Nonlinear Dynamics with Applications in Physical Systems

小行星9704554 项目部分的第一部分涉及数学的研究 Josephson Junction的相关文章最近发现， 在某些超导层交错的晶体中自然发生 具有绝缘层。建议导出并研究一个数学 这种晶体的模型和发展理论的行波离散 媒体 在项目的第二部分，建议制作 本文对最近提出的一个弛张振子系统的定性分析 发现的类型。项目的第三部分涉及各种应用 研究人员最近对曲率在高频平均中的作用的观察。这些应用包括研究的议案， 刚性体（例如嵌入流体中的颗粒）或耦合系统， 粒子，例如受到快速变化的力场的分子。 另一个应用包括谐振介质的行为，该谐振介质受到 电磁脉冲。 最后，项目的最后一部分涉及 在两类系统之间建立连接： 测地线 一方面是表面上的运动，另一方面是表面内的台球运动 另一边的表面。 第一部分是关于约瑟夫森的研究 结--具有巨大潜力的超导器件 用于应用程序。约瑟夫森结的优点在于它们的小 体积小，能耗低，这就大大减少了 电路尺寸，但缺点是低功率输出。克服 这个缺点，结可以放在阵列中。值得注意的是， 阵列自然发生：一些晶体被发现由堆叠的 约瑟夫森结层。最近的发现提供了潜在的新的 应用程序，特别是 用于超快光学器件之间的接口 通信线路和电路，以及用于产生高 通信和射电天文用频率振荡 意见。理解这样的数组的行为是一个至关重要的 重要性 这里提出的项目就是为了实现这种理解。 的 该项目的第二部分涉及弛张振荡器-这些发生在 振荡由缓慢漂移组成的生物和电子系统 被快速跳跃打断。 一大类这样的振荡器 混沌行为是最近发现的， 分析了理解耦合振荡器系统（本文的目标 项目）提供了巨大的挑战，在一些生物和 电力系统。 在这一努力的第三部分中， 将探索高频振荡。这些影响是 操作的保罗陷阱（发明者被授予诺贝尔奖）和 最近的“激光镊子”，允许移动细胞内的物体， 这项研究将扩展数学分析， 这些现象的目的是可能的应用，包括 振动控制以及环境应用，例如分离 过程（从空气中去除颗粒等） 第四部分 项目建议建立两个百年历史的连接 几何问题是并行开发的，但没有直接的 连接，通过显示一个是另一个的限制情况。 这两 问题在理论的发展中发挥了核心作用， 动力系统，因此间接但至关重要的发展， 粒子加速器，在天文学和光学中。

9704554 Levi The first part of the project part deals with the study of mathematical aspects of Josephson junctions. It was discovered recently that the junctions occur naturally in certain crystals where superconducting layers interlace with insulating layers. It is proposed to derive and to study a mathematical model of such crystals and to develop theory of traveling waves in discrete media. In the second part of the project it is proposed to produce the qualitative analysis of systems of relaxation oscillators of a recently discovered type. The third part of the project deals with various applications of the investigator's recent observation on the role of curvature in high- frequency averaging. These applications include the study of the motion of rigid bodies (such as particles embedded in fluid) or coupled systems of particles, such as molecules subjected to rapidly varying force fields. Another application includes the behavior of a resonant medium subjected to an electromagnetic pulse. Finally, the last part of the project deals with establishing a connection between two classes of systems: the geodesic motion on a surface on the one hand and the billiard motion inside that surface on the other. The first of the four parts of the project deals with the study of Josephson junctions -- superconducting devices whose properties offer great potential for applications. The advantages of Josephson junctions lie in their small size and in low energy consumption, which allow for a great reduction of the circuit size, but the disadvantage is the low power output. To overcome that disadvantage, the junctions can be put in arrays. Remarkably, such arrays occur naturally: some crystals were discovered to consist of stacked Josephson junction layers. This recent discovery offers potential new applications, notably for use in interfaces between ultrafast optical communication lines and electric circuits as well as for generation of high frequency oscillations for use in communications and radio-astronomical observations. Understanding the behavior of such arrays takes on a crucial importance. The project proposed here aims at such understanding. The second part of the project addresses relaxation oscillators - these occur in biological and electronic systems whose oscillations consist of slow drifts punctuated by fast jumps. A large class of such oscillators exhibiting chaotic behavior was discovered recently, and an isolated oscillator analyzed. Understanding systems of coupled oscillators (the goal of this project) offers great challenges and is of importance in some biological and electric systems. In the third part of this effort, the dynamical effects of high-frequency oscillations will be explored. Such effects underlie the operation of the Paul trap (the inventor was awarded a Nobel prize) and the more recent "laser tweezers" which allow to move objects inside the cell without damaging it. This research will extend mathematical analysis of these phenomena with the aim of possible applications which include vibrational control as well as environmental applications such as separation processes (removing particles from the air, etc.) In the fourth part of the project it is proposed to establish a connection between two century-old geometry problems which were developed in parallel but without a direct connection, by showing that one is a limiting case of another. Both of these problems played a central role in the development of the theory of dynamical systems, and thus indirectly but crucially in the development of particle accelerators, in astronomy and in optics.