THREE-DIMENSIONAL MODELS OF THE BELOUSOV-ZHABOTINSKII CHEMICAL REACTION.

摘要

Two three dimensional kinetic models of the Belousov-Zhabotinskii chemical reaction are studied and compared. The Field-Noyes model is compared with a new model, called the Quadratic model, whose chemistry is based upon an observation made by Tyson 1979 concerning the stoichiometry of the Field-Noyes kinetics. Both models are shown to possess a unique nontrivial steady state at which a Hopf bifurcation occurs as a parameter passes through a critical point. A locus of marginal stability is computed for physically permissible parameter ranges and the direction and stability of the Hopf bifurcation is determined for both cases. A change in the direction and stability of the Hopf bifurcation is found numerically for both models. It is this point in the Field-Noyes kinetic parameter space that is crucial to the discovery of a secondary Hopf bifurcation resulting from the two critical mode interaction in the reaction-diffusion equation discussed in Chapter III.; Nonuniform disturbances in the vincinity of the uniform steady state are studied for the two kinetic models in Chapter II. A static bifurcation from the uniform steady state when the kinetics are stable is shown to occur and those wave numbers and wave lengths signaling the onset of this bifuraction are computed. An asymptotic expansion of the bifurcating standing wave solution is computed resulting in the determination of its direction and stability of bifurcation. A proof is given which precludes the existence of oscillatory diffusive instabilities and also shows that both kinetic mechanisms sustain oscillations through self-inhibition and cross-catalysis rather than by auto-catalysis of their chemical concentrations.; An analysis of the interaction of static and Hopf bifurcations in the Field-Noyes reaction-diffusion model is performed in Chapter III using center manifold theory and the theory of normal forms. A spectral splitting process is used to reduce the co-dimension two problem from one defined on an infinite dimensional function space to one defined on a finite dimensional space of amplitudes. Secondary bifurcations of the original partial differential equation are investigated. Periodic or quasiperiodic solutions on the surface of a torus are found to bifurcate from the kinetic oscillation.