Abstract
A second-order accurate numerical method has been proposed for the solution of a coupled non-linear oscillator featuring in chemical kinetics. Although implicit by construction, the method enables the solution of the model initial-value problem (IVP) to be computed explicitly. The second-order method is constructed by taking a linear combination of first-order methods. The stability analysis of the system suggests the existence of a Hopf bifurcation, which is confirmed by the numerical method. Both the critical point of the continuous system and the fixed point of the numerical method will be seen to have the same stability properties. The second-order method is more competitive in terms of numerical stability than some well-known standard methods (such as the Runge–Kutta methods of order two and four).
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References
R. Lefever and G. Nicolis, Chemical instabilities and sustained oscillations, J. Theor. Biol. 30 (1971) 267.
G. Nicolis and I. Prigogine, Self-Organization in Non-Equilibrium Systems (Wiley-Interscience, 1977).
J. Tyson, Some further studies of non-linear oscillations in chemical systems, J. Chem. Phys. 58 (1994) 3919.
G. Adomian, The diffusion Brusselator equation, Comput. Math. Appl. 29(5) (1995) 1-3.
J.D. Lambert, Numerical Methods for Ordinary Differential Systems: The Initial-Value Problem (Wiley, 1991).
I.E. Marsden and M. McCracken, The Hopf bifurcation and Its Applications (Springer-Verlag, New York, 1976).
B.D. Hassard, N.D. Kazarinoff and Y.-H. Wan, Theory and Applications of Hopf bifurcation (Cambridge University Press, 1981).
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Gumel, A., Langford, W., Twizell, E. et al. Numerical solutions for a coupled non-linear oscillator. Journal of Mathematical Chemistry 28, 325–340 (2000). https://doi.org/10.1023/A:1011025104111
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DOI: https://doi.org/10.1023/A:1011025104111