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Vol. 7, No. 3

Vol. 7, No. 3, 2011
On the 60th birthday of A.P.Ivanov

Kuznetsov A. P.,  Sataev I. R.,  Turukina L. V.
The problem of external driving by the harmonic signal of two coupled self-oscillators is investigated. Comparison with the synchronization picture for phase oscillators is given. We discuss the configuration of periodic, two- and three-frequency regimes in the parameter space of external signal. The illustrations of three-frequency tori and resonance two-frequency tori are given. A number of significant differences from the bifurcation mechanisms for the destruction of synchronization are found compared with the case of phase oscillators.
Keywords: synchronization, bifurcations, quasi-periodic dynamics, chaos
Citation: Kuznetsov A. P.,  Sataev I. R.,  Turukina L. V., Forced synchronization of two coupled van der Pol self-oscillators, Rus. J. Nonlin. Dyn., 2011, Vol. 7, No. 3, pp. 411-425
Afonin V. V.,  Boletskaya T. K.
The investigation of II and III type intermittency in dynamical systems described by discrete maps using wavelet analysis in the presence and the absence of noise is carried out.
Keywords: intermittency, wavelet analysis
Citation: Afonin V. V.,  Boletskaya T. K., Wavelet analysis of II and III type intermittency, Rus. J. Nonlin. Dyn., 2011, Vol. 7, No. 3, pp. 427-436
Kondrashov R. E.,  Morozov A. D.
The problem of global behavior of solutions in system of two Duffing–Van der Pole equations close to nonlinear integrable is considered. For regions without unperturbed separatrixes we give partially averaged systems which describe the behavior of solutions of original system in resonant zones. The finiteness of number of non-trivial resonant structures is established. Also we give fully averaged systems which describe the behavior of solutions outside of neighborhoods of nontrivial resonant structures. The results of numerically investigation of these systems are resulted.
Keywords: limit cycles, resonances, averaging
Citation: Kondrashov R. E.,  Morozov A. D., On global behaviour of the solutions of system of two Duffing–Van der Pole equations, Rus. J. Nonlin. Dyn., 2011, Vol. 7, No. 3, pp. 437-449
Georgievsky D. V.
This work deals with tensor-nonlinear constitutive relations connecting the deviators of stress tensor and strain rate tensor in incompressible isotropic media which are called in continuum mechanics as Reiner–Rivlin fluids. The connections of quadratic and cubic invariants of two tensors, where two material functions involve, are presented. The main attention is given to one-dimensional shear flows in various curvilinear coordinate systems. The scheme of obtaining of the material functions for shear on the basis of the steady Poiseuille flow in a plane layer is described. The self-similar solutions corresponding to the generalized diffusion of vortex layer both in plane and axially symmetric cases are derived.
Keywords: tensor nonlinearity, invariant, material function, constitutive relation, Reiner–Rivlin fluid, shear, diffusion of vortex, vortex layer
Citation: Georgievsky D. V., Tensor-nonlinear shear flows: Material functions and the diffusion-vortex solutions, Rus. J. Nonlin. Dyn., 2011, Vol. 7, No. 3, pp. 451-463
Kurakin L. G.
We investigate the stability problem for a system of five stationary rotation identical point vortices located at the vertices of a regular pentagon inside a circular domain. The main result is the proof of theorems which have been announced the author in paper (Doklady Physics, 2004, vol. 49, no 11, pp. 658–661).
Keywords: point vortex, stationary motion, stability, resonance
Citation: Kurakin L. G., On the Stability of Thomson’s Vortex Pentagon Inside a Circular Domain, Rus. J. Nonlin. Dyn., 2011, Vol. 7, No. 3, pp. 465-488
Gutkin E.
The survey of the subject, emphasizing the open problems.

With a supplement written by the author for a Russian translation. (The 2003 English original: Eugene Gutkin, Billiard dynamics: A survey with the emphasis on open problems on billiards, Regul. Chaotic Dyn., 2003, 8 (1), pp. 1–13.)
Keywords: billiard flow, billiard map, periodic orbits, elliptic dynamics, hyperbolic dynamics, parabolic dynamics, Birkhoff conjecture, phase locking, security
Citation: Gutkin E., Billiard dynamics: a survey with the emphasis on open problems, Rus. J. Nonlin. Dyn., 2011, Vol. 7, No. 3, pp. 489-512
Citation: Alexander Pavlovich Ivanov. On his 60th Birthday, Rus. J. Nonlin. Dyn., 2011, Vol. 7, No. 3, pp. 515-520
Ivanov A. P.,  Shuvalov N. D.
