Sergey Gonchenko

We consider two-dimensional diffeomorphisms with homoclinic orbits to nonhyperbolic fixed points. We assume that the point has arbitrary finite order degeneracy and is either of saddlenode or weak saddle type. We consider two cases when the homoclinic orbit is transversal and when a quadratic homoclinic tangency takes place. In the first case we give a complete description of orbits entirely lying in a small neighborhood of the homoclinic orbit. In the second case we study main bifurcations in one-parameter families that split generally the homoclinic tangency but retain the degeneracy type of the fixed point.

We consider a nonholonomic model of movement of celtic stone on the plane. We show that, for certain values of parameters characterizing geometrical and physical properties of the stone, a strange Lorenz-like attractor is observed in the model. We have traced both scenarios of appearance and break-down of this attractor.

We study chaotic dynamics of a nonholonomic model of celtic stone movement on the plane. Scenarious of appearance and development of chaos are investigated.

We study questions of chaotic dynamics of three-dimensional smooth maps (diffeomorphisms). We show that there exist two main scenarios of chaos developing from a stable fixed point to strange attractors of various types: a spiral attractor, a Lorenz-like strange attractor or a «figure-8» attractor. We give a qualitative description of these attractors and define certain condition when these attractors can be «genuine» ones (pseudohyperbolic strange attractors). We include also the corresponding results of numerical analysis of attractors in three-dimensional Hénon maps.

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 Ω-explosion gains, in a sense, complete description in the case of two-dimensional diffeomorphisms.

Recently, Smale horseshoes of new types, the so called half-orientable horseshoes, were found in [1]. Such horseshoes may exist for endomorphisms of the disk and for diffeomorphisms of nonorientable two-dimensional manifolds as well.They have many interesting properties different from those of the classical orientable and non-orientable horseshoes. In particular, half-orientable horseshoes may have boundary points of arbitrary periods. It is shown from this fact that there are infinitely many types of such horseshoes with respect to the local topological congugacy. To prove this and similar results, an effective geometric construction is used.

We study bifurcations of of three-dimensional diffeomorphisms with non-transversal heteroclinic cycles which lead to the birth of wild hyperbolic Lorenz-like attractors. As known, such attractors can be appeared under small periodic perturbations of the classical Lorenz attractor and they allow homoclinic tangencies, however, do not contain stable periodic orbits.

We consider the problem of classification of Smale horseshoes from point of view of the local topological conjugacy of two-dimensionalmaps which generate the horseshoes.We show that there are 10 different types of linear horseshoes. As it was established in the recent paper [4], there are infinitelymany different types of nonlinear horseshoes. All of them belong to the class of the so-called half-orientable horseshoes and can be realized for endomorphisms (not one-to-one maps) of disk or for diffeomorphisms of non-orientable two-dimensional manifolds. We give also a short review of related results from [4].

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.