In this work we address the question of proving the stability of elliptic 2-periodic orbits for strictly convex billiards. Even though it is part of a widely accepted belief that ellipticity implies stability, classical theorems show that the certainty of stability relies upon more precise conditions. We present a review of the main results and general theorems and describe the procedure to fulfill the supplementary conditions for strictly convex billiards.
"The first Birkhoff coefficient and the stability of 2-periodic orbits on billiards." Experiment. Math. 14 (3) 299 - 306, 2005.