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

NSF奖摘要-DMS-0072596数学科学：模式、气泡和化学反应的应用动力学系统和奇异微扰理论这项研究项目包括化学模式形成、化学动力学和气泡的非线性动力学问题。自复制脉冲最近被发现是一种新的化学模式，而强的非线性脉冲相互作用在自复制中起着核心作用。脉冲，如一维和二维中的凸起、环形环和圆形光斑，是控制耦合反应扩散方程中全局稳定的齐次状态的局域大幅度扰动。具体目标包括定位控制分裂的鞍结点(消失)分支的层次结构，例如，当环解分裂成两个环、两个环分为四个环时，以及确定在两个维度上的潜在分裂机制。这项研究的另一个方面是关注这些模式的稳定性。在物理上重要的反应-扩散耦合方程系统中，通过改变慢抑制场到快激活场的耦合强度，以及利用最近在非局部本征值问题中发现的零极点抵消，实现了对脉冲稳定化的控制。在化学反应理论中，这个项目专注于涉及许多物种和反应的大规模系统，以及减少需要模拟的有效物种和反应数量的还原方法的开发。该项目研究使用几何奇异摄动理论在反应扩散方程组中寻找低维流形的迭代数值方法。最后，该项目开发并分析了液体中气泡相互作用的完全非线性模型。长期以来，化学和流体力学领域对数学的发展产生了强烈的影响；反过来，数学导致了化学和流体力学的许多有用的发展。本研究项目运用数学理论，特别是应用非线性动力系统理论，对化学中图样形成和大反应系统中的基本问题以及流体力学中气泡之间的非线性相互作用进行了新的见解和定量的预测。一种稳定模式的非线性控制机制正在开发中，在这种机制中，相互作用的化合物的浓度在局部区域保持在理想的水平。此外，该项目还设计、实施和测试还原方法，众所周知，这些方法对于对燃烧、反应流动和其他重要技术问题中出现的大规模化学反应系统进行建模至关重要。最后，对液体中气泡的非线性相互作用进行了基础理论研究。从长远来看，这项工作将加深对气泡云复杂问题的理解，气泡云会在潜艇后面产生噪音，并损坏涡轮叶片。

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.