Alexander Gonchenko

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.

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 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].