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.

拟议的研究是试图了解小噪声对动态系统分叉的影响。尽管分叉涉及行为的急剧变化，但噪声具有平滑效应（概率密度函数解决了一个抛物线PDE，而不是传输方程）。这是在我们感兴趣的双面行为和小噪声之间的主要竞争。我们试图通过考虑许多特定问题来阐明竞争。我们特别寻求研究canard现象（ISA分叉和多个尺度的ISA组合），在同层轨道附近的两个点型，以及某些高维系统的行为。在应用数学中，一种中央主题的一种高度差异是一种问题。 这样做的维护之一是微分方程。微分方程的理论划分具有数学吸引力，并且将定性效应转换为管理Ode的矢量领域非常简单。对任何行为的诚实评估当然也必须包括。 通常，这种噪音足够小，可以忽略不计，但是在某些情况下，即使很小的噪声也可能主导系统的行为。在这些情况下，我们的兴趣在动态系统中导致宏观观察到的随机过渡。通过某些精心挑选的示例的手段，我们在某些精心挑选的示例中寻求多个不同的现象，这些现象发生了这种情况的发生。 我们在许多领域中学习的特定方程：生物学，洋流研究和材料科学。