Linear Stability of Periodic Trajectories in Inverse Magnetic Billiards

TL;DR

This paper investigates the linear stability of periodic trajectories in inverse magnetic billiards, a system with straight-line motion inside a domain and circular arcs outside, providing explicit examples and stability comparisons.

Contribution

It introduces the analysis of stability for inverse magnetic billiards and offers explicit examples in various shapes, comparing results with classical billiard systems.

Findings

Explicit stability conditions for inverse magnetic billiard trajectories.

Comparison of stability properties with classical billiard and magnetic billiard systems.

Identification of stable and unstable periodic trajectories in specific geometries.

Abstract

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 $\Omega$ and circular arcs outside $\Omega$. 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.