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Vol. 5, No. 4

Vol. 5, No. 4, 2009

Borisov A. V.,  Kilin A. A.,  Mamaev I. S.
We consider the motion of a material point on the surface of a sphere in the field of 2n+1 identical Hooke centers (singularities with elastic potential) lying on a great circle. Our main result is that this system is superintegrable. The property of superintegrability for this system has been conjectured by us in [3], where the structure of a superintegral of arbitrarily high odd degree in momemnta was outlined. We also indicate an isomorphism between this system and the one-dimensional N-particle system discussed in the recent paper [13] and show that for the latter system an analogous superintegral can be constructed.
Keywords: superintegrable systems, systems with a potential, Hooke center
Citation: Borisov A. V.,  Kilin A. A.,  Mamaev I. S., New superintegrable system on a sphere, Rus. J. Nonlin. Dyn., 2009, Vol. 5, No. 4, pp. 455-462
Grigoryev Y. A.,  Tsiganov A. V.
The paper deals with superintegrable $N$-degree-of-freedom systems of Richelot type, for which $n\leqslant N$ equations of motion are the Abel equations on a hyperelliptic curve of genus $n−1$. The corresponding additional integrals of motion are second-order polynomials in momenta.
Keywords: superintegrable systems, separation of variables, Abel equations
Citation: Grigoryev Y. A.,  Tsiganov A. V., On the Abel equations and the Richelot integrals, Rus. J. Nonlin. Dyn., 2009, Vol. 5, No. 4, pp. 463-478
Ivanov A. P.
Examples of irregular behavior of dynamical systems with dry friction are discussed. A classification of frictional contacts with respect to their dimensionality, associativity, and the possibility of interruptions is proposed and basic models showing typical features are stated. In particular, bifurcation conditions for equilibrium families are obtained and formulas for the monodromy matrix for systems with friction are constructed. It is shown that systems with non-associated contacts possess singularities that lead to the nonexistence or nonuniqueness of phase trajectories; these results generalize the paradoxes of Painlev´e and Jellett. Owing to such behavior, a number of earlier results, including the problem on the motion of a rigid body on a rough plane, require an improvement.
Keywords: non-smooth dynamical systems, dry friction, discontinuous bifurcation
Citation: Ivanov A. P., Bifurcations in systems with friction: basic models and methods, Rus. J. Nonlin. Dyn., 2009, Vol. 5, No. 4, pp. 479-498
Kosenko I.,  Aleksandrov E. B.
In frame of the Hertz contact problem an approximate model to compute resulting wrench of the dry friction tangent forces is built up. The wrench consists of the total friction force and the drilling friction torque. An approach under consideration develops in a natural way the contact model built up earlier. The dry friction forces and torque are integrated over the contact elliptic spot. Generally an analytic computation of the integrals mentioned leads to the cumbersome calculation, decades of terms, including rational functions depending in turn on complete elliptic integrals. To implement the elastic bodies contact interaction computer model fast enough one builds up the approximate model in the direction as it was proposed by Contensou. The model under construction is one derived from the Contensou simplified model in the following directions: (a) the model is anisotropic: the total friction forces along ellipse axes are different; (b) for the translatory and almost translatory relative motions one uses the Coulomb friction law regularization; © the approximate model for the drilling torque also has been constructed. To verify the model built the results obtained by several authors were used. The Tippe-Top dynamic model is used as a an example under testing. It turned out the top revolution process is identical to one simulated using the set-valued functions approach. The ball bearing dynamic model is used to verify different approaches to the tangent forces computational implementation in details. The model objects corresponding to contacts between the balls and raceways were replaced by ones of new class developed here. Then the old friction model of the regularized Coulomb type and the new one, approximate Contensou, each embedded into the whole bearing dynamic model were thoroughly tested and compared. It turned out the simplified Contensou approach provides the computer model even faster in compare with the case of the point contact.
Keywords: the Hertz contact model, the Contensou–Erismann model, the Contensou simplified model, the Vilke model, the Tippe-Top, the ball bearing model
Citation: Kosenko I.,  Aleksandrov E. B., Implementation of the Contensou-Erismann model for tangent forces in the Hertz contact problem, Rus. J. Nonlin. Dyn., 2009, Vol. 5, No. 4, pp. 499-517
Rodnikov A. V.
