Skip to main content
SpringerLink
Log in

Time-Dependent Behavior of the Concentrations

  • Chapter
  • First Online:
Book cover Reaction Kinetics: Exercises, Programs and Theorems

  • 1272 Accesses

Abstract

Here we investigate in detail how the trajectories of the induced kinetic differential equations evolve. They may or may not be defined for all positive times, but they are always nonnegative; they remain in certain subsets of the state space determined by linear and nonlinear first integrals. We formulate the three most important theorems on the time evolution of concentrations: the theorem on detailed balanced mechanisms, the zero-deficiency theorem and also the theorem on reactions with acyclic Volpert graphs. Next, we turn to behaviors which were considered exotic in the second half of the twentieth century: oscillation, oligo-oscillation, chaos, etc. Symmetries of induced kinetic differential equations will also be touched upon.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 49.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 64.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 89.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  • Alberty RA (2004) Principle of detailed balance in kinetics. J Chem Educ 81(8):1206–1209

    Article  CAS  Google Scholar 

  • Amand MMS, Tran K, Radhakrishnan D, Robinson AS, Ogunnaike BA (2014) Controllability analysis of protein glycosylation in cho cells. Plos One 9:1–16

    Article  CAS  Google Scholar 

  • Anderson DF (2011) A proof of the global attractor conjecture in the single linkage class case. SIAM J Appl Math 71(4):1487–1508

    Article  CAS  Google Scholar 

  • Andronov AA, Leontovich EA, Gordon II, Maier AG (1973) Qualitative theory of second-order dynamic systems. Wiley, New York

    Google Scholar 

  • Angeli A (2010) A modular criterion for persistence of chemical reaction networks. IEEE Trans Aut Control 55(7):1674–1679

    Article  Google Scholar 

  • Angeli D, De Leenheer P, Sontag ED (2007) A Petri net approach to study the persistence in chemical reaction networks. Math Biosci 210(2):598–618

    Article  CAS  PubMed  Google Scholar 

  • Angeli D, De Leenheer P, Sontag ED (2011) Persistence results for chemical reaction networks with time-dependent kinetics and no global conservation laws. SIAM J Appl Math 71(1):128–146

    Article  CAS  Google Scholar 

  • Antonov V, Dolicanin D, Romanovski VG, Tóth J (2016) Invariant planes and periodic oscillations in the May–Leonard asymmetric model. MATCH Commun Math Comput Chem 76(2):455–474

    Google Scholar 

  • Bamberger A, Billette E (1994) Quelques extensions d’un théorème de Horn et Jackson. Comptes rendus de l’Académie des sciences Série 1, Mathématique 319(12):1257–1262

    Google Scholar 

  • Bautin NN (1954) On periodic solutions of a system of differential equations. Prikl Mat Mekh 18:128–134

    Google Scholar 

  • Beck MT (1990) Mechanistic and parametric conditions of exotic chemical kinetics. React Kinet Catal Lett 42(2):317–323

    Article  CAS  Google Scholar 

  • Beck MT (1992) Mechanistic and parametric conditions of exotic chemical kinetics. Why are there so few oscillatory reactions? Acta Chir Hung 129:519–529

    CAS  Google Scholar 

  • Beklemisheva LA (1978) Classification of polynomial systems with respect to birational transformations i. Differentsial’nye Uravneniya 14(5):807–816

    Google Scholar 

  • Belousov BP (1958) A periodic reaction and its mechanism. Sb Ref Radiats Med Moscow pp 145–147

    Google Scholar 

  • Bertolazzi E (1996) Positive and conservative schemes for mass action kinetics. Comput Math 32:29–43

    Google Scholar 

  • Bohner B, Endrődi B, Horváth D, Tóth Á (2016) Flow-driven pattern formation in the calcium-oxalate system. J Chem Phys 144(16):164504

    Article  CAS  PubMed  Google Scholar 

  • Boros B (2012) Notes on the deficiency one theorem: multiple linkage classes. Math Biosci 235(1):110–122

    Article  PubMed  Google Scholar 

  • Boros B (2013a) On the dependence of the existence of the positive steady states on the rate coefficients for deficiency-one mass action systems: single linkage class. J Math Chem 51(9):2455–2490

    Article  CAS  Google Scholar 

  • Boros B (2013b) On the existence of the positive steady states of weakly reversible deficiency-one mass action systems. Math Biosci 245(2):157–170

