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.

9704589库斯克这个项目的第一个目标是了解小噪声如何影响非线性行为的质的变化。例如，小噪声可以消除慢-快动力学中的混沌行为。最近，首席研究人员发展了一种新的方法来研究这些系统的概率密度。这些技术将被用来研究可激发系统和化学反应中延迟现象中的小噪声效应。这些问题的比较将为描述各种噪声现象提供基础。第二个目标是发现耦合振子系统中空间局域振荡的条件。特别是，首席研究人员将考虑激光阵列中定域和非定域动力学以及结构动力学的双稳性。数值模拟和奇异摄动方法相结合，探讨了系统参数、初始条件和共振的相互影响。第三个目标是探索反应扩散系统中的模式动力学。用互补的解析数值方法推导了演化方程，研究了调制方向图的动力学和稳定性。这些技术将应用于一般的反应扩散问题和燃烧器稳定火焰的特定应用。它们还将被用于研究空间非均匀系统中调制的非等边六角形图案和二维图案的跃迁。这个项目的第一个目标是由具有相互作用的组件的系统中的微小波动的戏剧性影响所推动的。在一些混沌相互作用的例子中，如对流、等离子体中的波、湍流和激光动力学，小噪声导致了近周期的行为，取代了大的波动和混沌。在神经脉冲和激光强度的传播中，噪声也可以改变从平坦态到振荡态的转变。当系统参数中的时间变化控制过渡时，过渡中可能会有延迟或滞后。正如实验所观察到的，在控制参数中引入小的、快速的波动可以减少这种延迟。实现第一个目标将促进数学模型的发展，这些模型可以解释系统噪声并描述真实的物理行为。第二个目标是由几个相互作用的振荡例子中的局域行为的影响所驱动的。振荡量包括在生物和化学系统中作为定时或反应机制的化学浓度和电流。这些振荡的局部化可以限制电脉冲或化学反应的传播。耦合振荡器还模拟激光阵列中的波动，用于需要高光功率输出的应用，如空间通信和高速光学记录。耦合激光的功率受到强度和同步变化的影响，这些变化发生在关于稳定强度的局域振荡中。重复工程结构中的振动也是耦合振动，例如在大型空间结构和带叶片的圆盘和梁组件中。工程结构中的不规则性会导致振动在空间上局部化。局部化可能会导致局部应力增加和开裂，或者起到理想的能量衰减作用。了解局部化的原因可以通过调整物理系统来控制现象。这个项目的第三个目标是研究在许多物理或化学系统中偏离平衡的二维模式的动力学，包括对流、化学反应和河流中的沉积。模式是温度、浓度或速度等成分的周期波的叠加。几何或物理控制参数的空间变化会导致图案的空间间歇性。例如，在一条蜿蜒的河流中，波纹沉积物在某些地区存在，而在其他地区则没有。在反应扩散系统中，控制物种的微小空间变化导致反应有限区域内化学浓度的周期性变化。由于描述模式演化的方程出现在各种应用程序中，因此理解一个应用程序中的这些模式会导致对其他应用程序中的模式的理解。