Abstract
Limit cycles are sought in a mathematical model of a simple hypothetical chemical reaction involving essentially only two reacting species. Physically, these limit cycles correspond to time-periodic oscillations in the concentrations of the two chemicals. A combination of analytical and numerical methods reveals that limit-cycle behaviour is only possible in a restricted region of the parameter space. Strong numerical evidence is presented to assert that the limit cycle is unique and stable to infinitesimal perturbations. Numerical solutions are displayed and discussed.
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References
R.J. Field and R.M. Noyes, Oscillations in chemical systems. IV. Limit cycle behavior in a model of a real chemical reaction,J. Chem. Phys. 60 (1974) 1877–1884.
J. Grasman,Asymptotic methods for relaxation oscillations and applications, Applied Mathematical Sciences, Vol. 63, Springer-Verlag, New York (1987).
B.F. Gray and M.J. Roberts, A method for the complete qualitative analysis of two coupled ordinary differential equations dependent on three parameters,Proc. Roy. Soc. London, Ser. A, 416 (1988) 361–389.
B.F. Gray and M.J. Roberts, Analysis of chemical kinetic systems over the entire parameter space. I. The Sal'nikov thermokinetic oscillator,Proc. Roy. Soc. London, Ser. A, 416 (1988) 391–402.
B.F. Gray and R.A. Thuraisingham, The cubic autocatalator: the influence of degenerate singularities in a closed system,J. Engineering Maths. 23 (1989) 283–293.
P. Gray, S.R. Kay and S.K. Scott, Oscillations of an exothermic reaction in a closed system. I. Approximate (exponential) representation of Arrhenius temperature-dependence,Proc. Roy. Soc. London, Ser. A, 416 (1988) 321–341.
P. Gray and S.K. Scott, Autocatalytic reactions in the CSTR: oscillations and instabilities in the system A + 2B → 3B; B → C,Chem. Eng. Sci. 39 (1984) 1087–1097.
C.A. Holmes and D. Wood, Studies of a complex Duffing equation in nonlinear waves on plane Poiseuille flow, inNonlinear Phenomena and Chaos, (S. Sarkar, editor), Malvern Physics Series, Adam Hilger Ltd., Bristol and Boston (1986).
D.W. Jordan and P. Smith,Nonlinear ordinary differential equations, Clarendon Press, Oxford (1977).
S.R. Kay and S.K. Scott, Oscillations of simple exothermic reactions in a closed system. II. Exact Arrhenius kinetics,Proc. Roy. Soc. London, Ser. A, 416 (1988) 343–359.
J.H. Merkin, D.J. Needham and S.K. Scott, Oscillatory chemical reactions in closed vessels,Proc. Roy. Soc. London, Ser. A 406 (1986) 299–323.
J.H. Merkin, D.J. Needham and S.K. Scott, On the creation, growth and extinction of oscillatory solutions for a simple pooled chemical reaction scheme,SIAM J. Appl. Math. 47 (1987) 1040–1060.
D.A. Sánchez,Ordinary differential equations and stability theory: An introduction, W.H. Freeman and Company, San Francisco (1968).
J.J. Tyson,The Belousov-Zhabotinskii reaction, Lecture Notes in Biomathematics, Vol. 10, Springer-Verlag, Berlin (1976).
M. Urabe, Galerkin's procedure for non-linear periodic series,Arch. Rational Mech. Anal. 20 (1965) 120–152.
Ye Yan-Qian,Theory of limit cycles, AMS Translations of Mathematical Monographs, Volume 66, American Mathematical Society, Providence, Rhode Island (1986).
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Forbes, L.K., Holmes, C.A. Limit-cycle behaviour in a model chemical reaction: the cubic autocatalator. J Eng Math 24, 179–189 (1990). https://doi.org/10.1007/BF00129873
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DOI: https://doi.org/10.1007/BF00129873