Mathematical Sciences: Theory and Applications of Homoclinicand Heteroclinic Bifurcation

研究人员将继续开发应用动力系统几个重要领域的同宿和异宿分叉的新的分析方法。特别是，他们将继续发展一种新的分支函数方法，以求同宿轨和异宿环附近的周期和非周期解。他们将利用异宿分支技术和一个版本的影子引理，继续发展奇异摄动问题的内层统一理论。他们将继续使用异宿分支技术处理非严格双曲偏微分方程组和改变类型的偏微分方程组的激波解。他们将扩展一种新的严格算法，用于同宿和异宿轨道的数值延续，以包括延续到平衡点变为非双曲的点。最后，他们将研究由常微分方程组的同宿轨道或异宿循环表示的空间齐次解附近的反应扩散方程的非齐次解。

The investigators will continue development of new analytic approaches to homoclinic and heteroclinic bifurcation in several important areas of applied dynamical systems. In particular, they will continue development of a new bifurcation function approach to periodic and aperiodic solutions near homoclinic orbits and heteroclinic cycles. They will to continue development of a unified theory of interior layers of singular perturbation problems using heteroclinic bifurcation techniques and a version of the shadowing lemma. They will continue working with shock wave solutions of nonstrictly hyperbolic PDE's and PDE's that change type using heteroclinic bifurcation techniques. They will extend a new rigorous algorithm for numerical continuation of homoclinic and heteroclinic orbits to include continuation to the point where an equilibrium becomes nonhyperbolic. Finally, they will investigate nonhomogeneous solutions of reaction-diffusion equations near spatially homogeneous solutions represented by homoclinic orbits or heteroclinic cycles of an ODE.