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

研究者们继续研究同斜和异斜分岔以及同斜和异斜分岔理论至关重要的几个重要应用问题。他们继续Lin开始的工作，研究接近空间齐次解的反应-扩散方程的空间模式解，用反应方程的异斜循环上的同斜轨道表示。他们研究了一种基于早期工作的新方法，用于由规则层和过渡层组成的小扩散反应-扩散方程的稳定解的稳定性，其中稳定性将逐层检查。他们继续谢克特开始的工作，利用异斜分岔理论研究守恒律系统的解。他们还继续研究寻找非双曲平衡点的同斜和异斜轨道的数值方法的稳定性和收敛性。如果一个动态过程在它的一个稳态附近开始，这个过程的最终轨道会远离那个状态，然后回到相同或不同的稳态。这样的轨道分别称为同斜轨或异斜轨。这些轨道是理解许多科学现象的关键。研究人员继续研究同斜轨道和异斜轨道，以及它们在应用中重要的数学方面。他们研究的一个课题是两个相互作用的分散种群如何自发地组织成不同种群密度随时间波动的空间模式，因为潜在动力学中的同斜或异斜轨道。第二个主题是林的一个想法，用于理解传播锋面的稳定性，例如，在化学反应中，通过将锋面视为异斜轨道。第三个主题是复杂的波动，这些波动出现在方程的解中，比如那些从地下油藏中抽水迫使石油流出的石油开采模型。这些波可以看作是异斜轨道。第四个主题是用于计算同斜和异斜轨道的数值方法的鲁棒性和准确性。

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.