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Vol. 6, No. 2

Vol. 6, No. 2, 2010

Kondrashov R. E.,  Morozov A. D.
Abstract
We consider a problem about interaction of the two Duffing—van der Pol equations close to nonlinear integrable. The average systems describing behaviour of the solutions of the initial equation in resonant zones are deduced. The conditions of existence of not trivial resonant structures are established. The results of research in cases are resulted, when at the uncoupled equations exist and there are no limiting cycles.
Keywords: limit cycles, resonances
Citation: Kondrashov R. E.,  Morozov A. D., On investigation of resonances in system of two Duffing–van der Pol equations, Rus. J. Nonlin. Dyn., 2010, Vol. 6, No. 2, pp. 241-254
DOI:10.20537/nd1002001
Potapov V. I.
Abstract
In paper the computer research of four-dimensional dynamic system with three parameters adequately describing behavior of model coupled Dynamo in view of viscous friction is carried out. Is shown, that in this system there are five equilibrium states: four stable are focuses–node and one is saddle (3, 1). Are established the bifurcations of the spatial overwound cycles appropriate to doubling of the period of oscillations dynamic variable and resulting to chaotic oscillations at increase of the relation of factors friction.
Keywords: Rikitake system, equilibrium states, limit cycles, chaos
Citation: Potapov V. I., Visualization of phase trajectories of the Rikitake dynamic system, Rus. J. Nonlin. Dyn., 2010, Vol. 6, No. 2, pp. 255-265
DOI:10.20537/nd1002002
Malyaev V. S.,  Vadivasova T. E.
Abstract
In the present paper possibilities of parameters estimation are considered in dynamical systems (DS) with additive noise. Simple and effective algorithms, optimal parameter values of numeric simulation and data filtration methods are proposed that enable one to find the controlling parameter value of a noisy DS with a high accuracy. Different DS are studied, and the accuracy of parameter estimation is examined for various dynamical modes and for different noise intensities.
Keywords: dynamical system, fluctuations, noise, parameter estimation, bifurcations, chaos
Citation: Malyaev V. S.,  Vadivasova T. E., Parameter estimation in dynamical systems with additive noise, Rus. J. Nonlin. Dyn., 2010, Vol. 6, No. 2, pp. 267-276
DOI:10.20537/nd1002003
Koblyanskiy S. A.,  Shabunin A. V.,  Astakhov V. V.
Abstract
The phenomenon of forced synchronization of periodic oscillations in the multistable system is studied by the example of two linear coupled modified oscillators with inertial nonlinearity. It was found out that external forcing at certain amplitudes can sufficiently change the structure of the phase space of the system. As a result, the synchronous regime breaking for in-phase and non-in-phase oscillations proceeds in accordance with different scenarios.
Keywords: synchronization, multistability
Citation: Koblyanskiy S. A.,  Shabunin A. V.,  Astakhov V. V., Forced synchronization of periodic oscillations in a system with phase multistability, Rus. J. Nonlin. Dyn., 2010, Vol. 6, No. 2, pp. 277-289
DOI:10.20537/nd1002004
Guzev M. A.,  Izrailsky Y. G.,  Koshel' K. V.
Abstract
The appearance of chaotic regimes near elliptic point in a cell of particles’ chain interacting by means of Lennard–Jones potential is studied. The threshold nature of chaotization advent in the case of single-frequency cell excitation is demonstrated. A method of global chaotization based on multifrequency external excitation is proposed. The results of numerical experiments show that in this case the formation of global chaos is achieved at essentially lower values of external excitation amplitude and frequency, than in the case of single frequency excitation.
Keywords: nonlinear dynamics, molecular dynamics, Lennard–Jones potential, chaotic dynamics, Chirikov’s criterion
Citation: Guzev M. A.,  Izrailsky Y. G.,  Koshel' K. V., Global chaotization effect in particles chain, Rus. J. Nonlin. Dyn., 2010, Vol. 6, No. 2, pp. 291-305
DOI:10.20537/nd1002005
Gudimenko A. I.,  Zakharenko A. D.
Abstract
Qualitative structure of relative motion of three point vortices on the unbounded plain is studied. A classification of phase portraits is proposed.
Keywords: point vortices, relative equilibria, stability, algebraic reduction
Citation: Gudimenko A. I.,  Zakharenko A. D., Qualitative analysis of relative motion of three vortices, Rus. J. Nonlin. Dyn., 2010, Vol. 6, No. 2, pp. 307-326
