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Volume 7, Number 3

Volume 7, Number 3, 2002

Nekhoroshev N. N.
Generalizations of Gordon Theorem
Abstract
In this article we present some generalizations of the Gordon theorem for Hamiltonian systems that are integrable on submanifolds of the phase space. The main result of the article is a generalization of the Gordon theorem. More precisely we consider a system that is integrable in "Hamiltonian" sense on such a submanifold and prove that for the conditionally periodic motion on invariant isotropic $k$-dimensional tori foliating this submanifold, the frequencies of this motion depend only on the values of $k$ "central" integrals of the system on a torus. Here, the central integrals are $k$ functions defined on the whole phase space that are integrals of the system on a submanifold on which the system is integrable. They are in involution on this submanifold and determine the foliation of the submanifold. That means that the Hamiltonian vector fields corresponding to these functions are tangent to the invariant tori foliating the submanifold of integrability. In addition, on some weaker assumptions (e.g. we do not postulate the existence of any Hamiltonian system at all), we have proved the following. Consider $k$ circular functions that correspond to the foliation of the submanifold into isotropic tori. The foliation is determined by $k$ functions in involution. Then, these $k$ circular functions are determined by the $k$ functions restricted to the submanifold.
Citation: Nekhoroshev N. N., Generalizations of Gordon Theorem, Regular and Chaotic Dynamics, 2002, vol. 7, no. 3, pp. 239-247
DOI:10.1070/RD2002v007n03ABEH000207
Fedotov A. V.,  Furta S. D.
On Stability of Motion of a Chain of $n$ Driven Bodies
Abstract
The article deals with the problem of stability of rectilinear motions of multi-link chains of rigid bodies moving in a resisting medium. The authors propose a simple mechanical model that allows them to give a reasonable explanation of instability phenomena observed in reality. The main idea of the article is based on the assumption that lateral forces acting upon lengthy bodies are large enough. This assumption results in the consideration of a nonintegrable constraint as a limiting case. It is well-known that in studies of nonintegrable constraints two different approaches are distinguished. The classical one leads to a nonholonomic constraint, while the other one developed recently by V.V. Kozlov leads to a vakonomic constraint. The authors have shown that instability is typical (in some sense) in both cases though the phase portraits are completely different from the topological point of view.
Citation: Fedotov A. V.,  Furta S. D., On Stability of Motion of a Chain of $n$ Driven Bodies, Regular and Chaotic Dynamics, 2002, vol. 7, no. 3, pp. 249-268
DOI:10.1070/RD2002v007n03ABEH000208
Byachkov A. B.,  Suslonov V. M.
Maggi's Equations in terms of Quasi-Coordinates
Abstract
The article is devoted to the developing of methods of nonholonomic mechanics suitable for solving problems of multibody dynamics. A new form of Maggi's equations, Maggi's equations in terms of quasi-coordinates, is represented. The application of this form of equations of motion to some problems of holonomic and nonholonomic multibody dynamics is discussed.
Citation: Byachkov A. B.,  Suslonov V. M., Maggi's Equations in terms of Quasi-Coordinates, Regular and Chaotic Dynamics, 2002, vol. 7, no. 3, pp. 269-279
DOI:10.1070/RD2002v007n03ABEH000209
Morozov A. D.
On Hypercomplex Dynamics
Abstract
Square mappings in a three-dimensional hyperspace are considered. The properties of the closure of a set of repelling points are studied. In some sense, this closure is analogous to a Julia set. Computer-aided visualizations of this set are given.
Citation: Morozov A. D., On Hypercomplex Dynamics, Regular and Chaotic Dynamics, 2002, vol. 7, no. 3, pp. 281-289
DOI:10.1070/RD2002v007n03ABEH000210
Ramodanov S. M.
Motion of a Circular Cylinder and $N$ Point Vortices in a Perfect Fluid
Abstract
