Brake orbits and heteroclinic connections for first order Mean Field Games

We consider first order variational MFG in the whole space, with aggregative interactions and density constraints, such that the stationary states of the game are contained in two isolated compact sets of mass distributions with finite quadratic moments. Under general assumptions on the interaction potential, we provide a method for the construction of periodic in time solutions for the MFG, which oscillate among the two sets of static equilibria. 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 modeled via a Riesz-type aggregative potential with spatial preferences.