    Article  CAS  PubMed  Google Scholar 

  • Boros B (2013c) On the positive steady states of deficiency-one mass action systems. PhD thesis, School of Mathematics. Director: Miklós Laczkovich, Doctoral Program of Applied Mathematics, Director: György Michaletzky, Department of Probability Theory and Statistics, Institute of Mathematics, Faculty of Science, Eötvös Loránd University, Budapest

    Google Scholar 

  • Brenig L, Goriely A (1994) Painlevé analysis and normal forms. In: Tournier E (ed) Computer algebra and differential equations. London mathematical society lecture note series. Cambridge University Press, Cambridge, pp 211–238

    Chapter  Google Scholar 

  • Britton NF (1986) Reaction-diffusion equations and their applications to biology. Academic, London

    Google Scholar 

  • Castets V, Dulos E, Boissonade J, De Kepper P (1990) Experimental evidence of a sustained standing turing-type nonequilibrium chemical pattern. Phys Rev Lett 64(24):2953

    Article  CAS  PubMed  Google Scholar 

  • Cavani M, Farkas M (1994) Bifurcations in a predator-prey model with memory and diffusion II: turing bifurcation. Acta Math Hung 63(4):375–393

    Article  Google Scholar 

  • Craciun G (2016) Toric differential inclusions and a proof of the global attractor conjecture. arXiv:150102860

    Google Scholar 

  • Craciun G, Feinberg M (2006) Multiple equilibria in complex chemical reaction networks: extensions to entrapped species models. IEE Proc-Syst Biol 153(4):179–186

    Article  CAS  Google Scholar 

  • Craciun G, Feinberg M (2010) Multiple equilibria in complex chemical reaction networks: semiopen mass action systems. SIAM J Appl Math 70(6):1859–1877

    Article  CAS  Google Scholar 

  • Csikja R, Tóth J (2007) Blow up in polynomial differential equations. Enformatika Int J Appl Math Comput Sci 4(2):728–733

    Google Scholar 

  • De Leenheer P, Angeli D, Sontag ED (2006) Monotone chemical reaction networks. J Math Chem 41(3):295–314

    Article  CAS  Google Scholar 

  • Dilao R (2005) Turing instabilities and patterns near a Hopf bifurcation. Appl Math Comput 164(2):391–414

    Google Scholar 

  • Domijan M, Pécou E (2012) The interaction graph structure of mass-action reaction networks. J Math Biol 65(2):375–402

    Article  PubMed  Google Scholar 

  • Drexler DA, Tóth J (2016) Global controllability of chemical reactions. J Math Chem 54(6):1327–1350

    Article  CAS  Google Scholar 

  • Drexler DA, Virágh E, Tóth J (2017) Controllability and reachability of reactions with temperature and inflow control. Fuel. Published online 7 October 2017

    Google Scholar 

  • Edelstein-Keshet L (2005) Mathematical models in biology. In: O’Malley RE Jr (ed) Classics in applied mathematics, vol 46. Society for Industrial and Applied Mathematics, Philadelphia

    Google Scholar 

  • Epstein I, Pojman J (1998) An introduction to nonlinear chemical dynamics: oscillations, waves, patterns, and chaos. Topics in physical chemistry series. Oxford University Press, New York. http://books.google.com/books?id=ci4MNrwSlo4C

    Google Scholar 

  • Ervadi-Radhakrishnan A, Voit EO (2005) Controllability of non-linear biochemical systems. Math Biosci 196(1):99–123

    Article  CAS  PubMed  Google Scholar 

  • Escher C (1980) Models of chemical reaction systems with exactly evaluable limit cycle oscillations and their bifurcation behaviour. Berichte der Bunsengesellschaft für physikalische Chemie 84(4):387–391

    Article  CAS  Google Scholar 

  • Escher C (1981) Bifurcation and coexistence of several limit cycles in models of open two-variable quadratic mass-action systems. Chem Phys 63(3):337–348

    Article  CAS  Google Scholar 

  • Farkas M (1995) On the distribution of capital and labour in a closed economy. SE Asian Bull Math 19(2):27–36

    Google Scholar 

  • Farkas G (1998a) Local controllability of reactions. J Math Chem 24:1–14

    Article  CAS  Google Scholar 

  • Farkas G (1998b) On local observability of chemical systems. J Math Chem 24:15–22

