Solutions of a Lagrangian system on 𝕋(2)

A Lagrangian system on 𝕋(2) that has been studied earlier under a geometrical condition and found to possess a pair of solutions, H(±), homoclinic to periodic solutions, v(±), of a given homotopy type, is considered further. It is shown with the aid of H(±) and variational arguments that, in fact, there is a much richer structure of homoclinics and heteroclinics to v(±). Indeed, the system admits chaotic solutions.