论文标题
空间等同步中的截面和周期轨道的全球表面三个身体问题
Global Surfaces of Section and Periodic Orbits in The Spatial Isosceles Three Body Problem
论文作者
论文摘要
储层计算是预测湍流的有力工具，其简单的架构具有处理大型系统的计算效率。然而，其实现通常需要完整的状态向量测量和系统非线性知识。我们使用非线性投影函数将系统测量扩展到高维空间，然后将其输入到储层中以获得预测。我们展示了这种储层计算网络在时空混沌系统上的应用，该系统模拟了湍流的若干特征。我们表明，使用径向基函数作为非线性投影器，即使只有部分观测并且不知道控制方程，也能稳健地捕捉复杂的系统非线性。最后，我们表明，当测量稀疏、不完整且带有噪声，甚至控制方程变得不准确时，我们的网络仍然可以产生相当准确的预测，从而为实际湍流系统的无模型预测铺平了道路。
We study the spatial isosceles three body problem, which is a system with two degrees of freedom after modulo the rotation symmetry. For certain choices of energy and angular momentum, we find some disk-like global surfaces of section with the Euler orbit as their common boundary, and a brake orbit passing through them. By considering the Poincaré maps of these global surfaces of section, we prove the existence of all kinds of different periodic orbits under certain assumption. Moreover, we are able to prove, for generic choices of masses, the system always has infinitely many periodic orbits. One of the key is to estimate the rotation numbers of the Euler orbit and the brake orbit with respect to the Poincaré map. For this, we establish formulas connected these numbers with the mean indices of the corresponding orbits using the Maslov-type index.
