Inverse-Magnetic Billiards on a Square

Linear Stability of Periodic Trajectories in Inverse Magnetic Billiards

We study the stability of periodic trajectories of planar inverse magnetic billiards, a dynamical system whose trajectories are straight lines inside a connected planar domain $Ω$ and circular arcs outside $Ω$. Explicit examples are calculated in circles, ellipses, and the one parameter family of curves $x^{2k}+y^{2k}=1$. Comparisons are made to the linear stability of periodic billiard and magnetic billiard trajectories.

Chaos in the square billiard with a modified reflection law

The purpose of this paper is to study the dynamics of a square billiard with a non-standard reflection law such that the angle of reflection of the particle is a linear contraction of the angle of incidence. We present numerical and analytical arguments that the nonwandering set of this billiard decomposes into three invariant sets, a parabolic attractor, a chaotic attractor and a set consisting of several horseshoes. This scenario implies the positivity of the topological entropy of the billiard, a property that is in sharp contrast with the integrability of the square billiard with the stand…

方形和矩形台球的周期轨道

Periodic orbits for square and rectangular billiards

本文研究方形与矩形数学台球，分类其周期轨道，证明碰撞数必为偶数，并将轨道类别数联系到 Euler 函数，同时分析给定碰撞数的构造及终止于角点的奇异轨迹。

Dissipative outer billiards: a case study

We study dissipative polygonal outer billiards, i.e. outer billiards about convex polygons with a contractive reflection law. We prove that dissipative outer billiards about any triangle and the square are asymptotically periodic, i.e. they have finitely many global attracting periodic orbits. A complete description of the bifurcations of the periodic orbits as the contraction rates vary is given. For the square billiard, we also show that the asymptotic periodic behavior is robust under small perturbations of the vertices and the contraction rates. Finally, we describe some numerical experime…

Polygonal billiards with one side scattering

We study polygonal billiards with one-sided vertical mirror scattered on a square billiard table. We associate trajectories of these kinds of billiards with double rotations and study orbit behavior and questions of complexity.

On the Dynamics of Inverse Magnetic Billiards

Consider a strictly convex set $Ω$ in the plane, and a homogeneous, stationary magnetic field orthogonal to the plane whose strength is $B$ on the complement of $Ω$ and $0$ inside $Ω$. The trajectories of a charged particle in this setting are straight lines concatenated with circular arcs of Larmor radius $μ$. We examine the dynamics of such a particle and call this inverse magnetic billiards. Comparisons are made to standard Birkhoff billiards and magnetic billiards, as some theorems regarding inverse magnetic billiards are consistent with each of these billiard variants while others are not…