Homoclinic and Heteroclinic Bifurcations, Shock Waves, and Singular Perturbations

9973105 schecter研究者建议继续对奇异微扰、守恒律系统和非守恒形式的相关系统、同斜和异斜分岔现象进行研究。这些领域是相关的：奇异摄动问题中的锐锋，守恒定律系统和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斜（有时是同斜）解。提出的研究分为四个方面：(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)非守恒形式系统初值问题的激波解；(3)模拟地球磁场动力机制的扰动瑞利-贝纳德对流模型中的混沌；(4)其他涉及奇异微扰或同斜分岔的应用问题（强迫蔡氏回路、多阶段癌症模型、血浆和鞘模型）。尖锐的波锋出现在许多科学领域。在近似的第一级，它们是简单的行波，保持其形状并以恒定的速度移动。在数学上，行波对应于“常微分方程的异斜解”。它们可以被认为是移动的粒子，从一个不稳定的平衡开始，向另一个不稳定的平衡移动。这种解决方案的循环经常导致混乱。研究者建议在这些相关领域继续进行研究。例如，他们将研究在石油开采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”系统的冲击波，当原始系统被简化以使其更易于处理时，会出现在流体力学和弹性问题中。他们将研究地球磁场模型中的混沌现象，这被认为与过去发生的磁场不规则倒转有关。他们将尝试改进用于模拟各种工业情况的“等离子鞘”模型，例如荧光灯装置；解决方案包括一个尖锐的正面。

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.