Applied dynamical systems and singular perturbation theory for patterns, bubbles and chemical reactions

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NSF Award Abstract - DMS-0072596Mathematical Sciences: Applied Dynamical Systems and Singular Perturbation Theory for Patterns, Bubbles, and Chemical ReactionsAbstract0072596 KaperThis research project encompasses problems in chemical pattern formation, chemical kinetics, and nonlinear dynamics of gas bubbles. Self-replicating pulses have recently been discovered as new chemical patterns, and a central role in self-replication is played by strong, nonlinear pulse interactions. Pulses, such as bumps, annular rings, and circular spots in one and two dimensions, are localized large-amplitude perturbations of globally stable homogeneous states in the governing coupled reaction-diffusion equations. Specific aims include locating the hierarchies of saddle-node (disappearance) bifurcations that govern splitting, for example, when a ring solution splits into two rings, two rings into four, etc., and determining the underlying splitting mechanisms in two dimensions. Another aspect of this research focuses on stability of these patterns. Control over the stabilization of pulses in physically important systems of coupled reaction-diffusion equations is achieved by varying the strength of the coupling of the slow inhibitor field to the faster activator field, and by exploiting a recently discovered zero-pole cancellation in the nonlocal eigenvalue problems. In chemical reaction theory, this project focuses on large-scale systems involving many species and reactions and on the development of reduction methods that decrease the number of effective species and reactions that need to be modeled. The project investigates iterative numerical methods to find low dimensional manifolds in systems of reaction-diffusion equations using geometric singular perturbation theory. Finally, the project develops and analyzes a fully nonlinear model of the interactions of gas bubbles in liquids.The fields of chemistry and fluid mechanics have long had a strong influence on the development of mathematics; and in turn, mathematics has led to many useful developments in both chemistry and fluid mechanics. This research project uses mathematical theory, specifically applied nonlinear dynamical systems theory, to gain new insights and make quantitative predictions for fundamental problems in pattern formation and large-scale reaction systems in chemistry and for nonlinear interactions between gas bubbles in fluid mechanics. A nonlinear control mechanism for stabilizing patterns in which the concentrations of thereacting compounds are maintained at desirable levels in localized regions is under development. In addition, the project designs, implements, and tests reduction methods, known to be essential for modeling the large-scale systems of chemical reactions that arise in combustion, reacting flows, and other technologically important problems. Finally, the project carries out fundamental theoretical research on the nonlinear interaction of gas bubbles in liquids. Over the long term, this work will lead to deeper understanding of the complex problems of bubble clouds that generate noise behind submarines and damage turbine blades.