In this paper we discuss the dynamics of an axisymmetric rigid body whose circular area moves upon a horizontal rough surface. We investigate the interaction between the character of the law of friction and the curvature of the body’s trajectory. For the case of a curling stone it is shown that the observed effects can only be explained using the dependence of the friction coefficient on the Gumbel number. The procedure for constructing the law of friction based on experimental data is developed. It is shown that the available data can only be substantiated by means of anisotropic friction. The simplest model of such friction is constructed which provides quantitative coincidence with the experiment.
Keywords: mixed friction, Stribeck curve, curling rock dynamics
Citation: Ivanov A. P.,  Shuvalov N. D., On the motion of a heavy body with a circular base on a horizontal plane and riddles of curling, Rus. J. Nonlin. Dyn., 2011, Vol. 7, No. 3, pp. 521-530
Markeev A. P.
A nonlinear equation of motion for a 0pendulum-type system is investigated. It differs from the classical equation of a mathematical pendulum in the presence of a parametric disturbance. The potential energy of the «pendulum» is a two-stage periodic step function of time. The equation depends on two parameters that characterize the time-averaged value of a parametric disturbance and the depth of its «ripple». These parameters can take on arbitrary values. There exist two equilibrium configurations corresponding to the hanging and inverse «pendulum». The problem of stability of these equilibria is considered. In the first approximation it necessitates an analysis of the well-known linear Meissner equation. A detailed investigation of this equation is carried out supplementing and specifying the known results. The nonlinear problem of stability of equilibria is solved.
Keywords: parametric oscillations, stability, resonance, mapping
Citation: Markeev A. P., On nonlinear Meissner’s equation, Rus. J. Nonlin. Dyn., 2011, Vol. 7, No. 3, pp. 531-547
Fedichev O. B.,  Fedichev P. O.
We propose a qualitative theory of stopping dynamics of solids moving on a plane surface with an arbitrary distribution of normal stresses in the contact area. We studied the equations of motion describing the combined action of the dry friction acting on a sliding and spinning body all the way long before the motion ceases, calculated the movement time, and the distance traveled. Finally we identified the localization of the instantaneous center of rotation at the time of the complete stop, which depends on the mass distribution within the body and on the asymptotic behavior of the friction force and torque.
Keywords: dry friction, flat motion, instantaneous center of rotation
Citation: Fedichev O. B.,  Fedichev P. O., Stopping dynamics of sliding and spinning bodies on a rough plane surface, Rus. J. Nonlin. Dyn., 2011, Vol. 7, No. 3, pp. 549-558
Borisov A. V.,  Mamaev I. S.
The paper considers two new integrable systems due to Chaplygin, which describe the rolling of a spherical shell on a plane, with a ball or Lagrange’s gyroscope inside. All necessary first integrals and an invariant measure are found. The reduction to quadratures is given.
Keywords: non-holonomic constraint, integrability, invariant measure, gyroscope, quadrature, coupled rigid bodies
Citation: Borisov A. V.,  Mamaev I. S., Two non-holonomic integrable systems of coupled rigid bodies, Rus. J. Nonlin. Dyn., 2011, Vol. 7, No. 3, pp. 559-568
Beletsky V. V.,  Rodnikov A. V.
A particle steady motions in vicinity of dynamically symmetric precessing rigid body are studied in assumption that the body gravitational field is modeled as two centers gravitational field. The particle motion equations are written as two-parametric generalization for equations of Restricted Circular Problem of Three Bodies (RCP3B). Existence and number of the particle relative equilibria in the plane passing through the body axis of dynamical symmetry and through the vector of angular momentum are established. These equilibria called Coplanar Libration Points (CLP) are analogs of Eulerian Libration Points in RCP3B. Stability of CLP is studied for the first approximation in assumption that attracting centers have equal masses.
Keywords: problem of three bodies, libration points, stability, steady motions, asteroid
Citation: Beletsky V. V.,  Rodnikov A. V., Coplanar Libration Points in the Generalized Restricted Circular Problem of Three Bodies, Rus. J. Nonlin. Dyn., 2011, Vol. 7, No. 3, pp. 569-576
Tsiganov A. V.
We discuss the nonholonomic Chaplygin and the Borisov–Mamaev–Fedorov systems when the corresponding phase space is equivalent to cotangent bundle to dwo-dimensional sphere. In both cases Poisson bivectors are determined by L-tensors with non-zero torsion on the configurational space, in contrast with the well known Eisenhart–Benenti and Turiel constructions.
