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We consider first order variational MFG in the whole space, with aggregativeinteractions and density constraints, such that the stationary states of thegame are contained in two isolated compact sets of mass distributions withfinite quadratic moments. Under general assumptions on the interactionpotential, we provide a method for the construction of periodic in timesolutions for the MFG, which oscillate among the two sets of static equilibria.Moreover, as the period increases to infinity, we show that these periodicsolutions converge, in a suitable sense, to heteroclinic connections. As amodel example, we consider a MFG system where the interactions are modeled viaa Riesz-type aggregative potential with spatial preferences.

Brake orbits and heteroclinic connections for first order Mean Field Games