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

TL;DR

This paper constructs specific examples of autonomous Lagrangian and Hamiltonian systems with exactly two homoclinics or brake orbits, demonstrating the sharpness of previous estimates in potential regions.

Contribution

It provides explicit geometric constructions of systems with minimal homoclinics and brake orbits, confirming the sharpness of earlier theoretical bounds.

Findings

Examples with exactly two homoclinics in Lagrangian systems.

Examples with exactly two brake orbits in Hamiltonian systems.

Validation of the sharpness of previous estimates.

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.