截断椭圆台球的混沌特性
Chaotic properties of the truncated elliptical billiard
摘要：
用数值方法和解析方法研究了具有椭圆弧的对称二维体育场台球系统的混沌特性。对于两参数截断椭圆台球系统，在全参数空间中研究了几个最低阶周期轨道的存在性和线性稳定性。Poincar\' e图的计算和用于评价的混乱程度与盒计数法。完全混沌行为的极限被确定为圆弧。在这个极限之上，对于扁平椭圆弧，存在具有许多稳定椭圆岛的混合动力学，类似于椭圆体育场台球。在这个极限之下，完全的混沌扩展到整个细长形状的区域，现有的轨道要么不稳定，要么是中性的。这是显着不同的行为在椭圆形体育场台球，其中的混沌区域是严格有界的两侧。为了研究这种差异的机制，提出了一种新的三参数家庭的边界形状的推广，并建议进一步评估。
英文摘要：
Chaotic properties of symmetrical two-dimensional stadium-like billiards with elliptical arcs are studied numerically and analytically. For the two-parameter truncated elliptical billiard the existence and linear stability of several lowest-order periodic orbits are investigated in the full parameter space. Poincar\' e plots are computed and used for evaluation of the degree of chaoticity with the box-counting method. The limit of the fully chaotic behavior is identified with circular arcs. Above this limit, for flattened elliptical arcs, mixed dynamics with numerous stable elliptic islands is present, similarly as in the elliptical stadium billiards. Below this limit the full chaos extends over the whole region of elongated shapes and the existing orbits are either unstable or neutral. This is conspicuously different from the behavior in the elliptical stadium billiards, where the chaotic region is strictly bounded from both sides. To examine the mechanism of this difference, a generalization to a novel three-parameter family of boundary shapes is proposed and suggested for further evaluation.
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