Mathematical Sciences: Applied Dynamics of Near Integrable Systems

这个提议描述了两类物理系统中关于不规则动力学行为的预期研究。第一堂课是用常微分方程描述的机械系统。这项提议的工作将证明特殊轨迹的存在，称为同斜轨道，它在无限长的时间内将平衡位置彼此连接起来。这些轨迹将由一系列缓慢和快速的拉伸组成，并且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。其中一个系统描述了两个耦合数学钟摆振幅中的不规则节拍。所研究的第二类物理系统是激光光学。提出的研究将探讨在描述环形激光器的麦克斯韦-布洛赫偏微分方程中出现的类孤子光脉冲的不规则动力学。这些动态将导致间歇性闪烁，在很短的时间尺度，激光输出。分析方法和计算方法将在本研究过程中使用。本提案描述了拟在近可积动力系统的两个问题领域进行的研究。第一个提出的领域涉及多碰撞轨道同斜共振带。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。与共振带同斜的多碰撞轨道由连续的快速碰撞组成，其中穿插着轨道在时间上移动非常缓慢的段。除了这些理论研究外，还提出了计算验证和物理上有趣的可能显示轨道与共振带同斜现象的例子的研究。第二个建议是关于描述环形腔中激光材料样品的麦克斯韦-布洛赫偏微分方程的近可积区中的混沌动力学。这些动力学可能是由于在相应的无限维相空间中存在空间相关的同斜轨道而存在的。这些同斜轨道和混沌动力学的存在将用数值和解析相结合的方法来确定。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、李雅普诺夫指数和李雅普诺夫吸引子维。分析技术将包括次谐波Melnikov方法、微扰理论、Floquet理论、不变流形理论、逆谱理论和近可积偏微分方程的同宿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.