Geodesic
On the stability of periodic trajectories of a~planar Birkhoff billiard
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Modern problems of mechanics, Tome 295 (2016), pp. 206-217

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The inertial motion of a material point is analyzed in a plane domain bounded by two curves that are coaxial segments of an ellipse. The collisions of the point with the boundary curves are assumed to be absolutely elastic. There exists a periodic motion of the point that is described by a two-link trajectory lying on a straight line segment passed twice within the period. This segment is orthogonal to both boundary curves at its endpoints. The nonlinear problem of stability of this trajectory is analyzed. The stability and instability conditions are obtained for almost all values of two dimensionless parameters of the problem. 
@article{TM_2016_295_a11,
     author = {A. P. Markeev},
     title = {On the stability of periodic trajectories of a~planar {Birkhoff} billiard},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {206--217},
     publisher = {mathdoc},
     volume = {295},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2016_295_a11/}
}
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A. P. Markeev. On the stability of periodic trajectories of a~planar Birkhoff billiard. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Modern problems of mechanics, Tome 295 (2016), pp. 206-217. http://geodesic.mathdoc.fr/item/TM_2016_295_a11/