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    Alexander Gonchenko

    ul. Uljanova 10, Nizhny Novgorod 603005, Russia
    Institute for Applied Mathematics and Cybernetics,


    Gonchenko A. S., Gonchenko S. V.
    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.
    Keywords: celtic stone, nonholonomic model, the Lorenz attractor, Lorenz-like attractor for diffeomorphisms, chaotic dynamics
    Citation: Gonchenko A. S., Gonchenko S. V.,  On existence of Lorenz-like attractors in a nonholonomic model of Celtic stones, Rus. J. Nonlin. Dyn., 2013, Vol. 9, No. 1, pp.  77-89
    Gonchenko A. S., Gonchenko S. V., Kazakov A. O.
    On some new aspects of Celtic stone chaotic dynamics
    2012, Vol. 8, No. 3, pp.  507-518
    We study chaotic dynamics of a nonholonomic model of celtic stone movement on the plane. Scenarious of appearance and development of chaos are investigated.
    Keywords: nonholonomic model, strange attractor, symmetry, bifurcation, mixed dynamics
    Citation: Gonchenko A. S., Gonchenko S. V., Kazakov A. O.,  On some new aspects of Celtic stone chaotic dynamics, Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 3, pp.  507-518
    Gonchenko A. S., Gonchenko S. V., Shilnikov L. P.
    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.
    Keywords: strange attractor, chaotic dynamics, spiral attractor, torus–chaos, homoclinic orbit, invariant curve, three-dimensional Hénon map
    Citation: Gonchenko A. S., Gonchenko S. V., Shilnikov L. P.,  Towards scenarios of chaos appearance in three-dimensional maps, Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 1, pp.  3-28
    Gonchenko S. V., Gonchenko A. S., Malkin M. I.
    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.
    Keywords: Smale horseshoe, local topological conjugacy, hyperbolic set, standard and generalized Henon maps
    Citation: Gonchenko S. V., Gonchenko A. S., Malkin M. I.,  On classification of classical and half-orientable horseshoes in terms of boundary points, Rus. J. Nonlin. Dyn., 2010, Vol. 6, No. 3, pp.  549-566
    Gonchenko S. V., Gonchenko A. S.
    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].
    Keywords: Smale horseshoe, local topological conjugacy, hyperbolic set, standard and generalized Henon maps
    Citation: Gonchenko S. V., Gonchenko A. S.,  Towards a classification of linear and nonlinear Smale horseshoes, Rus. J. Nonlin. Dyn., 2007, Vol. 3, No. 4, pp.  423-443

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