Refraction Periodic Trajectories in Central Mass Galaxies

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.