Mathematical Sciences: Applied Dynamics of Near Integrable Systems

9403750 Kovacic这项建议描述了关于两类物理系统中不规则动力学行为的预期研究。第一类是由常微分方程式描述的机械系统。这项拟议的工作将证明存在特殊的轨道，称为同宿轨道，它们在无限长的时间内将平衡位置彼此联系起来。这些轨迹将由一个接一个的慢速和快速伸展序列组成，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的不规则行为。一个这样的系统描述了两个耦合的数学钟摆的振幅中的不规则节拍。正在研究的第二类物理系统是激光光学。这项研究将研究描述环形激光器的麦克斯韦-布洛赫偏微分方程组中类孤子光脉冲的不规则动力学现象。这些动态将导致激光输出在非常短的时间尺度上间歇性闪烁。在本研究过程中，将同时使用分析方法和计算方法。这项建议描述了关于近可积动力系统的两个问题领域的预期研究。第一个被提出的领域涉及到共振带的多个凸起轨道同宿。本文提出的工作将利用Melnikov方法、几何奇异摄动理论和交换引理来描述新的这类轨道，以及可以用来在具体物理问题中寻找这些轨道的新技术。多个凸起轨道与共振带同宿，由一连串的快速凸起组成，其中散布着轨迹在时间上移动非常缓慢的片段。除了这些理论研究外，还提出了计算验证和对可能显示轨道同宿到共振带的现象的物理有趣的例子的研究。第二个命题是关于描述环形腔中激光材料样品的Maxwell-Bloch偏微分方程近可积区域的混沌动力学。这些动力学的存在可能是因为在相应的无限维相空间中存在与空间相关的同宿轨道。这些同宿轨和混沌动力学的存在将通过结合使用数值和分析技术来确定。数值技术将包括使用分步和伪谱方法的模拟，以及各种诊断的计算，如功率谱、李亚普诺夫指数和李亚普诺夫吸引子维度。分析技术包括次调和Melnikov方法、摄动理论、Floquet理论、不变流形理论、逆谱理论和同宿Melnikov方法。

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.