    Article  CAS  Google Scholar 

  • Feinberg M (1977) Mathematical aspects of mass action kinetics. In: Lapidus L, Amundson N (eds) Chemical reactor theory: a review. Prentice-Hall, Englewood Cliffs, pp 1–78

    Google Scholar 

  • Feinberg M (1979) Lectures on chemical reaction networks. Notes of lectures given at the Mathematics Research Center. University of Wisconsin, Feinberg

    Google Scholar 

  • Feinberg M (1980) Chemical oscillations, multiple equilibria, and reaction network structure. In: Stewart W, Ray WH, Conley C (eds) Dynamics and modelling of reactive systems. Academic, New York, pp 59–130

    Chapter  Google Scholar 

  • Feinberg M (1987) Chemical reaction network structure and the stability of complex isothermal reactors—I. The deficiency zero and deficiency one theorems. Chem Eng Sci 42(10):2229–2268

    Article  CAS  Google Scholar 

  • Feinberg M, Horn FJM (1977) Chemical mechanism structure and the coincidence of the stoichiometric and kinetic subspaces. Arch Ratl Mech Anal 66(1):83–97

    Article  Google Scholar 

  • Fife PC (1979) Mathematical aspects of reacting and diffusing systems. Springer, Berlin

    Book  Google Scholar 

  • Frank-Kamenetskii DA (1947) Diffusion and heat transfer in chemical kinetics. USSR Academy of Science Press, Moscow

    Google Scholar 

  • Fu Z, Heidel J (1997) Non-chaotic behaviour in three-dimensional quadratic systems. Nonlinearity 10:1289–1303

    Article  Google Scholar 

  • Gaiko V (2013) Global bifurcation theory and Hilbert’s sixteenth problem. Mathematics and its applications, vol 562. Springer Science & Business Media, Berlin

    Google Scholar 

  • Getz WM, Jacobson DH (1977) Sufficiency conditions for finite escape times in systems of quadratic differential equations. J Inst Math Applics 19:377–383

    Article  Google Scholar 

  • Glitzky A, Hünlich R (1997) Global estimates and asymptotics for electro-reaction-diffusion systems in heterostructures. Appl Anal 66:205–226

    Article  Google Scholar 

  • Glitzky A, Hünlich R (2000) Electro-reaction-diffusion systems including cluster reactions of higher order. Math Nachr 216:95–118

    Article  Google Scholar 

  • Glitzky A, Gröger K, Hünlich R (1994) Existence, uniqueness and asymptotic behaviour of solutions to equations modelling transport of dopants in semiconductors. Bonner Mathematische Schriften 258:49–78

    Google Scholar 

  • Glitzky A, Gröger K, Hünlich R (1996) Free energy and dissipation rate for reaction diffusion processes of electrically charged species. Appl Anal 60:201–217

    Article  Google Scholar 

  • Gonzales-Gascon F, Salas DP (2000) On the first integrals of Lotka-Volterra systems. Phys Lett A 266(4–6):336–340

    Article  Google Scholar 

  • Gray P, Scott SK (1986) A new model for oscillatory behaviour in closed systems: the autocatalator. Berichte der Bunsengesellschaft für physikalische Chemie 90(11):985–996. http://dx.doi.org/10.1002/bbpc.19860901112

    Article  CAS  Google Scholar 

  • Gröger K (1992) Free energy estimates and asymptotic behaviour of reaction-diffusion processes. IAAS-Preprint 20

    Google Scholar 

  • Guckenheimer J, Holmes P (1983) Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Applied mathematical sciences, vol 42. Springer, New York

    Google Scholar 

  • Gunawardena J (2003) Chemical reaction network theory for in-silico biologists. http://vcpmedharvardedu/papers/crntpdf

  • Györgyi L, Field RS (1991) Simple models of deterministic chaos in the Belousov–Zhabotinskii reaction. J Phys Chem 95(17):6594–6602

    Article  Google Scholar 

  • Halmschlager A, Tóth J (2004) Über Theorie und Anwendung von polynomialen Differentialgleichungen. In: Wissenschaftliche Mitteilungen der 16. Frühlingsakademie, Mai 19–23, 2004, München-Wildbad Kreuth, Deutschland, Technische und Wirtschaftswissenschaftliche Universität Budapest, Institut für Ingenieurweiterbildung, Budapest, pp 35–40

