Mathematical Sciences: Transport for Symplectic Mapping

这个项目将研究四维或更多维辛映射中的输运现象。我们的目标是将相空间划分成由部分势垒限定的区域，通过这些区域发生通量。这种区域的候选者是共振，定义为椭圆点附近的体积；它的边界将从振动周期轨道的限制中获得。我们将计算共振体积，并确定共振是否划分相空间。研究共振边界井附近轨道的关联函数，以确定扩散张量的级数是否收敛。输运将根据跨越共振边界的通量来定义，包括从一个共振到另一个共振的转变的分支比，以及沿可公度通道的漂移。为了计算后者，我们将引入并研究一种限制辛映射。数值技术将包括频率滤波和寻找周期轨道。

This project will study transport phenomena in symplectic mappings of four or more dimensions. The goal is to partition phase space into regions bounded by partial barriers through which flux occurs. A candidate for such a region is a resonance, defined as a volume in the neighborhood of an elliptic point; its boundary will be obtained from limits of librational periodic orbits. Resonance volumes will be computed and we will determine whether resonances partition phase space. The correlation function for orbits in the neighborhood of a resonance boundary well be studied to determine whether the series for the diffusion tensor converges. Transport will be defined in terms of the flux across resonance boundaries, including branching ratios for transitions from one resonance to a neighboring one, as well as drift along a commensurability channel. To compute the latter a restricted symplectic map will be introduced and studied. Numerical techniques will include frequency filtering and finding periodic orbits.