Mathematical Sciences: Theory and Applications of Homo- clinic and Heteroclinic Bifurcation

研究人员继续研究同宿位和 异宿分支及其几个重要应用 同宿和异宿分歧理论 至关重要 他们继续着林在空间上的工作， 反应扩散方程的模式解 空间齐次解，用同宿轨道表示 关于反应方程的异宿环 他们 研究一种新的方法，根据以前的工作， 反应扩散方程定常解的稳定性 具有由规则和过渡组成的小扩散 分层，逐层检查稳定性。 他们继续由Schecter开始的工作， 分歧理论研究系统的解决方案 守恒定律 他们还继续致力于稳定和 收敛的数值方法寻找同宿和 异宿轨道到非双曲平衡点。 如果一个人在一点附近启动一个动态过程， 在其稳定状态中， 然后回到相同或不同的稳态 状态 这样的轨道称为同宿或异宿轨道 分别 这些轨道是理解 科学中的许多现象。 调查人员继续 他们对同宿轨道和异宿轨道的研究，以及 在应用中很重要的数学方面。 他们研究的一个主题是两个相互作用的物体如何分散 人口可以自发地组织成空间 不同的人口密度模式， 时间，因为同宿或异宿轨道， 潜在的动力学 第二个主题是林语堂的思想 了解传播前沿的稳定性，在化学 反应，例如，通过将前线视为异宿线， 轨道 第三个主题是复杂的波浪， 方程的解，例如模拟 当水被泵入地下水库时， 逼出来 这些波可以看作是异宿轨道。 第四个课题是数值模拟的鲁棒性和准确性 用于计算同宿和异宿轨道的方法。

The investigators continue research on homoclinic and heteroclinic bifurcation and on several important applied problems in which homoclinic and heteroclinic bifurcation theory are crucial. They continue work begun by Lin on spatially patterned solutions of reaction-diffusion wquations near spatially homogeneous solutions, represented by homoclinic orbits on heteroclinic cycles of the reaction equation. They investigate a new approach, based on earlier work, to the stability of stationary solutions of reaction-diffusion equations with small diffusion that consist of regular and transition layers, in which the stability would be checked layer by layer. They continue work begun by Schecter using heteroclinic bifurcation theory to study the solutions of systems of conservation laws. They also continue work on the stability and convergence of numerical methods for finding homoclinic and heteroclinic orbits to nonhyperbolic equilibria. It can happen that if one starts a dynamic process near one of its steady states, the resulting orbit of the process goes far from that state, then returns to the same or a different steady state. Such an orbit is called homoclinic or heteroclinic respectively. These orbits turn out to be key to understanding numerous phenomena in the sciences. The investigators continue their studies of homoclinic and heteroclinic orbits, and of mathematical aspects of them that are important in applications. One subject they investigate is how two interacting dispersed populations can spontaneously become organized into spatial patterns of varying population densities that fluctuate over time, because of homoclinic or heteroclinic orbits in the underlying dynamics. The second subject is an idea of Lin for understanding the stability of propagating fronts, in chemical reactions, for example, by viewing the fronts as heteroclinic orbits. The third subject is complicated waves that arise in solutions of equations such as those that model the recovery of oil from an underground reservoir when water is pumped in to force it out. These waves can be viewed as heteroclinic orbits. The fourth subject is the robustness and accuracy of numerical methods used to compute homoclinic and heteroclinic orbits.