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

点击翻译按钮获取中文摘要

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.