Ulyanova st. 10, 603005, Nizhny Novgorod, Russia
Research Institute of Applied Mathematics and Cybernetics, Nizhny Novgorod State University
Gonchenko S. V., Sten'kin O. V.
Homoclinic Ω-explosion: hyperbolicity intervals and their boundaries
2011, Vol. 7, No. 1, pp. 3-24
It has been established by Gavrilov and Shilnikov in  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  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 Ω-explosion gains, in a sense, complete description in the case of two-dimensional diffeomorphisms.
Gonchenko S. V., Sten'kin O. V., Shilnikov L. P.
On the existence of infinitely many stable and unstable invariant tori for systems from Newhouse regions with heteroclinic tangencies
2006, Vol. 2, No. 1, pp. 3-25
Let a $C^r$-smooth $r \geqslant 5$ two-dimensional diffeomorphism $f$ have a non-transversal heteroclinic cycle containing several saddle periodic and heteroclinic orbits and, besides, some of the heteroclinic orbits are non-transversal, i.e. at the points of these orbits the invariant manifolds of the corresponding saddles intersect non-transversally. Suppose that a cycle contains at least two saddle periodic orbits such that the saddle value (the absolute value of product of multipliers) of one orbit is less than 1 and it is greater than 1 for the other orbit. We prove that in any neighbourhood (in $C^r$-topology) of $f$ in the space of $C^r$-diffeomorphisms, there are open regions (so-called Newhouse regions with heteroclinic tangencies) where diffeomorphisms with infinitely many stable and unstable invariant circles are dense. For three-dimensional flows, this result implies the existence of Newhouse regions where flows having infinitely many stable and unstable invariant two-dimensional tori are dense.