Mathematical Sciences: The Dynamics of Singular Reduction and SubRiemannian Geometry

首席研究人员将探索在传统方法不存在或未被完全理解的情况下，机械系统平衡和稳定运动的存在和稳定性的各种变分标准。他们将尝试推导出在存在非常小的阻尼值的情况下，形式不稳定性准确地预测线性不稳定性的条件。另一个目标是发展一种系统的方法来分析相空间中对称状态的稳定性和分叉。该奖项将支持动力系统一般领域的研究。一个在当地非常简单和容易理解的过程可能会变得极其复杂、非线性和难以在全球范围内分析。动力系统是研究这种局部到全球关系的学科。许多物理系统都可以使用这一数学领域进行建模，包括流体流动和湍流、复杂的生物系统、机械系统和化学反应。

The principal investigators will explore various variational criteria for the existence and stability of equilibria and steady motions of mechanical systems in contexts where traditional approaches do not exist or are not completely understood. They will try to derive conditions under which formal instability accurately predicts linear instability in the presence of very small damping. Another objective is the development of a systematic approach to the stability and bifurcation analysis of symmetric states in phase space. This award will support research in the general area of dynamical systems. A process which is very simple and easy to understand locally can become extremely complicated, nonlinear, and difficult to analyze globally. Dynamical systems is the study of this local to global relationship. Many physical systems can best be modeled using this area of mathematics including fluid flow and turbulence, complex biological systems, mechanical systems, and chemical reactions.