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Multiple Time Scale Dynamics Of Chemical Oscillators

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Abstract

In this thesis, we analyze the multiple time scale dynamics of two chemical oscillator models: the "autocatalator," a three-dimensional, two time scale vector field that satisfies the the law of mass action for an autocatalytic chemical reaction, and a four-dimensional model of the Belousov-Zhabotinskii (BZ) reaction taking place in a continuous-flow stirred tank called Model D. For each model, we concentrate on the multiple time scale nature of the reaction and the mechanisms that create mixed-mode oscillations (MMOs) in the models. In the analysis of the autocatalator, we show that a Poincar´ return map sie multaneously exhibits full rank and rank deficient behavior for different regions of phase space. Canard trajectories that follow a two-dimensional repelling slow manifold separate these regions. This allows us to compute a one-dimensional induced map from approximations of the return maps. The bifurcations of these induced maps are used to characterize the bifurcations of the mixed mode oscillations of the full three-dimensional system. We also analyze a four-dimensional model of the BZ reaction called Model D, first proposed by L. Gyorgyi and R. Field. Using experimental parameters as ¨ model parameters, we investigate the dynamic mechanisms shaping behavior in the low flow rate and high flow rate complexity regimes. We use geometric singular perturbation theory to interpret the behavior of the system in regions of phase space with a clear separation of time scales. At low flow rates, we show that a dynamic Hopf bifurcation is responsible for the creation of the small am- plitude oscillations of the MMOs. At high flow rates, the dynamics are shaped by interactions with an equilibrium point. Finally, we show that Model D is capable of replicating experimentally observed behaviors.

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2011-08-31

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Dynamical systems; Geometric singular perturbation theory; Mixed-mode oscillations

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Guckenheimer, John Mark

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Rand, Richard Herbert
Smillie, John D

Degree Discipline

Applied Mathematics

Degree Name

Ph. D., Applied Mathematics

Degree Level

Doctor of Philosophy

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Government Document

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dissertation or thesis

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