    Google Scholar 

  • Halmschlager A, Szenthe L, Tóth J (2003) Invariants of kinetic differential equations. Electron J Qual Theory Differ Equ 2003(14):1–14. Proceedings of the 7’th Colloquium on the Qualitative Theory of Differential Equations

    Google Scholar 

  • Hanusse P (1972) De l’existence d’un cycle limite dans l’évolution des systémes chimiques ouverts. C R Acad Sci Ser C 274:1245–1247

    CAS  Google Scholar 

  • Hanusse P (1973) Simulation des systémes chimiques par une methode de Monte Carlo. C R Acad Sci Ser C 277:93

    CAS  Google Scholar 

  • Heidel J, Fu Z (1999) Non-chaotic behaviour in three-dimensional quadratic systems, II. The conservative case. Nonlinearity 12:617–633

    Google Scholar 

  • Hernández-Bermejo B, Fairén V (1995) Nonpolynomial vector fields under the Lotka–Volterra normal form. Phys Lett A 206(1):31–37

    Article  Google Scholar 

  • Herschkowitz-Kaufman M (1975) Bifurcation analysis of nonlinear reaction-diffusion equations ii. Steady state solutions and comparison with numerical simulations. Bull Math Biol 37(6):589–636

    Article  Google Scholar 

  • Higgins J (1968) Some remarks on Shear’s Liapunov function for systems of chemical reactions. J Theor Biol 21:293–304

    Article  CAS  PubMed  Google Scholar 

  • Hirsch MW, Smale S, Devaney RL (2004) Differential equations, dynamical systems, and an introduction to chaos. Pure and applied mathematics, vol 60. Elsevier—Academic, Amsterdam

    Chapter  Google Scholar 

  • Hollis S (2004) Reaction-diffusionlab.m. http://www.math.armstrong.edu/faculty/hollis/mmade/RDL/

  • Hollis SL, Martin RH Jr, Pierre M (1987) Global existence and boundedness in reaction-diffusion systems. SIAM J Math Anal 18(3):744–761

    Article  Google Scholar 

  • Horn F (1973) On a connexion between stability and graphs in chemical kinetics. II. Stability and the complex graph. Proc R Soc Lond A 334:313–330

    Article  CAS  Google Scholar 

  • Horn FJM (1974) The dynamics of open reaction systems. In: SIAM-AMS proceedings, vol VIII. SIAM, Philadelphia, pp 125–137

    Google Scholar 

  • Horn F, Jackson R (1972) General mass action kinetics. Arch Ratl Mech Anal 47:81–116

    Article  Google Scholar 

  • Horváth Z (2002) Effect of lumping on controllability and observability. Poster presented at the colloquium on differential and difference equations dedicated to Prof. František Neuman on the occasion of his 65th birthday, Brno, September, pp 4–6

    Google Scholar 

  • Horváth Z (2002–2008) Effect of lumping on controllability and observability. arxivorg http://arxiv.org/PS_cache/arxiv/pdf/0803/0803.3133v1.pdf

  • Horváth D, Tóth Á (1998) Diffusion-driven front instabilities in the chlorite–tetrathionate reaction. J Chem Phys 108(4):1447–1451

    Article  Google Scholar 

  • Hsü ID (1976) Existence of periodic solutions for the Belousov–Zaikin–Zhabotinskiı̆ reaction by a theorem of Hopf. J Differ Equ 20(2):399–403

    Article  Google Scholar 

  • Huang Y, Yang XS (2006) Numerical analysis on the complex dynamics in a chemical system. J Math Chem 39(2):377–387

    Article  CAS  Google Scholar 

  • Isidori A (1995) Nonlinear control systems. Springer, Berlin

    Book  Google Scholar 

  • Jeffries C, Klee V, van den Driessche (1987) Qualitative stability of linear systems. Linear Algebra Appl 87:1–48

    Article  Google Scholar 

  • Johnston MD, Siegel D (2011) Linear conjugacy of chemical reaction networks. J Math Chem 49(7):1263–1282

    Article  CAS  Google Scholar 

  • Joshi B (2015) A detailed balanced reaction network is sufficient but not necessary for its Markov chain to be detailed balanced. Discrete Contin Dyn Syst Ser B 20(4):1077–1105

    Article  Google Scholar 

  • Kádas Z, Othmer H (1979a) Erratum: Stable limit cycles in a two-component bimolecular reaction system. J Chem Phys 72(12):1845

