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

研究人员继续研究同宿和异宿分岔以及同宿和异宿分岔理论至关重要的几个重要应用问题。 他们继续由 Lin 开始研究反应扩散方程的空间图案解，该解接近空间均匀解，由反应方程异宿循环上的同宿轨道表示。 他们基于早期的工作，研究了一种新方法，用于解决由规则层和过渡层组成的小扩散反应扩散方程的稳态解的稳定性，其中稳定性将逐层检查。 他们继续谢克特开始的工作，利用异宿分岔理论来研究守恒定律系统的解。 他们还继续研究数值方法的稳定性和收敛性，以寻找同宿和异宿轨道到非双曲平衡。 可能会发生这样的情况：如果在接近其稳态之一的情况下开始动态过程，则该过程的最终轨道会远离该状态，然后返回到相同或不同的稳态。 这样的轨道分别称为同宿或异宿。 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和异宿轨道，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种群如何自发地组织成不同种群密度的空间模式，这些密度随时间波动。 第二个主题是林的想法，用于理解化学反应中传播前沿的稳定性，例如，通过将前沿视为异宿轨道。 第三个主题是方程解中出现的复杂波，例如模拟从地下水库中泵入水以将其抽出时从地下水库中回收石油的方程。 这些波可以被视为异宿轨道。 第四个主题是用于计算同宿和异宿轨道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.