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Vol. 3, No. 1

Vol. 3, No. 1, 2007

Gledzer A. E.
Abstract
The oscillatory types of motion in frame of 6-modes model of convection are considered into nearly axially symmetrical ellipsoidal volume. The pseudo-Prandtl, Rayleigh numbers and vertical aspect ratio are changed in large ranges. The regimes with different types of vorticity oscillations are determined, that are connected with the turning of the fluid rotation axis or with the alteration of rotation. For some of the regimes the analytical solution and formulae of oscillation periods were obtained.
Keywords: Rayleigh-Benard convection, low-order models, linear space, thermal plumes
Citation: Gledzer A. E., The convection regimes in a low-order model of fluid motion in nearly axially symmetrical ellipsoidal cavity, Rus. J. Nonlin. Dyn., 2007, Vol. 3, No. 1, pp. 3-31
DOI:10.20537/nd0701001
Gudimenko A. I.
Abstract
A Hamiltonian dynamical system describing a rotating incompressible two-dimensional flow, disturbed by an oscillating point vortex, is studied numerically and analytically. It is shown numerically that under perturbation the region of strongly mixed trajectories of the system forms. As the amplitude of perturbation increases, the region grows in size due to the destruction and absorption of the nearest resonances. The order and multiplicity of the resonances are determined mainly by the relation ω/Ω, where ω is the perturbation frequency and Ω is the rotation frequency of the flow. The patterns of the resonances differ essentially whether this quantity is integer or fractional. The results of the numerical experiment are justified analytically. In the domain that is sufficiently far from the vortex, the Hamiltonian is represented in the angle-action variables. Based on the representation, the arrangement of the resonances on the phase plane is analyzed. In particular, a classification of the resonances, which is adequate to the numerical patterns, is proposed. The widths of the resonances are calculated. It is shown that, at large distances from the vortex, global chaotization of trajectories of the system is impossible.
Keywords: point vortex, chaotic dynamics, nonlinear resonances
Citation: Gudimenko A. I., Chaos and resonances in a rotating flow disturbed by а periodic motion of a point vortex, Rus. J. Nonlin. Dyn., 2007, Vol. 3, No. 1, pp. 33-48
DOI:10.20537/nd0701002
Borisov A. V.,  Mamaev I. S.
Abstract
We consider trajectory isomorphisms between various integrable systems on an $n$-dimensional sphere $S^n$ and a Euclidean space $R^n$. Some of the systems are classical integrable problems of Celestial Mechanics in plane and curved spaces. All the systems under consideration have an additional first integral quadratic in momentum and can be integrated analytically by using the separation of variables. We show that some integrable problems in constant curvature spaces are not essentially new from the viewpoint of the theory of integration, and they can be analyzed using known results of classical Celestial Mechanics.
Keywords: integrable systems, two-center problem, isomorphisms
Citation: Borisov A. V.,  Mamaev I. S., On isomorphisms of some integrable systems on a plane and a sphere, Rus. J. Nonlin. Dyn., 2007, Vol. 3, No. 1, pp. 49-56
DOI:10.20537/nd0701003
Bardin B. S.
Abstract
We deal with an autonomous Hamiltonian system with two degrees of freedom. We assume that the Hamiltonian function is analytic in a neighborhood of the phase space origin which is an equilibrium point. We consider the case when two imaginary eigenvalues of the matrix of the linearized system are in the ratio 3:1. We study nonlinear conditionally-periodic motions of the system in the vicinity of the equilibrium point. Omitting the terms of order higher then five in the normalized Hamiltonian we analyze the so-called truncated system in detail. We show that its general solution can be given in terms of elliptic integrals and elliptic functions. The motions of truncated system are either periodic, or asymptotic to a periodic one, or conditionally-periodic. By using the KAM theory methods we show that the most of conditionally-periodic trajectories of the truncated systems persist also in the full system. Moreover, the trajectories that became not conditionally-periodic in the full system belong to a subset of exponentially small measure. The results of the study are applied for the analysis of nonlinear motions of a symmetric satellite in a neighborhood of its cylindric precession.
Keywords: Hamiltonian system, periodic orbits, normal form, resonance, action-angel variables, KAM theory
Citation: Bardin B. S., On nonlinear oscillations of Hamiltonian system in case of fourth order resonance, Rus. J. Nonlin. Dyn., 2007, Vol. 3, No. 1, pp. 57-74
DOI:10.20537/nd0701004
Zotev D. B.
Abstract
Consider a Hamiltonian system, restricted onto an invariant surface. Does it have an integral, which may be explicitly expressed through the equations, determining this submanifold? A simple criterion of the existence of partial integral, equal to their Poisson matrix determinant, has been found. This integral is not trivial iff the induced Poisson structure is nondegenerate at least at one point. Particularly, the submanifold is to be even-dimensional.
Keywords: Hamiltonian system, partial integral, invariant submanifold
Citation: Zotev D. B., A criterion for a Poisson matrix determinant to be a partial integral of the Hamiltonian system, Rus. J. Nonlin. Dyn., 2007, Vol. 3, No. 1, pp. 75-80
DOI:10.20537/nd0701005
Zermelo E.
Abstract
Citation: Zermelo E., Hydrodynamische Untersuchungen uber die Wirbelbewegungen in einer Kugelflache, Rus. J. Nonlin. Dyn., 2007, Vol. 3, No. 1, pp. 81-110
DOI:10.20537/nd0701006

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