    Article  Google Scholar 

  • Kádas Z, Othmer H (1979b) Stable limit cycles in a two-component bimolecular reaction system. J Chem Phys 70(4):1845–1850

    Article  Google Scholar 

  • Kádas Z, Othmer H (1980) Reply to comment on stable limit cycles in a two-component bimolecular reaction system. J Chem Phys 72(4):2900–2901

    Article  Google Scholar 

  • Kertész V (1984) Global mathematical analysis of the Explodator. Nonlinear Anal 8(8):941–961

    Article  Google Scholar 

  • King RB (1982) The flow topology of chemical reaction networks. J Theor Biol 98(2):347–368

    Article  CAS  Google Scholar 

  • Kiss K, Kovács S (2008) Qualitative behaviour of n-dimensional ratio-dependent predator prey systems. Appl Math Comput 199(2):535–546

    Google Scholar 

  • Kiss K, Tóth J (2009) n-Dimensional ratio-dependent predator-prey systems with memory. Differ Equ Dyn Syst 17(1–2):17–35

    Article  Google Scholar 

  • Klee V, van den Driessche P (1977) Linear algorithms for testing the sign stability of a matrix and for finding Z-maximum matchings in acyclic graphs. Numer Math 28(3):273–285

    Article  Google Scholar 

  • Kordylewski W, Scott SK, Tomlin AS (1990) Development of oscillations in closed systems. J Chem Soc Faraday Trans 86(20):3365–3371

    Article  CAS  Google Scholar 

  • Kovács K, Vizvári B, Riedel M, Tóth J (2004) Computer assisted study of the mechanism of the permanganate/oxalic acid reaction. Phys Chem Chem Phys 6(6):1236–1242

    Article  Google Scholar 

  • Ladics T (2007) Application of operator splitting in the solution of reaction-diffusion equations. Proc Appl Math Mech 7(1):2020135–2020136. http://dx.doi.org/10.1002/pamm.200701017

    Article  Google Scholar 

  • Lagzi I, Kowalczyk B, Wang D, Grzybowski B (2010a) Nanoparticle oscillations and fronts. Angew Chem 122(46):8798–8801

    Article  Google Scholar 

  • Lagzi I, Soh S, Wesson PJ, Browne KP, Grzybowski BA (2010b) Maze solving by chemotactic droplets. J Amer Chem Soc 132(4):1198–1199

    Article  CAS  Google Scholar 

  • Lente G, Bazsa G, Fábian I (2007) What is and what isn’t a clock reaction? New J Chem 31:1707–1707

    Article  CAS  Google Scholar 

  • Lewin DR, Bogle D (1996) Controllability analysis of an industrial polymerization reactor. Comp Chem Engng 20:871–876

    Article  Google Scholar 

  • Li G, Rabitz H, Tóth J (1994) A general analysis of exact nonlinear lumping in chemical kinetics. Chem Eng Sci 49(3):343–361

    Article  Google Scholar 

  • Liggett TM (2010) Continuous time Markov processes: an introduction, vol 113. American Mathematical Society, Providence

    Google Scholar 

  • Martin RH Jr (1987) Applications of semigroup theory to reaction-diffusion systems. In: Gill TL, Zachary WW (eds) Nonlinear semigroups, partial differential equations and attractors. Lecture notes in mathematics, vol 1248. Springer, Berlin, pp 108–126

    Chapter  Google Scholar 

  • Maya-Yescas R, Aguilar R (2003) Controllability assessment approach for chemical reactors: nonlinear control affine systems. Chem Eng J 92:69–79

    Article  CAS  Google Scholar 

  • Mincheva M, Siegel D (2004) Stability of mass action reaction-diffusion systems. Nonlinear Anal 56(8):1105–1131

    Article  Google Scholar 

  • Mincheva M, Siegel D (2007) Nonnegativity and positiveness of solutions to reaction-diffusion systems. J Math Chem 42(4):1135–1145

    Article  CAS  Google Scholar 

  • Morales MF (1944) On a possible mechanism for biological periodicity. Bull Math Biophys 6:65–70

    Article  CAS  Google Scholar 

  • Müller-Herold U (1975) General mass-action kinetics. Positiveness of concentrations as structural property of Horn’s equation. Chem Phys Lett 33(3):467–470

    Article  Google Scholar 

  • Murphy MD, Ogle CA, Bertz SH (2005) Opening the “black box”: oscillations in organocuprate conjugate addition reactions. Chem Commun (7):854–856

