Brake orbits and heteroclinic connections for first order mean field games

We consider first order variational mean field games (MFG) in the whole space, with aggregative interactions and density constraints, having stationary equilibria consisting of two disjoint compact sets of distributions with finite quadratic moments. Under general assumptions on the interaction potential, we provide a method for the construction of periodic in time solutions to the MFG, which oscillate between the two sets of static equilibria, for arbitrarily large periods. Moreover, as the period increases to infinity, we show that these periodic solutions converge, in a suitable sense, to heteroclinic connections. As a model example, we consider a MFG system where the interactions are of (aggregative) Riesz-type.