Mathematical Sciences: Computational Methods for Global Analysis of Hemoclinic and Heterclinic Orbits: Theoretical Analysis and Applications

最近，Doedel和研究人员发展了一种精确、稳健和系统的方法来计算自治常微分方程组在解指数逼近不动点的情况下的异宿轨道分支，并在AUTO软件包中实现了该方法。在这个项目中，研究者在几个方向推广了上述结果：1)发展了不稳定流形和稳定流形的高阶近似，这在延拓过程中失去指数衰减率的情况下特别有用，即出现中心流形；2)考虑同宿轨和异宿轨分叉的几个问题，包括中心流形的情况，3)建立后验误差估计和自适应过程。目标是为自适应技术奠定基础，这些技术对于开发健壮可靠的全球同宿轨和异宿轨分析软件是必不可少的。无论是同宿轨还是异宿轨，连接常微分方程不动点的轨道的存在在各种应用中都具有特殊的意义。这些轨迹已被证明是流体力学中的间歇现象和数学生物学中的“爆发”现象的基础；它们出现在电路、激光以及光纤应用中的光脉冲的混沌行为中；它们出现在结构力学和各种化学反应中的混沌行为中，在这些反应中，反应物浓度的混沌振荡是由于同宿轨道的存在。

Recently Doedel and the investigator have developed an accurate, robust, and systematic method for computing branches of heteroclinic orbits for a system of autonomous ordinary differential equations in the case that the solution approaches the fixed points exponentially, and they implemented the method in the software package AUTO. In this project the investigator extends the above results in several directions: 1) developing higher order approximation of the unstable and stable manifolds, which is especially useful in the case when during the continuation process the exponential rate of decay is lost, i.e. a center manifold appears; 2) considering several problems with bifurcations of homoclinic and heteroclinic orbits, including the case of center manifolds, 3) developing a posteriori error estimates and the adaptive procedure. The goal is to lay the foundation for adaptive techniques that are essential in the development of robust and reliable software for the global analysis of homoclinic and heteroclinic orbits. The existence of a trajectory connecting fixed points of an ordinary differential equation, a homoclinic or heteroclinic orbit, is of special significance in a variety of applications. These trajectories have been shown to underlie the phenomena of intermittency in fluid mechanics and of "bursting" phenomena in mathematical biology; they appear in the chaotic behavior of electrical circuits, of lasers as well as light pulses in fiberoptics applications; they arise in chaotic behavior in structural mechanics and in a variety of chemical reactions where chaoticoscillations in the reactant concentrations are due to the presence of a homoclinic orbit.