Mathematical Sciences: Applied Dynamics of Near Integrable Systems

数学科学：近可积系统的应用动力学

• 批准号: 9403750
• 负责人: Gregor Kovacic
• 金额: $ 4万
• 依托单位: Rensselaer Polytechnic Institute
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-07-15 至 1996-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9403750&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Applied; Dynamics; Near

9403750 Kovacic This proposal describes intended research concerning irregular dynamical behavior in two classes of physical systems. The first class will be mechanical systems described by ordinary differential equations. The proposed work will show the existence of special trajectories, called homoclinic orbits, that connect equilibrium positions to each other in an infinite amount of time. These trajectories will consist of a sequence of slow and fast stretches following one another, and may be responsible for irregular behavior of nearby trajectories, and thus of the mechanical systems under investigation. One such system describes irregular beats in the amplitudes of two coupled mathematical pendula. The second class of physical systems under investigation is laser optics. The proposed research will investigate the occurrence of irregular dynamics of soliton-like optical pulses in the Maxwell-Bloch partial differential equations describing ring lasers. These dynamics will result in intermittent blinking, on a very short time scale, of the laser output. Both analytical and computational methods will be used in the course of this research. This proposal describes intended research in two problem areas concerning near-integrable dynamical systems. The first proposed area concerns multi-bump orbits homoclinic to resonance bands. The work proposed here will utilize the Melnikov method, geometric singular perturbation theory, and the exchange lemma to describe new classes of such orbits as well as new techniques that can be used to find these orbits in concrete physical problems. Multi-bump orbits homoclinic to resonance bands consist of a succession of fast bumps interspersed with segments on which trajectories move very slowly in time. Besides these theoretical studies, a computational verification and investigations of physically interesting examples that may exhibit the phenomenon of orbits homoclinic to resonance bands are also proposed. The second propos ed are concerns chaotic dynamics in the near-integrable regime of the Maxwell-Bloch partial differential equations describing samples of lasing material in a ring cavity. These dynamics may be present due to the existence of space-dependent homoclinic orbits in the corresponding infinite-dimensional phase space. The existence of these homoclinic orbits and chaotic dynamics will be ascertained with the use of a combination of numerical and analytical techniques. The numerical techniques will include simulations using split-step and pseudo-spectral methods, as well as calculations of various diagnostics such as power spectra, Lyapunov exponents, and Lyapunov attractor dimension. The analytical techniques will include the subharmonic Melnikov method, perturbation theory, Floquet theory, invariant manifold theory, inverse spectral theory, and the homoclinic Melnikov method for near-integrable partial differential equations.

9403750 Kovacic本提案描述了有关两类物理系统中不规则动力学行为的预期研究。一流将是由普通微分方程描述的机械系统。拟议的工作将显示出特殊的轨迹的存在，称为同型轨道，这些轨道轨道将平衡位置在无限的时间内相互连接。这些轨迹将包括一系列缓慢而快速的伸展，并可能导致附近轨迹的不规则行为，从而导致正在研究的机械系统。一个这样的系统描述了两个耦合数学吊长的幅度中的不规则节拍。正在研究的第二类物理系统是激光光学元件。拟议的研究将研究描述环激光器的Maxwell-Bloch部分微分方程中孤子样光脉冲的不规则动力学的发生。这些动力学将导致激光输出的间歇性闪烁。在这项研究过程中，将使用分析方法和计算方法。 该提案描述了在两个问题领域的预期研究，涉及近乎综合的动力学系统。第一个提议的区域涉及与共振频段的多重叉轨道轨道斜角。这里提出的工作将利用Melnikov方法，几何奇异扰动理论和交换引理来描述此类轨道的新类别以及可用于在具体物理问题中找到这些轨道的新技术。多重颠簸的轨道在共振带上的同型轨道轨道频段由一系列的快速颠簸组成，这些快速凸起散布在段落上，轨迹及时移动非常缓慢。除了这些理论研究外，还提出了对物理上有趣的例子的计算验证和研究，这些示例可能表现出与共振带的轨道同型轨迹现象的现象。 第二个提议是在麦克斯韦 - 柏拉希部分微分方程的近乎整合性方程中的混乱动力学，这些方程描述了环腔中的激光材料样品。 这些动力学可能存在，因为在相应的无限维相空间中存在空间依赖性的同型轨道。通过使用数值和分析技术的组合，将确定这些同质轨道和混沌动力学的存在。数值技术将包括使用拆分和伪谱方法的模拟，以及对各种诊断（例如功率谱，Lyapunov指数）和Lyapunov吸引子维度等各种诊断的计算。分析技术将包括亚谐波梅尔尼科夫方法，摄动理论，浮点理论，不变的歧管理论，逆光谱理论以及用于近乎综合部分偏微分方程的同型梅尔尼科夫方法。