Homoclinic $\Omega$-explosion: hyperbolicity intervals and their boundaries

It has been established by Gavrilov and Shilnikov in [1] that, at the bifurcation boundary separating Morse–Smale systems from systems with complicated dynamics, there are systems with homoclinic tangencies. Moreover, when crossing this boundary, infinitely many periodic orbits appear immediately, just by “explosion”. Newhouse and Palis have shown in [2] that in this case there are infinitely many intervals of values of the splitting parameter corresponding to hyperbolic systems. In the present paper, we show that such hyperbolicity intervals have natural bifurcation boundaries, so that the phenomenon of homoclinic $\Omega$-explosion gains, in a sense, complete description in the case of two-dimensional diffeomorphisms.