Three-dimensional Dissipative Diffeomorphisms with Codimension Two Homoclinic Tangencies and Generalized H Enon Maps

We study bifurcations of xed points of rst return maps in two-parameter families of three-dimensional diieomorphisms close to a diieomorphism with a codimension two homoclinic tangency. We suppose that the initial dif-feomorphism has a saddle xed point O with multipliers 1 ; 2 ; such that 0 < j 2 j < j 1 j < 1 < jj and j 1 2 j < 1; the invariant manifolds W s (O) and W u (O) has a quadratic tangency at points of a homoclinic orbit ? 0 ; this tangency is degenerate in that sense that the extended unstable manifold of O is nontransversal to leaves of the strong stable foliation on W s (O) at the homoclinic points (so-called, the generalized homoclinic tangency). We show that in the case j 1 j > 1 the rst return maps can be rescaled to maps which are asymptotically close to the H enon map. These generalized H enon maps make possible to recover the bifurcations of xed points with multipliers e i' .