标题： 波茨中混沌的表现：混合曲率的作用 显示英文标题

标题： Manifestations of chaos in billiards: the role of mixed curvature

摘要： 弹道系统边界在塑造其动力学方面起着关键作用，这可能是可积的、混合的或完全混沌的。 当边界具有变化的曲率时，它提供了一个独特的环境来研究经典混沌与量子行为之间的关系。 在本研究中，我们引入了两种几何上不同的弹道系统：一种是豆形边界和一种卡西尼卵形的花生形变体。 这些系统包含聚焦和发散壁，没有中性段。 我们的研究表明经典动力学和量子动力学之间存在强烈的关联。 我们对弹道流图的分析证实了对初始条件（ICs）的敏感性——这是混沌的一个特征。 庞加莱映射进一步显示相空间中充满了混沌运动区域和稳定岛屿。 转向量子领域，我们采用最近邻间距分布和能级间距比作为统计量来表征混沌。 光谱复杂性的早期时间饱和也支持这些系统中的遍历层次结构。 我们观察到一种显著的量子现象，即本征函数的疤痕。 这项工作连接了几何边界效应、经典双曲性和量子遍历性，为在受限系统中设计混沌提供了框架。 显示英文摘要

摘要： The boundary of a billiard system plays a crucial role in shaping its dynamics, which may be integrable, mixed, or fully chaotic. When a boundary has varying curvature, it offers a unique setting to study the relation between classical chaos and quantum behaviour. In this study, we introduce two geometrically distinct billiards: a bean-shaped boundary and a peanut-shaped variant of Cassini ovals. These systems incorporate both focusing and defocusing walls with no neutral segments. Our study reveals a strong correlation between classical and quantum dynamics. Our analysis of billiard flow diagrams confirms sensitivity to initial conditions(ICs)- a defining feature of chaos. Poincar\'e maps further show the phase space intricately woven with regions of chaotic motion and stability islands. Moving to the quantum domain, we employ nearest-neighbour spacing distribution and level spacing ratio as statistical measures to characterise chaos. Early time saturation in spectral complexity also supports an ergodic hierarchy in these systems. We observe a striking quantum phenomenon, i.e. eigenfunction scarring. This work bridges geometric boundary effects, classical hyperbolicity, and quantum ergodicity, offering a framework to engineer chaos in confined systems.