Problems in Nonlinear Dynamics

非线性动力学问题

• 批准号: 9704589
• 负责人: Rachel Kuske
• 金额: $ 1.92万
• 依托单位: Tufts University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-07-01 至 2000-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9704589&HistoricalAwards=false
• 关键词: Problems; Nonlinear; Dynamics

9704589 Kuske The first objective of this project is understanding how small noise can effect qualitative changes in nonlinear behavior. For example, small noise can eliminate chaotic behavior in slow-fast dynamics. Recently the principal investigator has developed a new approach for studying these systems in terms of probability densities. These techniques will be adapted to investigate small noise effects in delay phenomena in excitable systems and chemical reactions. Comparison of these problems will provide a basis for characterizing a variety of noisy phenomena. The second objective is to discover conditions for spatially localized oscillations in systems of coupled oscillators. In particular, the principal investigator will consider bistability of localized and nonlocalized dynamics in laser arrays and structural dynamics. Numerical simulations and singular perturbation methods are combined to explore competing influences of the system parameters, initial conditions, and resonance. The third objective is to explore pattern dynamics in reaction-diffusion systems. Complementary analytical numerical methods are used to derive evolution equations and study the dynamics and stability of modulated patterns. These techniques will be applied in general reaction-diffusion problems and a specific application of burner-stabilized flames. They will also be applied to investigate modulated nonequilateral hexagonal patterns and transitions of two- dimensional patterns in spatially inhomogeneous systems. The first objective of this project is motivated by the dramatic effects of small fluctuations in systems with interacting components. In some examples of chaotic interactions, such as convection, waves in plasma, turbulence and laser dynamics, small noise causes nearly periodic behavior to replace large fluctuations and chaos. Noise can also change the transition from a flat state to an oscillatory state in the propagation of nerve impulses an d in laser intensities. When a time variation in a system parameter controls the transition, there can be a delay or lag in the transition. The introduction of small, rapid fluctuations in the control parameter reduces this delay, as observed experimentally. Realizing the first objective will facilitate the development of mathematical models which account for system noise and describe realistic physical behavior. The second goal is motivated by the effects of localized behavior in several examples of interacting oscillations. Oscillating quantities include chemical concentrations and electrical currents which act as timing or reaction mechanisms in biological and chemical systems. Localization of these oscillations can limit the propagation of an electrical impulse or chemical reaction. Coupled oscillators also model fluctuations in laser arrays, used for applications which require a high optical power output, such as space communications and high speed optical recording. The power of coupled lasers is influenced by variations in intensities and synchronization, which occur in localized oscillations about a steady intensity. Vibrations in repetitive engineering structures are also coupled oscillations, such as in large space structures and bladed disk and beam assemblies. Irregularities in engineering structures can cause vibrations to be spatially localized. Localization can result in increased local stresses and cracking or serve as a desirable damping of energy. Understanding the causes of localization allows the control of the phenomena by adjusting the physical system. The third objective of this project focuses on the dynamics of two dimensional patterns which occur away from equilibrium in many physical or chemical systems, including convection, chemical reactions, and sedimentation in rivers. Patterns occur as superpositions of periodic waves of components such as temperature, concentration or velocity. Spatial variation of geometry or physical control parameters can cause spatial intermittency of the patterns. For example, in a meandering river, rippled sediment deposits are found in certain regions and not in others. In reaction-diffusion systems, slight spatial variation of control species causes periodic variation of chemical concentration in limited regions of the reaction. Since the equations describing pattern evolution appear in a variety of applications, understanding these patterns in one application leads to the understanding of patterns in others.

小行星9704589 这个项目的第一个目标是了解如何小噪音可以 影响非线性行为的质的变化。例如，小噪音 可以消除慢-快动力学中的混沌行为。最近，校长 研究人员开发了一种研究这些系统的新方法， 概率密度的概念。这些技术将适用于 研究可激发系统中延迟现象中的小噪声效应， 化学反应。这些问题的比较将为 表征各种噪声现象。第二个目标是 发现耦合系统中空间局部振荡的条件 振荡器特别是，主要研究者将考虑双稳态 激光阵列和结构中的局部和非局部动力学 动力学数值模拟和奇异摄动方法， 结合探索系统参数的竞争影响，初始 条件和共振。第三个目标是探索模式 反应扩散系统的动力学 补充分析 数值方法被用来推导演化方程，并研究 调制模式的动力学和稳定性。这些技术将 应用于一般的反应扩散问题和特定的应用 燃烧器稳定的火焰。它们也将被应用于研究调制 非等边六边形图案和二维的过渡 空间非均匀系统中的模式。 这个项目的第一个目标是由戏剧性的影响， 系统中相互作用的组件的小波动。在一些 混沌相互作用的例子，如对流，等离子体中的波， 湍流和激光动力学，小的噪声引起近周期性行为 来取代大的波动和混乱。噪音也可以改变过渡 在神经脉冲的传播中从平坦状态到振荡状态 和激光强度d。当系统参数的时间变化 控制过渡，过渡中可能存在延迟或滞后。的 在控制参数中引入小的、快速的波动减少了 这种延迟，正如实验所观察到的。实现第一个目标将 促进数学模型的发展， 系统噪声并描述真实物理行为。第二个目标是 在几个例子中， 交互振荡振荡量包括化学 浓度和电流作为定时或反应 生物和化学系统中的机制。本地化这些 振荡可以限制电脉冲或化学物质的传播 反应耦合振荡器也模拟激光器阵列中的波动， 对于需要高光功率输出的应用，例如空间 通信和高速光学记录。耦合的力量 激光受强度变化和同步性的影响， 发生在稳定强度的局部振荡中。中的振动 重复性工程结构也是耦合振动，例如 大空间结构和叶片盘和梁组件。的不规则性 工程结构可导致振动在空间上局部化。 局部化可能导致局部应力增加和开裂，或作为 理想的能量衰减。了解本地化的原因 允许通过调整物理系统来控制现象。的 本项目的第三个目标侧重于二维动力学 在许多物理或化学过程中， 系统，包括对流，化学反应和沉积， 河流图案出现的成分的周期波的叠加 例如温度、浓度或速度。 空间变化 几何或物理控制参数可导致 模式例如，在一条蜿蜒的河流中， 在某些区域发现，而在其他区域则没有。在反应扩散系统中， 控制物种的轻微空间变化引起周期性变化， 在反应的有限区域中的化学浓度。以来 描述图案演化的方程出现在各种应用中， 在一个应用程序中理解这些模式， 其他模式。