    Article  Google Scholar 

  • Murray JD (2002) Mathematical biology (I. An introduction), interdisciplinary applied mathematics, vol 17, 3rd edn. Springer, New York

    Google Scholar 

  • Nagy I, Tóth J (2014) Quadratic first integrals of kinetic differential equations. J Math Chem 52(1):93–114

    Article  CAS  Google Scholar 

  • Nagy I, Póta G, Tóth J (2006) Detailed balanced and unbalanced triangle reactions (manuscript)

    Google Scholar 

  • Nagy-Ungvárai Z, Tyson JJ, Hess B (1989) Experimental study of the chemical waves in the cerium-catalyzed Belousov–Zhabotinskii reaction. 1. Velocity of trigger waves. J Phys Chem 93(2):707–713

    Article  Google Scholar 

  • Noszticzius Z, Farkas H, Schelly ZA (1984) Explodator and Oregonator: parallel and serial oscillatory networks. A comparison. React Kinet Catal Lett 25:305–311

    Article  CAS  Google Scholar 

  • Noszticzius Z, Horsthemke W, McCormick WD, Swinney HL, Tam WY (1987) Sustained chemical waves in an annular gel reactor: a chemical pinwheel. Nature 329(6140):619–620

    Article  CAS  Google Scholar 

  • Ogg RAJ (1947) The mechanism of nitrogen pentoxide decomposition. J Chem Phys 15(5):337–338

    Article  CAS  Google Scholar 

  • Otero-Muras I, Szederkényi G, Hangos KM, Alonso AA (2008) Dynamic analysis and control of biochemical reaction networks. Math Comput Simul 79:999–1009

    Article  Google Scholar 

  • Perko L (1996) Differential equations and dynamical systems. Springer, Berlin

    Book  Google Scholar 

  • Petrov V, Gaspar V, Masere J, Showalter K (1993) Controlling chaos in the Belousov—Zhabotinsky reaction. Nature 361(6409):240–243

    Article  CAS  Google Scholar 

  • Póta G (1983) Two-component bimolecular systems cannot have limit cycles: a complete proof. J Chem Phys 78:1621–1622

    Article  Google Scholar 

  • Póta G (1992) Exact necessary conditions for oscillatory behaviour in a class of closed isothermal reaction systems. J Math Chem 9(4):369–372

    Article  Google Scholar 

  • Póta G (1996) Chemical waves and spatial patterns in reaction-diffusion systems. Kossuth Egyetemi Kiadó, Debrecen (in Hungarian)

    Google Scholar 

  • Póta G (2016) Solutions defined on finite time intervals in a model of a real kinetic system. J Math Chem 54:1–5

    Article  CAS  Google Scholar 

  • Prigogine I, Lefever R (1968) Symmetry breaking instabilities in dissipative systems, II. J Chem Phys 48:1695–1700

    Article  Google Scholar 

  • Rábai G, Bazsa G, Beck MT (1979) Design of reaction systems exhibiting overshoot-undershoot kinetics. J Am Chem Soc 101(12):6746–6748

    Article  Google Scholar 

  • Romanovski V, Shafer D (2009) The center and cyclicity problems: a computational algebra approach. Birkhäuser, Boston

    Google Scholar 

  • Ronkin LI (1977) Elements of multivariate complex function theory. Naukova Dumka, Kiev

    Google Scholar 

  • Rosenbaum JS (1977) Conservation properties for numerical integration methods for systems of differential equations. 2. J Phys Chem 81(25):2362–2365

    Article  CAS  Google Scholar 

  • Rössler OE (1976) Chaotic behavior in simple reaction system. Zeitschrift für Naturforsch A 31:259–264

    Google Scholar 

  • Rössler OE (1976) An equation for continuous chaos. Phys Lett A 57(5):397–398

    Article  Google Scholar 

  • Rothe F (1984) Global stability of reaction-diffusion systems. Springer, Berlin

    Book  Google Scholar 

  • Schlosser PM, Feinberg M (1994) A theory of multiple steady states in isothermal homogeneous CFSTRs with many reactions. Chem Eng Sci 49(11):1749–1767

    Article  CAS  Google Scholar 

  • Schneider K, Wegner B, Tóth J (1987) Qualitative analysis of a model for synaptic slow waves. J Math Chem 1:219–234

    Article  Google Scholar 

  • Schuman B, Tóth J (2003) No limit cycle in two species second order kinetics. Bull Sci Math 127:222–230

