Mathematical Sciences: Applied Dynamics of Near Integrable Systems

小行星9403750 该建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。第一类是用常微分方程描述的力学系统。这项工作将展示特殊轨道的存在，称为同宿轨道，它在无限长的时间内将平衡位置相互连接起来。这些轨迹将包括一系列的慢和快的拉伸，彼此跟随，并可能负责附近的轨迹，从而在调查中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摆的振幅中的不规则节拍。研究中的第二类物理系统是激光光学。本研究将探讨描述环形雷射之Maxwell-Bloch偏微分方程式中类孤子光脉冲之不规则动力学现象。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。分析和计算方法将被用于本研究的过程中。 该建议描述了拟在两个问题领域的研究近可积动力系统。第一个提出的领域涉及多凸轨道同宿共振带。这里提出的工作将利用Melnikov方法，几何奇异摄动理论，和交换引理来描述新的类这样的轨道，以及新的技术，可以用来找到这些轨道在具体的物理问题。同宿于共振带的多凸点轨道由一系列快速凸点组成，这些凸点上散布着轨迹在时间上移动非常缓慢的段。除了这些理论研究，计算验证和调查物理上有趣的例子，可能会表现出同宿轨道共振带的现象也提出了。 第二个问题艾德是关于描述环形腔中激光材料样品的Maxwell-Bloch偏微分方程的近可积区域的混沌动力学。 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宿轨道。这些同宿轨道和混沌动力学的存在将确定与使用相结合的数值和分析技术。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数。分析技术将包括亚调和Melnikov方法，扰动理论，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.