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Vol. 8, No. 1

Vol. 8, No. 1, 2012

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
Anishchenko V. S.,  Astakhov S. V.,  Boev Y. I.,  Kurths J.
Statistical properties of Poincaré recurrences in a two-dimensional map with chaotic non-strange attractor have been studied in numerical simulations. A local and a global approaches were analyzed in the framework of the considered problem. It has been shown that the local approach corresponds to Kac’s theorem including the case of a noisy system in certain conditions which have been established. Numerical proof of theoretical results for a global approach as well as the Afraimovich–Pesin dimension calculation are presented.
Keywords: Poincaré recurrence, attractor dimension, Afraimovich–Pesin dimension
Citation: Anishchenko V. S.,  Astakhov S. V.,  Boev Y. I.,  Kurths J., Poincaré recurrences in a system with non-strange chaotic attractor, Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 1, pp. 29-41
Tsiganov A. V.
The main aim of the second part of the paper is a construction of the rational potentials, which may be added to the Hamiltonians of the Chaplygin and Borisov–Mamaev–Fedorov systems without loss of integrability. All these potentials may be considered as natural nonholonomic generalizations of the standard separable potentials associated with an elliptic (or sphero-conical) coordinate system on the sphere.
Keywords: nonholonomic mechanics, Chaplygin sphere, Poisson brackets
Citation: Tsiganov A. V., On the bi-Hamiltonian structure of the Chaplygin and Borisov–Mamaev–Fedorov systems at a zero constant of areas. II, Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 1, pp. 43-55
Kozlov V. V.
Invariant manifolds of equations governing the dynamics of conservative nonholonomic systems are investigated. These manifolds are assumed to be uniquely projected onto configuration space. The invariance conditions are represented in the form of generalized Lamb’s equations. Conditions are found under which the solutions to these equations admit a hydrodynamical description typical of Hamiltonian systems. As an illustration, nonholonomic systems on Lie groups with a left-invariant metric and left-invariant (right-invariant) constraints are considered.
Keywords: invariant manifold, Lamb’s equation, vortex manifold, Bernoulli’s theorem, Helmholtz’ theorem
Citation: Kozlov V. V., On invariant manifolds of nonholonomic systems, Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 1, pp. 57-69
Markeev A. P.
A material system consisting of a «carrying» rigid body (a shell) and a body «being carried» (a material point) is considered. The system moves in a uniform field of gravity over a fixed absolutely smooth horizontal plane. The central ellipsoid of inertia of the shell is an ellipsoid of rotation. The material point moves according to the harmonic law along a straight-line segment rigidly attached to the shell and lying on its axis of dynamical symmetry. During its motion, the shell may collide with the plane. The coefficient of restitution for an impact is supposed to be arbitrary. The periodic motion of the shell is found when its symmetry axis is situated along a fixed vertical, and the shell rotates around this vertical with an arbitrary constant angular velocity. The conditions for existence of this periodic motion are obtained, and its stability in the linear approximation is studied.
Keywords: rigid body dynamics, collision, periodic motion, stability
Citation: Markeev A. P., On a periodic motion of a rigid body carrying a material point in the presence of impacts with a horizontal plane, Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 1, pp. 71-81
Salnikova T. V.,  Treschev D. V.,  Gallyamov S. R.
We consider the problem of a disk sliding on a horizontal plane under the action of dry friction forces. The model is based on three hypotheses. The law of interaction of a small element of the disk’s surface with the plane is the Amonton–Coulomb law, the pressure distribution over the contact patch is a linear (generally speaking, time-dependent) function of Cartesian coordinates, the height of the disk is not high. The equations of motion possess a rich group of symmetry, which enables a detailed qualitative analysis of the problem.
