Solutions of a Lagrangian system on T squared

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