DOI:10.20537/nd1002006
Vaskin V. V.,  Vaskina A. V.,  Mamaev I. S.
Abstract
With the help of mathematical modelling, we study the dynamics of many point vortices system on the plane. For this system, we consider the following cases:
— vortex rings with outer radius $r = 1$ and variable inner radius $r_0$,
— vortex ellipses with semiaxes $a$, $b$.
The emphasis is on the analysis of the asymptotic $(t → ∞)$ behavior of the system and on the verification of the stability criteria for vorticity continuous distributions.
Keywords: vortex dynamics, point vortex, hydrodynamics, asymptotic behavior
Citation: Vaskin V. V.,  Vaskina A. V.,  Mamaev I. S., Problems of stability and asymptotic behavior of vortex patches on the plane, Rus. J. Nonlin. Dyn., 2010, Vol. 6, No. 2, pp. 327-343
DOI:10.20537/nd1002007
Moskvin A. Y.
Abstract
The paper deals with the rolling motion of a balanced, dynamically asymmetric ball on a plane without sliding and spinning. The problem is natural but was not considered by classicists. Generalizations of the problem are analyzed for the case where gyrostat and force Brun field are added. To investigate the dynamic behavior of the system some peculiar periodic solutions are described and their stability is examined. By integral mapping, bifurcation diagrams and bifurcation complexes are constructed.
Keywords: bifurcation complex, rubber ball, stability, nonholonomic system
Citation: Moskvin A. Y., Rubber ball on a plane: singular solutions, Rus. J. Nonlin. Dyn., 2010, Vol. 6, No. 2, pp. 345-358
DOI:10.20537/nd1002008
Fedichev O. B.,  Fedichev P. O.
Abstract
We report a novel general method for constructing an approximate solution of the planar motion of solids with an axially symmetric mass distribution and normal stresses over the contact area on a rough horizontal surface. For a disk characterized by Galin distribution of contact stresses we obtain explicit dependence of the angular and sliding velocity of the body as a function of time. The relative errors of the method do not exceed 1,5–2 %. The simplicity and high accuracy of the method let us recommend its applications in the practice of engineering calculations.
Keywords: dry friction, Galin disk, flat motion
Citation: Fedichev O. B.,  Fedichev P. O., An approximate solution of a 2D rigid body motion problem on a rough surface, Rus. J. Nonlin. Dyn., 2010, Vol. 6, No. 2, pp. 359-364
DOI:10.20537/nd1002009
Abstract
Citation: Comments on "Conservation Laws, Hierarchy of Dynamics and Explicit Integration of Nonholonomic Systems" by A. V. Borisov and I. S. Mamaev. Response of A. V. Borisov, Rus. J. Nonlin. Dyn., 2010, Vol. 6, No. 2, pp. 365-369
DOI:10.20537/nd1002010
Abstract
Citation: Problems of the impact theory: Billiards, Rus. J. Nonlin. Dyn., 2010, Vol. 6, No. 2, pp. 371-372
DOI:10.20537/nd1002011
Borisov A. V.,  Kilin A. A.,  Mamaev I. S.
Abstract
In this paper we develop a new model of non-holonomic billiard that accounts for the intrinsic rotation of the billiard ball. This model is a limit case of the problem of rolling without slipping of a ball without slipping over a quadric surface. The billiards between two parallel walls and inside a circle are studied in detail. Using the three-dimensional-point-map technique, the non-integrability of the non-holonomic billiard within an ellipse is shown.
Keywords: billiard, impact, point mapping, nonintegrability, periodic solution, nonholonomic constraint, integral of motion
Citation: Borisov A. V.,  Kilin A. A.,  Mamaev I. S., On the model of non-holonomic billiard, Rus. J. Nonlin. Dyn., 2010, Vol. 6, No. 2, pp. 373-385
DOI:10.20537/nd1002012
Darboux G.
Abstract
Citation: Darboux G., Etude geometrique sur les percussions et le choc des corps, Rus. J. Nonlin. Dyn., 2010, Vol. 6, No. 2, pp. 387-413
DOI:10.20537/nd1002013
Resal H.
Abstract
Citation: Resal H., Commentarie a la theorie mathematique du jeu de billard, Rus. J. Nonlin. Dyn., 2010, Vol. 6, No. 2, pp. 415-438
DOI:10.20537/nd1002014
Ivanov A. P.
Abstract
Citation: Ivanov A. P., On the mathematical treatment of the impact in billiards, Rus. J. Nonlin. Dyn., 2010, Vol. 6, No. 2, pp. 439-447
DOI:10.20537/nd1002015
Abstract
Citation: Review of the book «Integrable billiards, quadrics, and multidimensional Poncelet’s porisms» by V. Dragovic and M. Radnovic, Rus. J. Nonlin. Dyn., 2010, Vol. 6, No. 2, pp. 449-450
Abstract
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., 2010, Vol. 6, No. 2, pp. 451-456

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