Numerical analysis of bifurcation problems

我的主要研究兴趣是发展、分析、实现和应用数值方法来分析非线性方程的解行为。我研究过动力系统，特别是那些用常微分方程建模的系统，我也对非线性椭圆型和抛物型偏微分方程感兴趣。当参数变化时，非线性方程的解所表现出的令人着迷的行为变化激发了我的工作：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元素周期轨道分类到不变流形的计算，这解释了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的开发取得了显著的成功，该软件在全球网络中与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岔的软件，甚至在近奇异应用中也是如此。另一个重点是周期轨道（包括无限周期轨道）的延拓技术，以及耗散和保守系统中的不变流形。重要的应用将被研究。

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.