Mathematical Sciences: Chaos-Integrability Transition in Nonlinear Dynamical Systems: Exponental Asymptotics Approach

数学科学：非线性动力系统中的混沌可积性转变：指数渐近方法

• 批准号: 9500644
• 负责人: Alexander Tovbis
• 金额: $ 4万
• 依托单位: West Virginia University Research Corporation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-06-01 至 1997-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9500644&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Chaos; Integrability; Transition

9500644 Tovbis The objective of the project is to study the transition from integrable (predictable) to nonintegrable (chaotic) dynamics in perturbed nonlinear systems. This transition plays a fundamental role in a very wide variety of physical, biological, chemical, etc. problems (turbulence in incompressible fluids, scattering of atoms from metal surfaces, host - parasitoid models in population dynamics, complex formation in atom-diatom collisions, etc.). It is also crucial in understanding the nature of numerical instabilities (numerical chaos) in computational problems. The proposed approach to the problem combines classical analytic and modern asymptotic technique (exponential asymptotics). In our initial studies the model example shows an excellent agreement between the "theoretical" results and numerical simulations. %%% The objective of the project is to study the transition from integrable to nonintegrable (chaotic) dynamics in singularly perturbed nonlinear systems. This transition plays a fundamental role in a very wide variety of physical, biological, chemical, etc. problems. It is also crucial in understanding the nature of numerical instabilities (numerical chaos) il computational problems. Of our particular interest are situations when the effect of perturbation is exponentially small in the small parameter of the problem. We propose to apply the technique of exponential asymptotics to the study of the analytical mechanism of the onset of chaos in perturbed integrable systems and to approximate the dynamics of perturbed systems. This approach has proven fruitful in our initial investigation. ***

9500644 Tovbis 该项目的目标是研究扰动非线性系统中从可积（可预测）到不可积（混沌）动力学的转变。这种转变在各种物理、生物、化学等问题中发挥着基础作用（不可压缩流体中的湍流、金属表面原子的散射、群体动力学中的宿主-寄生模型、原子-硅藻碰撞中的复杂形成、 ETC。）。它对于理解计算问题中数值不稳定性（数值混沌）的本质也至关重要。所提出的解决问题的方法结合了经典分析和现代渐近技术（指数渐近学）。 在我们的初步研究中，模型示例显示“理论”结果与数值模拟之间非常吻合。 %%% 该项目的目标是研究奇扰动非线性系统中从可积到不可积（混沌）动力学的转变。这种转变在各种物理、生物、化学等问题中发挥着基础作用。它对于理解计算问题中数值不稳定性（数值混沌）的本质也至关重要。我们特别感兴趣的是当问题的小参数中扰动的影响呈指数小时的情况。我们建议应用指数渐近技术来研究摄动可积系统中混沌开始的解析机制，并近似摄动系统的动力学。 在我们的初步调查中，这种方法已被证明是卓有成效的。 ***