    Article  Google Scholar 

  • Scott SK (1991, 1993, 1994) Chemical chaos. International series of monographs on chemistry, vol 24. Oxford University Press, Oxford

    Google Scholar 

  • Shapiro A, Horn F (1979a) Erratum. Math Biosci 46(1–2):157

    Google Scholar 

  • Shapiro A, Horn F (1979b) On the possibility of sustained oscillations, multiple steady states, and asymmetric steady states in multicell reaction systems. Math Biosci 44(1–2):19–39

    Article  CAS  Google Scholar 

  • Shear D (1967) An analog of the Boltzmann H-theorem (a Liapunov function) for systems of coupled chemical reactions. J Theor Biol 16:212–225

    Article  CAS  PubMed  Google Scholar 

  • Simon LP (1995) Globally attracting domains in two-dimensional reversible chemical dynamical systems. Ann Univ Sci Budapest Sect Comput 15:179–200

    Google Scholar 

  • Sipos-Szabó E, Pál I, Tóth J, Zsély IG, Turányi T, Csikász-Nagy A (2008) Sensitivity analysis of a generic cell-cycle model. In: 2nd FUNCDYN meeting, Rothenburg oberhalb der Taube. http://www.math.bme.hu/~jtoth/pubtexts/SiposSzaboTothPal.pdf

  • Smith HL (2012) Global dynamics of the smallest chemical reaction system with Hopf bifurcation. J Math Chem 50(4):989–995

    Article  CAS  Google Scholar 

  • Smoller J (1983) Shock waves and reaction-diffusion equations. Springer, New York

    Book  Google Scholar 

  • Szederkényi G, Hangos KM, Magyar A (2005) On the time-reparametrization of quasi-polynomial systems. Phys Lett A 334(4):288–294

    Article  CAS  Google Scholar 

  • Szili L, Tóth J (1993) Necessary condition of the Turing instability. Phys Rev E 48(1):183–186

    Article  CAS  Google Scholar 

  • Szili L, Tóth J (1997) On the origin of Turing instability. J Math Chem 22(1):39–53

    Article  CAS  Google Scholar 

  • Thomas R (1978) Logical analysis of systems comprising feedback loops. J Theor Biol 73(4):631–656

    Article  CAS  PubMed  Google Scholar 

  • Tomlin AS, Pilling MJ, Turányi T, Merkin J, Brindley J (1992) Mechanism reduction for the oscillatory oxidation of hydrogen: sensitivity and quasi-steady-state analyses. Comb Flame 91:107–130

    Article  CAS  Google Scholar 

  • Tóth J (1987) Bendixson-type theorems with applications. Zeitschrift für Angewandte Mathematik und Mechanik 67(1):31–35

    Article  Google Scholar 

  • Tóth J, Érdi P (1978) Models, problems and applications of formal reaction kinetics. A kémia újabb eredményei 41:227–350

    Google Scholar 

  • Tóth J, Érdi P (1988) Kinetic symmetries: some hints. In: Moreau M, Turq P (eds) Chemical reactivity in liquids. Fundamental aspects, Paris, Sept. 7–11 1987. Plenum Press, New York, pp 517–522

    Chapter  Google Scholar 

  • Tóth J, Hárs V (1986a) Orthogonal transforms of the Lorenz- and Rössler-equations. Physica D 19:135–144

    Article  Google Scholar 

  • Tóth J, Hárs V (1986b) Specification of oscillating chemical models starting form a given linearized form. Theor Chim Acta 70:143–150

    Article  Google Scholar 

  • Tóth Á, Lagzi I, Horváth D (1996) Pattern formation in reaction-diffusion systems: cellular acidity fronts. J Phys Chem 100(36):14837–14839

    Article  Google Scholar 

  • Tóth J, Li G, Rabitz H, Tomlin AS (1997) The effect of lumping and expanding on kinetic differential equations. SIAM J Appl Math 57:1531–1556

    Article  Google Scholar 

  • Turányi T (1990) Sensitivity analysis of complex kinetic systems. Tools and applications. J Math Chem 5(3):203–248

    Article  Google Scholar 

  • Turányi T, Tomlin AS (2014) Analysis of kinetic reaction mechanisms. Springer, Berlin

