Refraction Periodic Trajectories in Central Mass Galaxies

TL;DR

This paper introduces a new dynamical system combining Keplerian and harmonic forces separated by a refraction interface, analyzing how interface geometry affects the stability and bifurcation of periodic orbits.

Contribution

It provides a local geometric condition for stability and a comprehensive analysis of bifurcations of periodic orbits in a novel two-region dynamical system.

Findings

Interface geometry influences orbit stability.

Complete bifurcation analysis for period one and two orbits.

System is integrable under radial symmetry of the interface.

Abstract

We consider a new type of dynamical systems of physical interest, where two different forces act in two complementary regions of the space, namely a Keplerian attractive center sits in the inner region, while an harmonic oscillator is acting in the outer one. In addition, the two regions are separated by an interface $\Sigma$, where a Snell's law of ray refraction holds. Trajectories concatenate arcs of Keplerian hyperbolae with harmonic ellipses, with a refraction at the boundary. When the interface also has a radial symmetry, then the system is integrable, and we are interested in the effect of the geometry of the interface on the stability and bifurcation of periodic orbits from the homotetic collision-ejection ones. We give local condition on the geometry of the interface for the stability and obtain a complete picture of stability and bifurcations in the elliptic case for period one and period two orbits.