Mathematical Sciences: Nonlinear Dynamics of Oscillator Arrays

DMS_9500948斯特罗加茨建议利用动力系统理论、分叉理论和微扰理论的数学方法，结合数值模拟，研究振荡器阵列的非线性动力学。其中一个项目涉及超导约瑟夫森结阵列，将与麻省理工学院的一个实验小组合作进行。这项工作的目的是了解串联阵列的锁相特性，一维平行阵列中扭结的传播和不稳定性，以及二维阵列中涡旋和行切换态的行为。第二个项目(与佐治亚理工学院的一个实验小组合作)将解决激光阵列的动力学问题。研究领域将包括在略微失谐的激光阵列中进行锁相；控制激光阵列中的混沌；以及混沌激光的相互同步，并可能应用于私人通信。首席研究员和他的实验合作者将研究超导约瑟夫森阵列和激光阵列的动力学。约瑟夫森阵列可用作磁场的非常灵敏的探测器(具有生物医学和天文应用)、电压标准、亚毫米波长的功率发生器(这是其他技术难以访问的频谱的一部分)，也可用作超快开关器件和传输线。所有这些技术都将受益于对约瑟夫森阵列中同步和涡旋传播的更深入的数学理解。激光阵列项目的目标是找到改善激光阵列的锁相特性(增加其功率输出)的方法，并基于混沌激光的相互同步开发用于私人通信的新型加密方案。

DMS_9500948 Strogatz The proposal is to study the nonlinear dynamics of oscillator arrays, using mathematical methods of dynamical systems theory, bifurcation theory, and perturbation theory, along with numerical simulation. One project concerns arrays of superconducting Josephson junctions, and will be carried out in collaboration with an experimental group at MIT. This work is aimed at understanding the phase-locking properties of series arrays; the propagation and instabilities of kinks in one-dimensional parallel arrays; and the behavior of vortices and row-switched states in two-dimensional arrays. A second project (in collaboration with an experimental group at Georgia Tech) will address the dynamics of laser arrays. Areas of investigation will include phase-locking in slightly detuned laser arrays; controlling chaos in laser arrays; and mutual synchronization of chaotic lasers, with possible applications to private communication. The Principal Investigator and his experimental collaborators will study the dynamics of superconducting Josephson arrays and laser arrays. Josephson arrays can be used as very sensitive detectors of magnetic fields (with biomedical and astronomical applications), voltage standards, power generators at submillimeter wavelengths (a part of the spectrum that is difficult to access by other technologies), and as ultrafast switching devices and transmission lines. All of these technologies would benefit from a deeper mathematical understanding of synchronization and vortex propagation in Josephson arrays. The goals of the laser array project are to find methods for improving the phase-locking characteristics of laser arrays (to increase their power output), and to develop novel encryption schemes for private communications, based on mutual synchronization of chaotic lasers.