Transition and pattern formation in physical and physiological systems

将数学思想应用于物理学和生物学的开放性问题是一个不断发展的研究领域。这种增长的催化剂是科学计算，它使我们能够解决直到最近还遥不可及的复杂问题。在目前的建议中，使用动力系统的数学理论来研究人脑动力学和流体湍流。尽管这些应用程序显然不相关，但它们有许多共同点。例如，它们都是强非线性的，它们都会产生在空间尺度和时间尺度范围内一致的信号。一种常用的测量大脑动态的方法是脑电图仪，它经常显示出在首选频率上的振荡，比如8-13Hz的α频率，也显示出“脑电波”，即在大脑皮层上扩散的同步活动。在流体湍流中，我们看到了垂直结构的形成、相互作用和破坏。这里提出的工作旨在解释这种连贯行为的动力学和起源。

The application of mathematical ideas to open problems in physics and biology is an ever growing area of research. The catalyst of this growth is scientific computing, which allows us to attack problems of a complexity that was, until recently, out of reach. In the current proposal, the mathematical theory of dynamical systems is used to study human brain dynamics and fluid turbulence. Although apparently unrelated, these applications have many traits in common. For instance, they are both known to be strongly nonlinear, and they both give rise to signals that can be coherent over a range of spatial scales and time scales. One common measurement of brain dynamics, the electroencephalograph, often shows oscillations at preferred frequencies, like the 8-13Hz alpha frequency, and also shows "brain waves", synchronized activity that spreads out over the cortex. In fluid turbulence we see vortical structures form, interact, and break down. The work proposed here aims to explain the dynamics and origin of such coherent behaviour.