da Costa Diogo Ricardo, Dettmann Carl P, de Oliveira Juliano A, Leonel Edson D
Instituto de Física da USP, Rua do Matão, Travessa R, 187, Cidade Universitária, CEP 05314-970 São Paulo, SP, Brazil.
School of Mathematics, University of Bristol, Bristol, United Kingdom.
Chaos. 2015 Mar;25(3):033109. doi: 10.1063/1.4915474.
Some dynamical properties for an oval billiard with a scatterer in its interior are studied. The dynamics consists of a classical particle colliding between an inner circle and an external boundary given by an oval, elliptical, or circle shapes, exploring for the first time some natural generalizations. The billiard is indeed a generalization of the annular billiard, which is of strong interest for understanding marginally unstable periodic orbits and their role in the boundary between regular and chaotic regions in both classical and quantum (including experimental) systems. For the oval billiard, which has a mixed phase space, the presence of an obstacle is an interesting addition. We demonstrate, with details, how to obtain the equations of the mapping, and the changes in the phase space are discussed. We study the linear stability of some fixed points and show both analytically and numerically the occurrence of direct and inverse parabolic bifurcations. Lyapunov exponents and generalized bifurcation diagrams are obtained. Moreover, histograms of the number of successive iterations for orbits that stay in a cusp are studied. These histograms are shown to be scaling invariant when changing the radius of the scatterer, and they have a power law slope around -3. The results here can be generalized to other kinds of external boundaries.
研究了内部带有一个散射体的椭圆形台球桌的一些动力学性质。该动力学过程包括一个经典粒子在一个内圆和由椭圆形、椭圆或圆形给出的外部边界之间碰撞，首次探索了一些自然的推广情况。这个台球桌实际上是环形台球桌的一种推广，对于理解在经典和量子（包括实验）系统中边缘不稳定周期轨道及其在规则区域和混沌区域边界中的作用具有重要意义。对于具有混合相空间的椭圆形台球桌，障碍物的存在是一个有趣的补充。我们详细展示了如何得到映射方程，并讨论了相空间中的变化。我们研究了一些不动点的线性稳定性，并通过解析和数值方法展示了直接和逆抛物线分岔的出现。得到了李雅普诺夫指数和广义分岔图。此外，还研究了停留在尖点处的轨道连续迭代次数的直方图。当改变散射体半径时，这些直方图显示出尺度不变性，并且它们在 -3 左右具有幂律斜率。这里的结果可以推广到其他类型的外部边界。
