Differential Dynamical Systems. SIAM Mathematical Modeling and Computation, vol. 14 (2007) CrossRefzbMATHGoogle Scholar
  68. 68.
    Murray, J.D.: Mathematical Biology. Vol. 1: An Introduction, 3rd edn. Interdisciplinary Applied Mathematics, vol. 17. Springer, New York (2002) Google Scholar
  69. 69.
    Murray, J.D.: Mathematical Biology. Vol. 2: Spatial Models and Biomedical Applications, 3rd edn. Interdisciplinary Applied Mathematics, vol. 18. Springer, New York (2003) Google Scholar
  70. 70.
    Nipp, K.: An algorithmic approach for solving singularly perturbed initial value problems. In: Kircraber, U., Walther, H.O. (eds.) Dynamics Reported, vol. 1. Teubner/Wiley, Stuttgart/New York (1988) CrossRefGoogle Scholar
  71. 71.
    Othmer, H.G.: Analysis of Complex Reaction Networks. Lecture Notes. University of Minnesota (2003) Google Scholar
  72. 72.
    Pandey, R., Flockerzi, D., Hauser, M.J.B., Straube, R.: Modeling the light- and redox-dependent interaction of PpsR/AppA Rhodobacter sphaeroides. Biophys. J. 100(10), 2347–2355 (2011) CrossRefGoogle Scholar
  73. 73.
    Pandey, R., Flockerzi, D., Hauser, M.J.B., Straube, R.: An extended model for the repression of photosynthesis genes by the AppA/PpsR system in Rhodobacter sphaeroides. FEBS J. 279(18), 3449–3461 (2012) CrossRefGoogle Scholar
  74. 74.
    Prüss, J.W., Schnaubelt, R., Zacher, R.: Mathematische Modelle in der Biologie. Mathematik Kompakt. Birkhäuser, Basel (2008) CrossRefzbMATHGoogle Scholar
  75. 75.
    Rhee, H.-K., Aris, R., Amundson, N.R.: First-Order Partial Differential Equations: Vol. I—Theory and Application of Single Equations. Dover, Mineola (2001) Google Scholar
  76. 76.
    Rhee, H.-K., Aris, R., Amundson, N.R.: First-Order Partial Differential Equations: Vol. II—Theory and Application of Hyperbolic Systems of Quasilinear Equations. Dover, Mineola (2001) Google Scholar
  77. 77.
    Robinson, C.: Dynamical Systems, 2nd edn. CRC Press, Boca Raton (1999) zbMATHGoogle Scholar
  78. 78.
    Rubiera Landa, H.O., Flockerzi, D., Seidel-Morgenstern, A.: A method for efficiently solving the IAST equations with an application to adsorber dynamics. AIChE J. 59(4), 1263–1277 (2012) CrossRefGoogle Scholar
  79. 79.
    Schaber, J., Lapytsko, A., Flockerzi, D.: Nesed auto-inhibitory feedbacks alter the resistance of homeostatic adaptive biochemical networks. J. R. Soc. Interface 11, 20130971 (2014) CrossRefGoogle Scholar
  80. 80.
    Segel, L.A., Goldbeter, A.: Scaling in biochemical kinetics: dissection of a relaxation oscillator. J. Math. Biol. 32, 147–160 (1994) CrossRefzbMATHGoogle Scholar
  81. 81.
    Segel, L.A., Slemrod, M.: The quasi–steady state assumption: a case study in perturbation. SIAM Rev. 31(3), 446–477 (1989) MathSciNetCrossRefzbMATHGoogle Scholar
  82. 82.
    Seydel, R.: Practical Bifurcation and Stability Analysis, 3rd edn. Interdisciplinary Applied Mathematics, vol. 5. Springer, New York (2010) CrossRefzbMATHGoogle Scholar
  83. 83.
    Smith, H., Waltman, P.: The Theory of the Chemostat. Cambridge Studies in Mathematical Biology, vol. 13. Cambridge University Press, Cambridge (1995) CrossRefzbMATHGoogle Scholar
  84. 84.
    Steerneman, T., van Perlo-ten Kleij, F.: Properties of the matrix AXY . Linear Algebra Appl. 410, 70–86 (2005) MathSciNetCrossRefzbMATHGoogle Scholar
  85. 85.
    Storti, G., Mazzotti, M., Morbidelli, M., Carra, S.: Robust design of binary countercurrent adsorption separation processes. AIChE J. 39, 471–492 (1993) CrossRefGoogle Scholar
  86. 86.
    Straube, R., Flockerzi, D., Müller, S.C., Hauser, M.J.B.: Reduction of chemical reaction networks using quasi-integrals. J. Phys. Chem. A 109, 441–450 (2005) CrossRefGoogle Scholar
  87. 87.
    Straube, R., Flockerzi, D., Müller, S.C., Hauser, M.J.B.: The origin of bursting pH oscillations in an enzyme model reaction system. Phys. Rev. B 72, 066205 (2005) CrossRefGoogle Scholar
  88. 88.
    Straube, R., Flockerzi, D., Hauser, M.J.B.: Sub-Hopf/fold-cycle bursting and its relation to (quasi-)periodic oscillations. J. Phys. Conf. Ser. 55, 214–231 (2006) CrossRefGoogle Scholar
  89. 89.
    Ung, S., Doherty, M.F.: Vapor-liquid phase equilibrium in systems with multiple chemical reactions. Chem. Eng. Sci. 50, 23–48 (1995) CrossRefGoogle Scholar
  90. 90.
    Ung, S., Doherty, M.F.: Synthesis of reactive distillation systems with multiple equilibrium chemical reactions. Ind. Eng. Chem. Res. 34, 2555–2565 (1995) CrossRefGoogle Scholar
  91. 91.
    Ung, S., Doherty, M.F.: Calculation of residue curve maps for mixtures with multiple equilibrium chemical reactions. Ind. Eng. Chem. Res. 34, 3195–3202 (1995) CrossRefGoogle Scholar
  92. 92.
    Vanderbauwhede, A.: Centre manifolds, normal forms and elementary bifurcations. In: Kirchgraber, U., Walther, H.O. (eds.) Dynamical Systems, vol. 2. Teubner/Wiley, Stuttgart/New York (1989) Google Scholar
  93. 93.
    Vu, T.D., Seidel-Morgenstern, A., Grüner, S., Kienle, A.: Analysis of ester hydrolysis reactions in a chromatographic reactor using equilibrium theory and a rate model. Ind. Eng. Chem. Res. 44, 9565–9574 (2005) CrossRefGoogle Scholar

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© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Max Planck Institute for Dynamics of Complex Technical SystemsMagdeburgGermany

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