一个三自由度八通道的哈密顿系统：大逃亡
A Hamiltonian system of three degrees of freedom with eight channels of escape: The Great Escape
摘要：
在这项工作中，我们试图揭示一些光的性质，在一个有八个可能的逃逸通道的扰动谐振子，这是一个有趣的例子，开放的三维哈密顿系统的三维潜力。特别是，我们进行了彻底的数值研究区分定期和混乱的轨道，以及被困和逃逸轨道，考虑无界运动的几个值的能量。在试图区分有序和混乱的运动之间的安全和确定性，我们使用较小的对准指数（SALI）检测器，通过数值积分的基本方程的运动以及变分方程计算。特别感兴趣的是，找到逃逸盆地朝向不同的逃逸通道，并将它们与轨道的相应逃逸周期联系起来。我们根据用于发射测试粒子的$z$坐标的初始值将我们的研究分为三种不同的情况。我们发现，当轨道开始非常接近的主要$(x,y)$平面的各自的网格表现出高度的分形，而另一方面，对于轨道具有相对较高的值的$z_0$几个良好形成的逃逸盆地出现，从而显着减少网格的分形。还观察到，对于非常接近逃逸能量的能量值，轨道的逃逸时间很大，而对于比逃逸能量高得多的能级，绝大多数轨道逃逸得非常快，甚至立即逃逸到无穷大。我们希望我们的结果是有用的进一步了解的逃逸过程中开放的三维哈密顿系统。
英文摘要：
In this work, we try to shed some light to the nature of orbits in a three-dimensional potential of a perturbed harmonic oscillator with eight possible channels of escape, which was chosen as an interesting example of open three-dimensional Hamiltonian systems. In particular, we conduct a thorough numerical investigation distinguishing between regular and chaotic orbits as well as between trapped and escaping orbits, considering unbounded motion for several values of the energy. In an attempt to discriminate safely and with certainty between ordered and chaotic motion, we use the Smaller ALingment Index (SALI) detector, computed by integrating numerically the basic equations of motion as well as the variational equations. Of particular interest, is to locate the basins of escape towards the different escape channels and connect them with the corresponding escape periods of the orbits. We split our study into three different cases depending on the initial value of the $z$ coordinate which was used for launching the test particles. We found, that when the orbits are started very close to the primary $(x,y)$ plane the respective grids exhibit a high degree of fractalization, while on the other hand for orbits with relatively high values of $z_0$ several well-formed basins of escape emerge thus, reducing significantly the fractalization of the grids. It was also observed, that for values of energy very close to the escape energy the escape times of orbits are large, while for energy levels much higher than the escape energy the vast majority of orbits escape extremely fast or even immediately to infinity. We hope our outcomes to be useful for a further understanding of the escape process in open 3D Hamiltonian systems.
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