  We introduce a one-parameter family of polymatrix replicators defined in a
three-dimensional cube and study its bifurcations. For a given interval of
parameters, this family exhibits suspended horseshoes and persistent strange
attractors. The proof relies on the existence of a homoclinic cycle to the
interior equilibrium. We also describe the phenomenological steps responsible
for the transition from regular to chaotic dynamics in our system (route to
chaos).
