Examples with minimal number of brake orbits and homoclinics in annular potential regions

Abstract: We use a geometric construction to exhibit examples of autonomous Lagrangian systems admitting exactly two homoclinics emanating from a nondegenerate maximum of the potential energy and reaching a regular level of the potential having the same value of the maximum point. Similarly, we show examples of Hamiltonian systems that admit exactly two brake orbits in an annular potential region connecting the two connected components of the boundary of the potential well. These examples show that the estimates proven in [R. Giamb`o, F. Giannoni, P. Piccione, Arch. Ration. Mech. Anal. 200, (2011) 691-724] are sharp.