Numerical analysis of bifurcation problems

分岔问题的数值分析

• 批准号: 4274-2007
• 负责人: Doedel, Eusebius
• 金额: $ 4.52万
• 依托单位: Concordia University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2007
• 资助国家: 加拿大
• 起止时间: 2007-01-01 至 2008-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=363357
• 关键词: Numerical; analysis; bifurcation; problems

My main research interest is the development, analysis, implementation, and application of numerical methods for analyzing the solution behavior of nonlinear equations. I have worked on dynamical systems, especially those modeled by ordinary differential equations, and I am also interested in nonlinear elliptic and parabolic partial differential equations.My work is motivated by the fascinating changes of behavior exhibited by solutions to nonlinear equations as parameters are varied: There is tremendous satisfaction in devising new numerical algorithms that help increase our understanding of these phenomena. Recent applications range from a classification of elemental periodic orbits of the Circular Restricted 3-Body Problem, which is relevant to space mission design, to the computation of invariant manifolds, which explains an apparent discontinuous response in a cardiac pacemaker cell model.I try to make the numerical methods available for use in widely varying research areas. A notable success has been the development of the research software AUTO, which continues to evolve in cooperation with researchers in a world-wide network.I will further develop high-order accurate, adaptive discretizations of elliptic PDEs, resulting in software to track solutions, and determine their bifurcations, even in near-singular applications. Another focus is continuation techniques for periodic orbits (including infinite period orbits) and invariant manifolds in dissipative as well as conservative systems. Important applications will be studied.

我的主要研究兴趣是用于分析非线性方程解行为的数值方法的开发、分析、实现和应用。我研究过动力系统，特别是那些用常微分方程建模的系统，而且我对非线性椭圆和抛物型偏微分方程也很感兴趣。我的工作动力来自于参数变化时非线性方程的解所表现出的令人着迷的行为变化：设计新的数值算法有助于增加我们对这些现象的理解，这是一种巨大的满足感。最近的应用范围从与空间任务设计相关的圆形受限三体问题的基本周期轨道的分类，到不变流形的计算（解释心脏起搏器细胞模型中明显的不连续响应）。我试图使数值方法可用于广泛不同的研究领域。研究软件 AUTO 的开发取得了显着的成功，该软件与全球网络中的研究人员合作不断发展。我将进一步开发椭圆偏微分方程的高阶精确自适应离散化，从而开发出跟踪解并确定其分叉的软件，即使在近奇异的应用中也是如此。另一个重点是耗散系统和保守系统中的周期轨道（包括无限周期轨道）和不变流形的连续技术。将研究重要的应用。