We consider the system moving in the Newtonian central force field and consisting of a dumbbell satellite and a particle. The particle coasts along on the cable with ends placed in the dumbbell endpoints. We call such cable a «leier». We suppose the system mass center describes circular orbit, the particle mass is small in comparison with the dumbbell mass and the cable length is small in comparison with orbit radius. Assuming the cable don’t leave the orbit plane we study the dumbbell rotations forced by the particle. We note that the particle sufficiently influence the dumbbell motion only in vicinity of the dumbbell rotation separatrix. We claim that there exist a set of the dumbbell unstable asymptotic motions tending to librations about the orbit tangent. Initial conditions for these motions compose a surface in the system phase space. We deduce an equation approximating this surface. We consider this equation as a criterion for the direction of the dumbbell rotation from the vicinity of unstable equilibria. We deduce formulae approximating the dumbbell near-separatrix motion if the cable is rather long and the dumbbell is composed of equal masses. Using numerical procedures, we analyse the dumbbell motion in two-dimensional transections of four-dimensional space of initial conditions if the cable is rather short.
Keywords: space tether system, unilateral constraint, central Newtonian field, Lagrangian, perturbed system
Citation: Rodnikov A. V., On the leier influence on a dumbbell motion in the central Newtonian force field, Rus. J. Nonlin. Dyn., 2009, Vol. 5, No. 4, pp. 519-533
Bardin B. S.
We deal with the problem of orbital stability of pendulum like periodic motions of a heavy rigid body with a fixed point. We suppose that the geometry of the mass of the body corresponds to the Bobylev—Steklov case. Unperturbed motion represents oscillations or rotations of the body around a principal axis, occupying a fixed horizontal position. The problem of the orbital stability is considered on the base of a nonlinear analysis.

In the case of oscillations with small amplitudes as well as in the case of rotations with high angular velocities we studied the problem analytically. In general case we reduce the problem to the stability study of fixed point of the symplectic map generated by equations of perturbed motion. We calculate coefficients of the symplectic map numerically. By analyzing of the coefficients mentioned we establish orbital stability or instability of the unperturbed motion. The results of the study are represented in the form of stability diagram.
Keywords: Hamiltonian system, periodic orbits, normal form, resonance, action-angel variables, KAM theory
Citation: Bardin B. S., On orbital stability of pendulum like motions of a rigid body in the Bobylev-Steklov case, Rus. J. Nonlin. Dyn., 2009, Vol. 5, No. 4, pp. 535-550
Sudakov S. N.
The paper of S.V. Jaques [1] is concerned with the problem of finding forms of rotation cavities in which there can exist a homogeneous vortex motion of an ideal incompressible fluid. This paper solves this problem without the assumption that the boundary of the cavity is the rotation surface.
Keywords: the homogeneous vortex flows, incompressible inviscid fluid, cavity, the theorems of Helmholtz
Citation: Sudakov S. N., On the form of a closed cavity in which there exist homogeneous vortex motions of an ideal incompressible fluid, Rus. J. Nonlin. Dyn., 2009, Vol. 5, No. 4, pp. 551-559
Vaskin V. V.,  Erdakova N. N.
This study is the continuation of the computer experiment [1] with particles of gas in a one-dimensional tube, described earlier. In this paper we give investigation results for the statistical properties of a relativistic gas in a one-dimensional tube. It is shown that this system reaches the state of thermodynamical equilibrium whose distribution function is determined by the relativistic energy of particles. The system of particles in a one-dimensional tube is described by analogy with the billiards in a polygon.
Keywords: relativistic gas, thermodynamical equilibrium, gas in a one-dimensional tube, Boltzmann distribution
Citation: Vaskin V. V.,  Erdakova N. N., Statistical mechanics of relativistic gas in a one-dimensional tube, Rus. J. Nonlin. Dyn., 2009, Vol. 5, No. 4, pp. 561-567
Lerman L. M.,  Slinyakova N. A.
In this paper we study the piecewise-linear model of the stationary Swift—Hohenberg equation well known in mathematical physics, which provides explicit front type solutions. Due to the reversibility relative to two involutions of the corresponding Hamiltonian system, this involves the existence of a heteroclinic contour connecting two saddle-foci. Using methods of symbolic dynamics, we give a description of all solutions lying in the neighborhood of the contour at the level of the Hamiltonian containing the contour.
