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

研究者们继续研究同宿和异宿分支以及几个重要的应用问题，其中同宿和异宿分支理论是关键。他们继续了由Lin开始的关于空间均匀解附近的反应扩散方程的空间模式化解的工作，由反应方程的异宿循环上的同宿轨道表示。在早期工作的基础上，他们研究了一种新的方法来研究具有小扩散的反应扩散方程驻定解的稳定性，该扩散方程由规则层和过渡层组成，其中稳定性将被逐层检查。他们继续了谢克特开始的工作，利用异宿分支理论研究守恒律系统的解。他们还继续研究寻找同宿轨和异宿轨到非双曲平衡的数值方法的稳定性和收敛。可能发生的情况是，如果一个人在其某个稳态附近启动一个动态过程，则该过程的结果轨道会远离该状态，然后返回到相同或不同的稳态。这样的轨道分别称为同宿轨或异宿轨。事实证明，这些轨道是理解科学中众多现象的关键。研究人员继续研究同宿轨和异宿轨，以及它们在应用中重要的数学方面。他们研究的一个主题是，由于潜在动力学中的同宿或异宿轨道，两个相互作用的分散人口如何自发地组织成不同人口密度的空间模式，这些模式随着时间的推移而波动。第二个主题是LIN的一种想法，他通过将传播前锋视为异宿轨道来理解传播前锋的稳定性，例如在化学反应中。第三个问题是在方程的解中出现的复杂波动，例如那些对地下储油层中的石油进行抽水以迫使其排出时的模型。这些波可以看作是异宿轨道。第四个主题是用于计算同宿轨和异宿轨的数值方法的稳健性和准确性。

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.