Keywords: dry friction, Amontons–Coulomb law
Citation: Salnikova T. V.,  Treschev D. V.,  Gallyamov S. R., On the motion of free disc on the rough horisontal plane, Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 1, pp. 83-101
Borisov A. V.,  Mamaev I. S.
We consider the problem of rolling of a ball with an ellipsoidal cavity filled with an ideal fluid, which executes a uniform vortex motion, on an absolutely rough plane. We point out the case of existence of an invariant measure and show that there is a particular case of integrability under conditions of axial symmetry.
Keywords: vortex motion, non-holonomic constraint, Chaplygin ball, invariant measure, integrability, rigid body, ideal fluid
Citation: Borisov A. V.,  Mamaev I. S., The dynamics of the Chaplygin ball with a fluid-filled cavity, Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 1, pp. 103-111
Borisov A. V.,  Kilin A. A.,  Mamaev I. S.
We consider the problem of the motion of axisymmetric vortex rings in an ideal incompressible fluid. Using the topological approach, we present a method for complete qualitative analysis of the dynamics of a system of two vortex rings. In particular, we completely solve the problem of describing the conditions for the onset of leapfrogging motion of vortex rings. In addition, for the system of two vortex rings we find new families of motions in which the mutual distances remain finite (we call them pseudo-leapfrogging). We also find solutions for the problem of three vortex rings, which describe both the regular and chaotic leapfrogging motion of vortex rings.
Keywords: ideal fluid, vortex ring, leapfrogging motion of vortex rings, bifurcation complex, periodic solution, integrability, chaotic dynamics
Citation: Borisov A. V.,  Kilin A. A.,  Mamaev I. S., The dynamics of vortex rings: Leapfrogging, choreographies and the stability problem, Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 1, pp. 113-147
Thompson S. P.
Citation: Thompson S. P., Gyrostatics and wave motion (Chapter XVIII of «The Life of William Thomson»), Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 1, pp. 149-153
Thomson W.
Citation: Thomson W., On the precessional motion of a liquid, Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 1, pp. 155-159
Maxwell J. C.
Citation: Maxwell J. C., Letter to William Thomson, 6 October 1868, Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 1, pp. 161-165
Ryabov P. E.
Citation: Ryabov P. E., A reply to «Comments» by A.V. Tsiganov (ND, 2011, no. 3, p. 715), Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 1, pp. 167-172
Khokhlov A. R.,  Brilliantov N. V.,  Belyakin S. T.,  Dzhanoev A. R.,  Zhuchkova E. A.,  Kotlyarov O. L.,  Krotov S. S.,  Larionov S. A.,  Postnikov E. B.,  Riznichenko, G. Y.,  Rybalko S. D.,  Ryabov A. B.
Citation: Khokhlov A. R.,  Brilliantov N. V.,  Belyakin S. T.,  Dzhanoev A. R.,  Zhuchkova E. A.,  Kotlyarov O. L.,  Krotov S. S.,  Larionov S. A.,  Postnikov E. B.,  Riznichenko, G. Y.,  Rybalko S. D.,  Ryabov A. B., Alexander Yurievich Loskutov (5.5.1959–5.11.2011), Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 1, pp. 175-177
Grinchenko V. T.,  Krasnopolskaya T. S.,  Borisov A. V.,  van Heijst G. J.
Citation: Grinchenko V. T.,  Krasnopolskaya T. S.,  Borisov A. V.,  van Heijst G. J., Viatcheslav Vladimirovich Meleshko (07.10.1951–14.11.2011), Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 1, pp. 179-182
Afraimovich V. S.,  Belyakov L. A.,  Bykov V. V.,  Gonchenko S. V.,  Lerman L. M.,  Lukyanov V. I.,  Malkin M. I.,  Morozov A. D.,  Turaev D. V.
Citation: Afraimovich V. S.,  Belyakov L. A.,  Bykov V. V.,  Gonchenko S. V.,  Lerman L. M.,  Lukyanov V. I.,  Malkin M. I.,  Morozov A. D.,  Turaev D. V., Leonid Pavlovich Shilnikov (17.12.1934–26.12.2011), Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 1, pp. 183-186
Anishchenko V. S.
Citation: Anishchenko V. S., In memory of Leonid Pavlovich Shilnikov, Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 1, pp. 187-190
Citation: New books, Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 1, pp. 191-193

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