Topological Torus Bifurcations and Applications to Ion Traps

提议：DMS-0407521PI：Andrew J Szeri研究所：加州大学伯克利分校标题：拓扑环分叉和离子捕获的应用ABSTRACTION陷阱是用于包含带电粒子和用于高选择性质谱学的设备。对于含有缓冲气体的二次端帽激励的非线性陷阱，其运动方程是耦合的、有阻尼项的、非线性的、准周期的Mathieu方程。该系统的吸引子在4维相空间中非平凡地“打结”。这位研究人员和他的同事们已经开发了工具来从拓扑和几何上描述它们，并发现了与它们类型变化相关的新分叉。重要的突出问题包括：通过环面辫子和穿过曲面的吸引子的拓扑等价是如何联系的；混沌是否与鞍型纽结托叶的稳定流形和不稳定流形的横交有关。一系列新发现的分叉导致了混沌动力学通过一条已观察到但尚未刻画的路线。在某些情况下，两个分叉结合了非常尖锐的共振，这在应用中是有用的。关于这个系统的新知识可以用来建议用于质谱学中物种鉴别的离子陷阱的新的操作机制。已经观察到的尖锐共振表明，离子陷阱被用作可调谐的机电过滤器。可以使用两个类似的离子陷阱系统来屏蔽和解除屏蔽信号以进行安全通信。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源于对基础数学的详细了解。该奖项由数学科学部的应用数学和拓扑学计划以及化学部的分析和表面化学计划共同资助，该计划隶属于NSF-Wide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.