AMC-SS: Noise-Induced Transitions in Multiscale Systems

这项研究试图理解小噪声对动力系统中分叉的影响。当分叉涉及行为的急剧变化时，噪声具有平滑效应(概率密度函数求解抛物型偏微分方程，而不是输运方程)。我们感兴趣的是主导分叉行为和小噪声之间的竞争。我们试图通过考虑一些具体的问题来解释这场竞赛。我们特别试图研究鸭子现象(分叉和多个尺度的组合)、同宿轨道附近的两点运动以及某些高维系统的行为。应用数学中的一个中心主题是对演化行为进行建模的问题。这样做的主要工具之一是微分方程式；微分方程式的理论基础在数学上很吸引人，将定性效应转化为支配常微分方程的矢量场是相当简单的。当然，对任何行为的诚实评估也必须包括噪声。通常这种噪声小到可以忽略不计的影响，但在某些情况下，即使是很小的噪声也可能支配系统的行为。我们感兴趣的是在这些情况下，噪声导致了动力学系统中宏观上可见的随机跃迁。通过某些精心挑选的例子，我们试图理解发生这种随机跃迁的一些不同的现象。我们研究的具体方程出现在许多领域：生物学、洋流研究和材料科学。

The proposed research is an attempt to understand the effectof small noise on bifurcations in dynamical systems.While bifurcations involve sharp changes in behavior, noise has a smoothingeffect (the probability density function solves a parabolic PDE,rather than a transport equation).It is this competition between dominant bifurcative behaviorand small noise which is of interest to us. We seek to illuminatethis competition by considering a number of specific problems.We in particular seek to study canard phenomena (which isa combination of bifurcation and multiple scales), two-pointmotion near a homoclinic orbit, and behavior of certain higher-dimensionalsystems.One of the central themes in applied mathematics is the problemof modelling evolving behavior. One of the maintools for doing so is differential equations; the theoreticalunderpinnings of differential equations are mathematicallyattractive, and it is fairly simple to convert qualitativeeffects into the vector fields which govern ODE's.An honest assessment of any behavior must, of course, also includenoise. Usually this noise is small enough to have negligible effect, but in certainsituations, even small noise may dominate the behavior of a system.Our interest is in these situations where noise causes macroscopicallyvisible random transitions in a dynamical system.By means of certain carefully-chosen examples, we seek tounderstand a number of different phenomena where suchrandom transitions occur. The specific equations we studyappear in a number of areas: biology, the study of ocean currents,and materials science.