    Book  Google Scholar 

  • Turányi T, Györgyi L, Field RJ (1993) Analysis and simplification of the GTF model of the Belousov–Zhabotinsky reaction. J Phys Chem 97:1931–1941

    Article  Google Scholar 

  • Turing AM (1952) The chemical basis of morphogenesis. Philos Trans R Soc Lond Ser B Biol Sci 237(641):37–72

    Article  Google Scholar 

  • Tyson JJ (1977) Analytic representation of oscillations, excitability, and traveling waves in a realistic model of the Belousov–Zhabotinskii reaction. J Chem Phys 66(3):905–915

    Article  CAS  Google Scholar 

  • Tyson JJ (1980) Comment on stable limit cycles in a two-component bimolecular reaction system. J Chem Phys 72(4):2898–2899

    Article  CAS  Google Scholar 

  • Tyson JJ, Light JC (1973) Properties of two-component bimolecular and trimolecular chemical reaction systems. J Chem Phys 59(8):4164–4273

    Article  CAS  Google Scholar 

  • Vandenberghe L, Boyd S (1996) Semidefinite programming. SIAM Rev 38(1):49–95

    Article  Google Scholar 

  • Várdai J, Tóth J (2008) Hopf Bifurcation in the Brusselator. http://demonstrations.wolfram.com/HopfBifurcationInTheBrusselator/, from The Wolfram Demonstrations Project

  • Volpert AI (1972) Differential equations on graphs. Mat Sb 88(130):578–588

    Google Scholar 

  • Volpert AI, Hudyaev S (1985) Analyses in classes of discontinuous functions and equations of mathematical physics. Martinus Nijhoff Publishers, Dordrecht (Russian original: 1975)

    Google Scholar 

  • Weber A, Sturm T, Seiler W, Abdel-Rahman EO (2010) Parametric qualitative analysis of ordinary differential equations: computer algebra methods for excluding oscillations. Lect Notes Comput Sci 6244:267–279. Extended Abstract of an Invited Talk, CASC 2010

    Google Scholar 

  • Wilhelm T, Heinrich R (1995) Smallest chemical reaction system with Hopf bifurcation. J Math Chem 17:1–14

    Article  CAS  Google Scholar 

  • Willamowski KD, Rössler OE (1980) Irregular oscillations in a realistic abstract quadratic mass action system. Zeitschrift für Naturforschung A 35(3):317–318

    Article  Google Scholar 

  • Winfree AT (2001) The geometry of biological time, vol 12. Springer Science & Business Media, Berlin

    Google Scholar 

  • Wolkowicz H, Saigal R, Vandenberghe L (2000) Handbook of semidefinite programming: theory, algorithms, and applications, vol 27. Springer Science & Business Media, Berlin

    Google Scholar 

  • Ye YQ, Lo CY (1986) Theory of limit cycles, vol 66. American Mathematical Society, Providence

    Google Scholar 

  • Yuan Z, Chen B, Zhao J (2011a) An overview on controllability analysis of chemical processes. AIChE J 57(5):1185–1201

    Article  CAS  Google Scholar 

  • Yuan Z, Chen B, Zhao J (2011b) Controllability analysis for the liquid-phase catalytic oxidation of toluene to benzoic acid. Chem Eng Sci 66:5137–5147

    Article  CAS  Google Scholar 

  • Zak DE, Stelling J, Doyle FJ (2005) Sensitivity analysis of oscillatory (bio) chemical systems. Comput Chem Eng 29(3):663–673

    Article  CAS  Google Scholar 

  • Zhabotinsky AM (1964) Periodic liquid-phase oxidation reactions. Doklady Akademii Nauk SSSR 157:392–395

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Analysis, Budapest University of Technology and Economics, Budapest, Hungary

    János Tóth

  2. Chemical Kinetics Laboratory, Eötvös Loránd University, Budapest, Hungary

    János Tóth

  3. Department of Stochastics, Budapest University of Technology and Economics, Budapest, Hungary

    Attila László Nagy

  4. Department of Mathematics, North Carolina State University, Raleigh, USA

    Dávid Papp

Authors
  1. János Tóth
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Attila László Nagy
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Dávid Papp
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer Science+Business Media, LLC, part of Springer Nature

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Tóth, J., Nagy, A.L., Papp, D. (2018). Time-Dependent Behavior of the Concentrations. In: Reaction Kinetics: Exercises, Programs and Theorems. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-8643-9_8

Download citation

Publish with us

Policies and ethics

Search

Navigation