Transition and pattern formation in physical and physiological systems

数学思想在物理学和生物学中的应用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无法解决的复杂问题。在目前的提议中，动力系统的数学理论被用来研究人脑动力学和流体湍流。虽然表面上看起来不相关，但这些应用程序有许多共同的特征。例如，已知它们都是强非线性的，并且它们都产生可以在空间尺度和时间尺度范围内相干的信号。一种常见的大脑动力学测量方法，脑电图仪，经常显示在首选频率的振荡，如8- 13赫兹的阿尔法频率，也显示“脑波”，同步活动，分布在皮层。在流体湍流中，我们看到旋涡结构的形成、相互作用和分解。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.