Topological Torus Bifurcations and Applications to Ion Traps

提案：DMS-0407521PI: Andrew J szeri机构：加州大学伯克利分校标题：拓扑环面分岔和离子阱的应用抽象阱是用于包含带电粒子和高选择性质谱的设备。对于受二次端盖激励并包含缓冲气体的非线性陷阱，其运动方程是耦合的、阻尼的、非线性的、准周期的马蒂厄方程。该系统的吸引子在四维相空间中是非平凡的“打结”。研究者和他的同事们已经开发了工具来表征它们的拓扑和几何特征，并确定了与它们类型变化相关的新分支。重要的突出问题包括：吸引子穿过环面辫和穿过曲面的拓扑等价是如何联系起来的；鞍型结环面稳定流形和不稳定流形的横向交叉是否与混沌有关。新识别的分岔序列通过观察到的路线导致混沌动力学，但尚未表征。在某些情况下，两个分岔结合了非常尖锐的共振，这在应用中是有用的。有关该系统的新知识可用于建议用于质谱中物种识别的离子阱的新操作制度。已经观察到的尖锐共振表明使用离子阱作为可调谐机电滤波器。两种类似的离子阱系统可用于屏蔽和解禁用于保密通信的信号。最后，结果表明，捕获的带电粒子或离子可以用作加速度计和陀螺仪。这些技术创新或改进直接来自对基础数学的详细理解。该奖项由数学科学部的应用数学和拓扑项目以及化学部门的分析和表面化学项目共同资助，隶属于美国国家科学基金会的数学科学优先领域。

Proposal: DMS-0407521PI: Andrew J SzeriInstitution: University of California BerkeleyTitle: Topological Torus Bifurcations and Applications to Ion TrapsABSTRACTIon traps are devices used to contain charged particles, and for highly selective mass spectroscopy. For a nonlinear trap subjected to secondary end-cap excitation and including a buffer gas, the equations of motion are coupled, damped, nonlinear, quasiperiodic Mathieu equations. The attractors for this system are non-trivially "knotted" in the 4-D phase space. The investigator and his colleagues have developed tools to characterize them topologically and geometrically, and have identified new bifurcations associated with their changes of type. Important outstanding problems include: how topological equivalence of attractors through torus braids and through surfaces is related; whether chaos is associated with transverse intersections of stable and unstable manifolds of saddle-type knotted tori.A sequence of the newly identified bifurcations leads to chaotic dynamics through a route observed, but not yet characterized. In some cases, two bifurcations bound extremely sharp resonances which are useful in applications.The new knowledge being developed about this system can be used to suggest new operating regimes for ion traps used in discrimination of species in mass spectroscopy. The sharp resonances already observed suggest the use of an ion trap as a tunable electromechanical filter. Two similar ion trap systems may be used for the purpose of masking and unmasking signals for secure communication. Finally, the results suggest that a trapped charged particle or ion can be used as an accelerometer and gyroscope. These technological innovations or improvements flow directly from a detailed understanding of the underlying mathematics.This award is co-funded by the Applied Mathematics and Topology programs of the Division of Mathematical Sciences and the Analytical and Surface Chemistry program of the Chemistry Division under the umbrella of the NSF-wide Mathematical Sciences Priority Area.