Homoclinic and Heteroclinic Bifurcations, Shock Waves, and Singular Perturbations

同宿和异宿分岔、冲击波和奇异扰动

• 批准号: 9973105
• 负责人: Stephen Schecter
• 金额: $ 12.5万
• 依托单位: North Carolina State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-08-01 至 2002-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9973105&HistoricalAwards=false
• 关键词: Homoclinic; Heteroclinic; Bifurcations; Shock; Waves

9973105SchecterThe investigators propose to continue their research on singular perturbations, systems of conservation laws and related systems not in conservation form, and homoclinic and heteroclinic bifurcation phenomena. These areas are related: sharp fronts in singular perturbation problems, and shock waves in systems of conservation laws and related systems not in conservation form, can be viewed as traveling waves, which correspond to heteroclinic (or sometimes homoclinic) solutions of an associated equation. The proposed research falls into four areas: (1) systems of conservation laws (failure of strict hyperbolicity, geometric singular perturbation theory and the Dafermos regularization); (2) shock solutions of initial-value problems for systems not in conservation form; (3) chaos in a perturbed Rayleigh-Benard convection model that models the dynamo mechanism responsible for the earth's magnetic field; (4) other applied problems involving singular perturbation or homoclinic bifurcation (forced Chua's circuit, multiple-stage cancer models, plasma and sheath models).Sharp wave fronts occur in many areas of science. At the first level of approximation, they are simple traveling waves that keep their shape and move at a constant speed. Mathematically, traveling waves correspond to "heteroclinic solutions of ordinary differential equations." These can be thought of as moving particles that start near an unstable equilibrium and move toward another unstable equilibrium. Cycles of such solutions often give rise to chaos. The investigators propose to continue their research in these related areas. For example, they will study certain unusual shock waves that arise in model equations for oil recovery. They will study shock waves for systems "not in conservation form," which arise in fluid mechanics and elasticity problems when the original systems are simplified to make them more tractable. They will study chaos in a model for the earth's magnetic field, which is believed to be related to irregular reversals of the magnetic field that have occurred in the past. They will try to improve the "plasma-sheath" models used to model various industrial situations, such as fluorescent light fixtures; the solutions include a sharp front.

9973105学者建议继续研究奇摄动、守恒律系统和非守恒形式的相关系统，以及同宿和异宿分支现象。这些领域是相关的：奇异摄动问题中的尖锋和守恒律系统中的激波以及非守恒形式的相关系统中的激波可以被视为行波，它们对应于相关方程的异宿(有时是同宿)解。这项研究分为四个方面：(1)守恒律系统(严格双曲性失效、几何奇异摄动理论和Dafermos正则化)；(2)非守恒型系统初值问题的激波解；(3)扰动Rayleigh-Benard对流模型中的混沌，该模型模拟了产生地球磁场的发电机机制；(4)其他涉及奇异摄动或同宿分叉的应用问题(强迫蔡氏电路、多阶段癌症模型、等离子体和鞘模型)。尖峰存在于许多科学领域。在第一级近似中，它们是简单的行波，保持其形状并以恒定的速度移动。从数学上讲，行波相当于“常微分方程异宿解”。这些可以被认为是运动的粒子，它们从一个不稳定的平衡附近开始，然后向另一个不稳定的平衡移动。这样的解决方案的循环往往会导致混乱。调查人员建议继续在这些相关领域进行研究。例如，他们将研究在采油模型方程中出现的某些不寻常的冲击波。他们将研究“非守恒形式”的系统的冲击波，这是在流体力学和弹性问题中出现的，当原始系统被简化以使其更容易处理时。他们将在地球磁场的模型中研究混沌，据信这与过去发生的磁场不规则逆转有关。他们将努力改进用于模拟各种工业情况的“等离子体鞘”模型，例如荧光灯灯具；解决方案包括尖锐的正面。