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

NSF奖摘要- DMS-0072596数学科学：应用动力学系统和奇异摄动理论的模式，气泡，和化学反应摘要0072596 Kaper这个研究项目包括化学图案形成，化学动力学和非线性动力学的气泡问题。自复制脉冲最近被发现作为新的化学模式，并在自我复制的中心作用是由强，非线性脉冲相互作用。在一维和二维空间中，脉冲（如凸块、圆环和圆形斑点）是耦合反应扩散方程中全局稳定的均匀态的局部化大振幅扰动。具体目标包括定位支配分裂的鞍节点（消失）分叉的层次，例如，当环解分裂成两个环，两个环分裂成四个环等时，并在二维中确定潜在的分裂机制。本研究的另一个方面重点关注这些模式的稳定性。通过改变慢抑制剂场到快激活剂场的耦合强度，以及利用最近发现的非局部特征值问题中的零极点相消，实现了在物理上重要的反应扩散方程耦合系统中脉冲稳定的控制。在化学反应理论中，该项目侧重于涉及许多物种和反应的大规模系统，以及减少需要建模的有效物种和反应数量的还原方法的开发。本计画研究利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。化学和流体力学领域长期以来对数学的发展产生了很大的影响;反过来，数学也导致了化学和流体力学领域的许多有用的发展。本研究课题利用数学理论，特别是应用非线性动力系统理论，对化学中的图案形成和大规模反应系统的基本问题，以及流体力学中气泡之间的非线性相互作用，进行新的见解和定量预测。一个非线性的控制机制，稳定的模式，其中反应化合物的浓度保持在所需的水平，在局部地区正在开发中。此外，该项目设计，实施和测试还原方法，已知对于模拟燃烧，反应流和其他技术重要问题中出现的化学反应的大规模系统至关重要。最后，本项目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，这项工作将导致更深入地了解气泡云的复杂问题，这些气泡云在潜艇后面产生噪音并损坏涡轮机叶片。

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.