The analytical and numerical analysis of a model of a chemical ocillator
Compelli, Susan (1989) The analytical and numerical analysis of a model of a chemical ocillator. Master of Science thesis, Dublin City University.
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This thesis concerns the analysis of the Exlpodator model for a Belousov-Zhabotinskn type oscillating chemical reaction. The chemical kinetics of the reaction is discussed in detail and a system of kinetic equations, the Explodator, modelling the system is derived. The equations are reduced to the system of non-dimensionahsed equations.
X i — 2fi2+Xi(l—3^3)—£1^2—3u1x21
£2=¡¿4—+3aX3—x1x2, x3=u3-2az3 + X\X+/i-[x\
The existence for all time and boundedness of solutions of the Explodator are proved. It is also proved that any trajectory solution which starts in the positive octant subsequently remains m it and that the model has a unique equilibrium point in the positive octant for a wide range of parameter values.
The theory of Hopf bifurcation is introduced Stability is defined and the Hopf bifurcation theorem is explained. The stability properties of the equilibrium solutions are examined. A result is then proved that gives simple necessary and sufficient conditions in terms of the kinetic parameters, for an equilibrium point of the system to a be Hopf bifurcation point, and thus for there to be a family of limit cycle solutions AUTO, a software package for continuation and bifurcation problems in ordinary differential equations, is used to solve the system and to determine the stabihty of the periodic solutions. The numerical solutions of the model agree very well with the chemical kinetics of the reaction and mathematical theory.
Centre Manifold theory is used to reduce the model to a two-dimensional system with the same stabihty properties as the full system AUTO is then used to verify that the linear stability of the stationary solutions of the reduced system agree with that of the solutions of the full model.
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