Keywords: Swift–Hohenberg equation, fronts, heteroclinic contour, Hamiltonian system, saddle focus, symbolic dynamics
Citation: Lerman L. M.,  Slinyakova N. A., On the dynamics of the piecewise-linear model of the Swift–Hohenberg equation, Rus. J. Nonlin. Dyn., 2009, Vol. 5, No. 4, pp. 569-583
Alfimov G. L.
Nonlocal sine-Gordon equation arises in numerous problems of modern mathematical physics, for instance, in Josephson junction models and lattice models with long-range interactions. Kink solutions of this equation correspond to physically relevant objects such as magnetic flux vortex in Josephson electrodynamics. In this paper the kink solutions for the nonlocal sine-Gordon equation are considered in weak nonlocality limit. In this limit the equation for travelling waves can be reduced to ordinary differential equation of 4th order with two governing parameters. A survey of possible kink solutions for this equation and for all combination of the governing parameters is presented. The collection of known results is given for the regions on the plane of model parameters which just have been investigated. New results of qualitative and numerical analysis are reported for other regions of the plane of model parameters.
Keywords: sine-Gordon equation, nonlocality, separatrices
Citation: Alfimov G. L., Nonlocal sine-Gordon equation: kink solutions in the weak nonlocality limit, Rus. J. Nonlin. Dyn., 2009, Vol. 5, No. 4, pp. 585-602
Bashkirtseva I. A.,  Zubarev A. Y.,  Iskakova L. Y.,  Ryashko L. B.
This paper is devoted to research of mathematical model of a suspension flow. For these flows, the transitions from stationary to the oscillatory regimes have been observed in experiments. Bifurcation analysis allows us to divide the space of parameters onto steady equilibria and limit cycles zones. Details of Hopf bifurcation depending on degree of system stiffness are investigated. On the basis of the stochastic sensitivity function technique, the parametrical analysis of influence of random disturbances on the system attractors is carried out. It is shown that as a system stiffness increases, the stochastic sensitivity of oscillations rises sharply. The narrow zone of super-sensitivity of oscillations was found. In this zone, even small background disturbances result in the essential fluctuations of their amplitude.
Keywords: reology, suspension, stability, sensitivity, stochastic oscillations
Citation: Bashkirtseva I. A.,  Zubarev A. Y.,  Iskakova L. Y.,  Ryashko L. B., Regular and stochastic auto-oscillations in the reological model, Rus. J. Nonlin. Dyn., 2009, Vol. 5, No. 4, pp. 603-620
Citation: Editorial remarks on the paper by G. M. Rozenblat «On the motion of a two-dimensional rigid body on a rough straight line» and «A method determination of the parameters of motion of a top that moves without bouncing on a smooth plane», Rus. J. Nonlin. Dyn., 2009, Vol. 5, No. 4, pp. 621-624
Vaskin V. V.,  Naymushina O. S.
We analyze domains of an axisymmetric ball with the shifted center mass motion without bouncing (i. e. in constant contact) on a smooth plane. We show that these domains belong to the region of parameters, corresponding to regimes of regular precession (the ball’s axis about axis z). We also give explicit formulas for domain boundaries.
Keywords: rigid body, motion without bouncing on a smooth plane
Citation: Vaskin V. V.,  Naymushina O. S., On the motion of a ball without bouncing on a smooth plane, Rus. J. Nonlin. Dyn., 2009, Vol. 5, No. 4, pp. 625-632
Frost V. A.
Citation: Frost V. A., Preface to the translation of the pioneering work by K. Hasselmann, Rus. J. Nonlin. Dyn., 2009, Vol. 5, No. 4, pp. 633-634
Hasselmann K.
Citation: Hasselmann K., Zur Deutung der dreifachen Geschwindigkeitskorrelationen der isot-ropen Turbulenz, Rus. J. Nonlin. Dyn., 2009, Vol. 5, No. 4, pp. 635-647
Citation: New books of the Scientific and Publishing Center «Regular and Chaotic Dynamics» and Institute of Computer Science (Moscow-Izhevsk). New issues of «Regular and Chaotic Dynamics», Rus. J. Nonlin. Dyn., 2009, Vol. 5, No. 4, pp. 649-654

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