2-Elliptic Periodic Orbits near a Nonsimple Homoclinic Tangency in Four-Dimensional Symplectic Maps

We show that bifurcations of four-dimensional symplectic diffeomorphisms with a quadratic homoclinic tangency to a saddle periodic orbit with real multipliers produce 2-elliptic periodic orbits if the tangency is not partially hyperbolic. We show that a normal form for the rescaled first-return maps near such tangency is given by a four-dimensional symplectic Hénon-like map and study bifurcations of the first-return maps in generic two-parameter families.