AMC-SS: Noise-Induced Transitions in Multiscale Systems

AMC-SS：多尺度系统中噪声引起的转变

• 批准号: 0604249
• 负责人: Richard Sowers
• 金额: $ 13万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0604249&HistoricalAwards=false
• 关键词: AMC; SS; Noise; Induced; Transitions

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.

本文的研究试图理解小噪声对动力系统分岔的影响。分岔涉及行为的急剧变化，而噪声具有平滑效应（概率密度函数求解的是抛物型偏微分方程，而不是输运方程）。我们感兴趣的正是这种主导分岔行为与小噪声之间的竞争。我们试图通过考虑一些具体的问题来阐明这一竞争。我们特别试图研究鸭式现象（它伊萨分叉和多尺度的结合），同宿轨道附近的两点运动，以及某些高维系统的行为。应用数学的中心主题之一是模拟演化行为的问题。 做到这一点的主要工具之一是微分方程;微分方程的理论基础在数学上很有吸引力，并且将定性效应转换为控制ODE的向量场相当简单。当然，对任何行为的诚实评估也必须包括降噪。 通常，这种噪声很小，其影响可以忽略不计，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.我们的兴趣是在这些情况下，噪声导致宏观上可见的随机跃迁的动力学系统.通过某些精心挑选的例子，我们试图了解一些不同的现象，这种随机跃迁发生. 我们研究的特定方程出现在许多领域：生物学，洋流研究和材料科学。