Keywords: nonholonomic mechanics, Chaplygin sphere, Poisson brackets
Citation: Tsiganov A. V., On deformations of the canonical Poisson bracket for the nonholonomic Chaplygin and the Borisov–Mamaev–Fedorov systems on zero-level of the area integral I, Rus. J. Nonlin. Dyn., 2011, Vol. 7, No. 3, pp. 577-599
Kuleshov A. S.,  Treschev D. V.,  Ivanova T. B.,  Naymushina O. S.
The paper considers two two-dimensional dynamical problems for an absolutely rigid cylinder interacting with a deformable flat base (the motion of an absolutely rigid disk on a base which in non-deformed condition is a straight line). The base is a sufficiently stiff viscoelastic medium that creates a normal pressure $p(x) = kY(x)+ν\dot{Y}(x)$, where $x$ is a coordinate on the straight line, $Y(x)$ is a normal displacement of the point $x$, and $k$ and $ν$ are elasticity and viscosity coefficients (the Kelvin—Voigt medium). We are also of the opinion that during deformation the base generates friction forces, which are subject to Coulomb’s law. We consider the phenomenon of impact that arises during an arbitrary fall of the disk onto the straight line and investigate the disk’s motion «along the straight line» including the stages of sliding and rolling.
Keywords: Kelvin–Voight medium, impact, viscoelasticity, friction
Citation: Kuleshov A. S.,  Treschev D. V.,  Ivanova T. B.,  Naymushina O. S., A rigid cylinder on a viscoelastic plane, Rus. J. Nonlin. Dyn., 2011, Vol. 7, No. 3, pp. 601-625
Kozlov V. V.
The structure of the Lorentz force and the related analogy between electromagnetism and inertia are discussed. The problem of invariant manifolds of the equations of motion for a charge in an electromagnetic field and the conditions for these manifolds to be Lagrangian are considered.
Keywords: Lorentz force, Maxwell equations, Coriolis force, symplectic structure, Lagrangian manifold
Citation: Kozlov V. V., The Lorentz force and its generalizations, Rus. J. Nonlin. Dyn., 2011, Vol. 7, No. 3, pp. 627-634
Ramodanov S. M.,  Tenenev V. A.
In the paper we consider the motion of a rigid body in a boundless volume of liquid. The body is set in motion by redistribution of internal masses. The mathematical model employs the equations of motion for the rigid body coupled with the hydrodynamic Navier–Stokes equations. The problem is mostly dealt with numerically. Simulations have revealed that the body’s trajectory is strongly governed by viscous effects.
Keywords: self-propulsion, Navier–Stokes equations, viscous vortical motion, numerical methods
Citation: Ramodanov S. M.,  Tenenev V. A., Motion of a body with variable distribution of mass in a boundless viscous liquid, Rus. J. Nonlin. Dyn., 2011, Vol. 7, No. 3, pp. 635-647
Bolsinov A. V.,  Borisov A. V.,  Mamaev I. S.
The paper is concerned with the use of bifurcation analysis and the Conley index in Hamiltonian dynamical systems. We give the proof of the theorem on the appearance (disappearance) of fixed points in the case of the Morse index change. New relative equilibria in the problem of the motion of point vortices of equal intensity in a circle are found.
Keywords: Morse index, Conley index, bifurcation analysis, bifurcation diagram, Hamiltonian dynamics, fixed point, relative equilibrium
Citation: Bolsinov A. V.,  Borisov A. V.,  Mamaev I. S., The bifurcation analysis and the Conley index in mechanics, Rus. J. Nonlin. Dyn., 2011, Vol. 7, No. 3, pp. 649-681
McGehee R.
Citation: McGehee R., Charles C. Conley, Rus. J. Nonlin. Dyn., 2011, Vol. 7, No. 3, pp. 683-690
Meleshko V. V.
Citation: Meleshko V. V., Hassan Aref (1950–2011), Rus. J. Nonlin. Dyn., 2011, Vol. 7, No. 3, pp. 691-713
Tsiganov A. V.
Citation: Tsiganov A. V., Comments on P.E. Ryabov «Explicit integration and topology of D.N. Goryachev case», Rus. J. Nonlin. Dyn., 2011, Vol. 7, No. 3, pp. 715-717
Citation: Book review: O.M.O’Reilly «Intermediate dynamics for engineers: A unified treatment of Newton–Euler and Lagrangian mechanics», Rus. J. Nonlin. Dyn., 2011, Vol. 7, No. 3, pp. 719-722
Citation: New books. New issues of «Regular and Chaotic Dynamics», Rus. J. Nonlin. Dyn., 2011, Vol. 7, No. 3, pp. 723-726

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