一类奇异扰动问题的严格数值解
Rigorous numerics for a singular perturbation problem
摘要：
众所周知，快-慢系统很难用严格的数值分析，因为当所谓的小参数$ε$从0变化到小的非零值时，解空间的定性性质会发生根本变化。在这篇论文中，我开发了一个计算机辅助的严格的方法，可以用来结合拓扑工具证明存在的周期和连接轨道在近零参数制度。作为应用，证明了具有显式$ε_0$的$ε\in (0,ε_0]$的FitzHugh-Nagumo系统的周期轨道和同宿轨道的存在性.这篇论文是在Piotr Zgliczyovski教授的监督下编写的，并于2016年6月在Jagiellonian大学数学与计算机科学系获得哲学博士学位的部分要求。本论文的部分结果已发表在A. Czechowski和P. Zgliczybowski。小参数显式取值范围下FitzHugh-Nagumo方程周期解的存在性arXiv：1502.02451，SIAM J. Appl. Dyn.系统：15（3），1615-1655，并且可能发生文本重叠。论文还包含了一些（尚未）发表的结果，特别是关于存在的同宿轨道的明确范围的$ε$。
英文摘要：
Fast-slow systems are notoriously difficult to analyze with rigorous numerics, since the qualitative properties of the solution space change fundamentally when the so-called small parameter $ε$ is varied from 0 to small non-zero values. In this dissertation I develop a computer-assisted rigorous method which can be used in combination with topological tools for proving the existence of period and connecting orbits in the near-zero parameter regime. As an application, I prove the existence of periodic and homoclinic orbits in the FitzHugh-Nagumo system, for $ε\in (0,ε_0]$ with an explicit $ε_0$. This dissertation was prepared under supervision of prof. Piotr Zgliczyński and submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy, awarded at the Jagiellonian University, Department of Mathematics and Computer Science in June 2016. Some of the results of this dissertation have been previously published in A. Czechowski and P. Zgliczyński. Existence of periodic solutions of the FitzHugh-Nagumo equations for an explicit range of the small parameter. arXiv:1502.02451, SIAM J. Appl. Dyn. Syst., 15(3), 1615-1655 and text overlap may occur. The dissertation also contains some (yet) unpublished results, in particular concerning the existence of homoclinic orbits for explicit ranges of $ε$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
