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

最近，Doedel和研究者开发了一种精确的、鲁棒的、系统的方法，用于计算一个自治常微分方程系统在解指数接近不动点的情况下的异斜轨道分支，并在软件包AUTO中实现了该方法。在这个项目中，研究者在几个方向上扩展了上述结果：1)发展了不稳定流形和稳定流形的高阶近似，这在延续过程中指数衰减率丢失的情况下特别有用，即出现中心流形；2)考虑了包括中心流形在内的同斜和异斜轨道分岔问题；3)发展了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，这些技术对于开发用于同斜轨道和异斜轨道全球分析的强大而可靠的软件至关重要。连接常微分方程不动点的轨道的存在性，即同斜轨道或异斜轨道，在各种应用中具有特殊的意义。这些轨迹已被证明是流体力学中的间歇性现象和数学生物学中的“爆发”现象的基础；它们出现在电路的混沌行为中，激光以及光纤应用中的光脉冲；它们出现在结构力学中的混沌行为和各种化学反应中，其中反应物浓度的混沌振荡是由于同斜轨道的存在。

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.