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Vol. 9, No. 4

Vol. 9, No. 4, 2013

Sysoeva M. V.,  Ponomarenko V. I.,  Prokhorov M. D.,  Sysoev I. V.
A method is proposed for the reconstruction of first-order time-delay systems under external periodic driving from their time series. The method takes into account the structure of the model equation of the system, while constructing the autoregressive model. The proposed method allows one to reconstruct the delay time, the parameter characterizing the system inertial properties, the nonlinear function, and the amplitude and frequency of the external periodic driving. The method efficiency is demonstrated in a numerical experiment by reconstructing a number of different nonautonomous time-delay systems.
Keywords: reconstruction of model equations, time-delay systems, time series analysis
Citation: Sysoeva M. V.,  Ponomarenko V. I.,  Prokhorov M. D.,  Sysoev I. V., Reconstruction of time-delay systems under external periodic driving, Rus. J. Nonlin. Dyn., 2013, Vol. 9, No. 4, pp. 613-625
Bolsinov A. V.,  Borisov A. V.,  Mamaev I. S.
This paper develops the theory of the reducing multiplier for a special class of nonholonomic dynamical systems, when the resulting nonlinear Poisson structure is reduced to the Lie–Poisson bracket of the algebra $e(3)$. As an illustration, the Chaplygin ball rolling problem and the Veselova system are considered. In addition, an integrable gyrostatic generalization of the Veselova system is obtained.
Keywords: nonholonomic dynamical system, Poisson bracket, Poisson structure, reducing multiplier, Hamiltonization, conformally Hamiltonian system, Chaplygin ball
Citation: Bolsinov A. V.,  Borisov A. V.,  Mamaev I. S., Geometrization of the Chaplygin reducing-multiplier theorem, Rus. J. Nonlin. Dyn., 2013, Vol. 9, No. 4, pp. 627-640
Markov P. V.
An application of Lie groups of transformations theory for analysis of discrete dynamical systems is showed in this article. Families of two-dimensional and three-dimensional discrete dynamical systems with two-parameter and three-parameter Lie groups of transformations as continuous symmetries were obtained with using of classifications of two-dimensional and three-dimensional Lie algebras.
Keywords: discrete dynamical system, continuous symmetry, group classification, Lie group
Citation: Markov P. V., Group classification of discrete dynamical systems, Rus. J. Nonlin. Dyn., 2013, Vol. 9, No. 4, pp. 641-649
Aristov S. N.,  Prosviryakov E. Y.
New exact steady-state solutions of the Oberbeck–Boussinesq system which describe laminar flows of the Benard–Marangoni convection are constructed. We consider two types of boundary conditions: those specifying a temperature gradient on one of the boundaries and those specifying it on both boundaries simultaneously. It is shown that when the temperature gradient is specified the problem is essentially two-dimensional: there is no linear transformation allowing the flows to be transformed into one-dimensional ones. The resulting solutions are physically interpreted and dimensions of the layers are found for which there is no friction on the solid surface and a change occurs in the direction of velocity on the free surface.
Keywords: laminar flow, analytical solution, polynomial solution, decrease in dimension, Benard–Marangoni convection
Citation: Aristov S. N.,  Prosviryakov E. Y., On laminar flows of planar free convection, Rus. J. Nonlin. Dyn., 2013, Vol. 9, No. 4, pp. 651-657
Kulik K. N.,  Tur A. V.,  Yanovsky V. V.
In this work considered the motion of a point dipole vortex in circular domain occupied by an ideal fluid. The motion equations for a dipole vortex in an domain bounded by solid wall, are obtained. These equations have the Hamiltonian form. Integrability in the quadratures of the motion equations for a point dipole vortex in a circular domain is proved. The character movement vortex is discussed.
Keywords: point dipole vortex, Hamiltonian, motion equations
