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# 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

Author(s): Gonchenko S. V., Sten'kin O. V., Shilnikov L. P.

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.
Keywords: nontransversal heteroclinic cycle, Newhouse region, invariant circle
Citation: 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, Rus. J. Nonlin. Dyn., 2006, Vol. 2, No. 1, pp.  3-25
DOI:10.20537/nd0601001