The motion of a rigid circular cylinder and $N$ point vortices in an unbounded volume of perfect fluid is treated here on the basis of a potential framework. The formulas for the hydrodynamic force and moment acting upon a cylinder of arbitrary cross section are obtained. The equations governing the motion of a circular cylinder interacting with vortices are derived. For the greater part this paper coincides with [4] in which, however, only the case of one vortex was treated. It so happened that (due to a makeup man`s fault maybe) [4] was printed with no pictures in it. In this paper we reproduce those gures and extend the previous results.
Citation: Ramodanov S. M., Motion of a Circular Cylinder and $N$ Point Vortices in a Perfect Fluid, Regular and Chaotic Dynamics, 2002, vol. 7, no. 3, pp. 291-298
DOI:10.1070/RD2002v007n03ABEH000211
Lakhno V. D.,  Fialko N. S.
Long-Range Charge Transfer in DNA
Abstract
The paper develops the general theory of excitation transfer dynamics in multi-site systems. The approach we used is based on a quantum-mechanical description and is used for calculating the dynamics of charge transfer in DNA over a long distance. A typical pattern of transfer, involving the possibilities of superexchange and hopping transfer, as well as transfer through excitations of polaron and soliton types, has been obtained from a large number of numerical experiments. The model under examination describes (in some particular cases) the dynamics of neural networks, the population dynamics in mathematical ecology, and evolution of wave packets in non-linear optics. This model can be used in various allied fields.
Citation: Lakhno V. D.,  Fialko N. S., Long-Range Charge Transfer in DNA, Regular and Chaotic Dynamics, 2002, vol. 7, no. 3, pp. 299-313
DOI:10.1070/RD2002v007n03ABEH000212
Zubelevich O. E.
On Exponentially Small Effects in Dynamical Systems with a Small Parameter
Abstract
In the present paper we obtain a theorem which enables us to treat different exponentially small effects of dynamics from a unified point of view. As an example, we discuss the problem of fast phase averaging in multi-frequency systems with slow variable belonging to Banach space.
Citation: Zubelevich O. E., On Exponentially Small Effects in Dynamical Systems with a Small Parameter, Regular and Chaotic Dynamics, 2002, vol. 7, no. 3, pp. 315-324
DOI:10.1070/RD2002v007n03ABEH000213
Kuznetsov S. P.,  Osbaldestin A. H.
Generalized Dimensions of Feigenbaum's Attractor from Renormalization-Group Functional Equations
Abstract
A method is suggested for the computation of the generalized dimensions of fractal attractors at the period-doubling transition to chaos. The approach is based on an eigenvalue problem formulated in terms of functional equations with coefficients expressed in terms of Feigenbaum's universal fixed-point function. The accuracy of the results depends only on the accuracy of the representation of the universal function.
Citation: Kuznetsov S. P.,  Osbaldestin A. H., Generalized Dimensions of Feigenbaum's Attractor from Renormalization-Group Functional Equations, Regular and Chaotic Dynamics, 2002, vol. 7, no. 3, pp. 325-330
DOI:10.1070/RD2002v007n03ABEH000214
Tsiganov A. V.
On Integrable Deformation of the Poincaré System
Abstract
An integrable deformation of the Poincaré system is considered. This system is derived by using the general $r$-matrix theory.
Citation: Tsiganov A. V., On Integrable Deformation of the Poincaré System, Regular and Chaotic Dynamics, 2002, vol. 7, no. 3, pp. 331-336
DOI:10.1070/RD2002v007n03ABEH000215
Sadetov S. T.
On the Regular Reduction of the $n$-Dimensional Problem of $N+1$ Bodies to Euler–Poincaré Equations on the Lie Algebra $sp(2N)$
Abstract
After the lowering of the order up to the location of the center of inertia the reduction of this paper is performed by the passage to the complete set of invariants of the action of linear group of rotations and reflections on the phase space $\mathrm{T}^{*}\mathbb{R}^n \otimes \mathbb{R}^N$. In distinction to known reductions, this reduction is homeomorphic. For $N + 1 = 3,\, n=3,2$ the orbits of the coadjoint representation of the group $Sp(4)$, on which real motions take place, have the homotopy type of projective space $\mathbb{R}P^3$, sphere $S^2$, homogeneous space $(S^2\times S^1)/\mathbb{Z}^2$.
Citation: Sadetov S. T., On the Regular Reduction of the $n$-Dimensional Problem of $N+1$ Bodies to Euler–Poincaré Equations on the Lie Algebra $sp(2N)$, Regular and Chaotic Dynamics, 2002, vol. 7, no. 3, pp. 337-350
DOI:10.1070/RD2002v007n03ABEH000216

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