Mathematical Sciences: Theory and Applications of Homoclinicand Heteroclinic Bifurcation

研究人员将继续开发新的 同宿和异宿分支的分析方法 应用动力系统的几个重要领域。 在 特别是，他们将继续开发一种新的 周期与非周期的分歧函数法 在同宿轨道和异宿环附近的解。 他们 将继续发展统一的理论内部 异宿层奇异摄动问题 分支技术和一个版本的阴影引理。 他们将继续研究 非严格双曲型偏微分方程和使用 异宿分歧技术 他们将延长一个 同宿线数值延拓的一个新的严格算法 和异宿轨道，包括延拓点 在那里平衡变成非双曲的。 最后他们 将研究反应扩散方程的非齐次解 近似空间齐次解的方程， 常微分方程的同宿轨道或异宿圈。

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.