Citation: Kulik K. N.,  Tur A. V.,  Yanovsky V. V., The evolution point dipole vortex in a domain with circular boundaries, Rus. J. Nonlin. Dyn., 2013, Vol. 9, No. 4, pp. 659-669
Krasilnikov P. S.
The equation of plane nonlinear oscillations of satellite in an weakly elliptical orbit is investigated. Suppose, that equation of motion contains two small parameters. Various kinds of procedure which reduce the equation to one small parameter case are investigated. Lacks of such procedure are described. New resonance effects of satellite’s rotation are described with the help of the generalized averaging method with independent small parameters.
Keywords: independent parameters, average, small oscillations, reductions
Citation: Krasilnikov P. S., Small plane oscillations of satellite in an weakly elliptical orbit, Rus. J. Nonlin. Dyn., 2013, Vol. 9, No. 4, pp. 671-696
Rodnikov A. V.
A particle relative equilibria near a rigid body in the plane passing through the body axes of precession and of dynamical symmetry are studied in assumption that the body gravitational field can be composed as gravitational field of two conjugate complex masses being on imaginary distance. Using terminology of the Generalized Restricted Circular Problem of Three Bodies, these equilibria are called Coplanar Libration Points (CLP). One can show that CLP set is divided into three subsets dependently on CLPs type of evolution. There are 2 «external» CLPs going from infinity to the rigid body if precession angular velocity goes from zero to infinity, from 2 to 6 «internal» CLPs between axis of precession and axis of dynamical symmetry, and from 0 to 3 «central» CLPs near singular points of gravitational potential. Numerical-analitical algorithm of CLPs coordinates computation is suggested. This algorithm is based on some special trigonometrical transformations of coordinates and parameters.
Keywords: problem of three bodies, libration points, relative equilibrium, rigid body, regular precession
Citation: Rodnikov A. V., Coplanar libration points of the generalized restricted circular problem of three bodies for conjugate complex masses of attracting centers, Rus. J. Nonlin. Dyn., 2013, Vol. 9, No. 4, pp. 697-710
Tsiganov A. V.
We discuss an application of the Lie integrability theorem to the nonholonomic system describing the rolling of a dynamically balanced ball on horizontal absolutely rough table without slipping or sliding.
Keywords: nonholonomic mechanics, integrable systems, Poisson geometry
Citation: Tsiganov A. V., On the absolute dynamics of the Chaplygin ball, Rus. J. Nonlin. Dyn., 2013, Vol. 9, No. 4, pp. 711-719
Borisov A. V.,  Kilin A. A.,  Mamaev I. S.
We investigate the motion of the point of contact (absolute dynamics) in the integrable problem of the Chaplygin ball rolling on a plane. Although the velocity of the point of contact is a given vector function of variables of a reduced system, it is impossible to apply standard methods of the theory of integrable Hamiltonian systems due to the absence of an appropriate conformally Hamiltonian representation for an unreduced system. For a complete analysis we apply the standard analytical approach, due to Bohl and Weyl, and develop topological methods of investigation. In this way we obtain conditions for boundedness and unboundedness of the trajectories of the contact point.
Keywords: nonholonomic constraint, absolute dynamics, bifurcation diagram, bifurcation complex, drift, resonance, invariant torus
Citation: Borisov A. V.,  Kilin A. A.,  Mamaev I. S., The problem of drift and recurrence for the rolling Chaplygin ball, Rus. J. Nonlin. Dyn., 2013, Vol. 9, No. 4, pp. 721-754
Beghin H.
Citation: Beghin H., Sur certains problèmes de frottement, Rus. J. Nonlin. Dyn., 2013, Vol. 9, No. 4, pp. 755-759
Beghin H.
Citation: Beghin H., Sur l’indétermination de certains problèmes de frottement, Rus. J. Nonlin. Dyn., 2013, Vol. 9, No. 4, pp. 761-763
Ivanov A. P.
Citation: Ivanov A. P., Comments on the H.Beghin papers, Rus. J. Nonlin. Dyn., 2013, Vol. 9, No. 4, pp. 765-766

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