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Vol. 3, No. 4

Vol. 3, No. 4, 2007

Gudimenko A. I.
Abstract
Dynamics of perturbed stable equilateral and collinear configurations of three point vortices in an incompressible ideal fluid is studied. The asymptotics of the perturbed motion to the unperturbed one is obtained. It is shown that in the first approximation in a appropriate coordinate system the vortices rotate about their undisturbed positions in elliptical orbits. The velocity of the rotation is calculated. It is shown that the eccentricities of the orbits are coincide. The ratio of major axes of any two orbits is calculated. In case of equilateral configuration this ratio is equal to the ratio of inverse intensities of the corresponding vortices. The angle between major axes of any two orbits of the vortices is calculated. In case of equilateral configuration this angle is ±120 degrees.
Keywords: point vortices, integrable dynamics, perturbation theory
Citation: Gudimenko A. I., Dynamics of perturbed equilateral and collinear configurations of three point vortices, Rus. J. Nonlin. Dyn., 2007, Vol. 3, No. 4, pp. 379-391
DOI:10.20537/nd0704001
Nagaev P. A.
Abstract
We investigate the problem of motion of two identical particles on the unit segment. Particularly it was proved using the Poincare’ method that in case of motion in any potential field one can find no additional first integrals except the full energy. We also found some conditions on the type potential under which the two-particles system is stable as the first approximation.
Keywords: nonintegrability, motion equations, billiard
Citation: Nagaev P. A., On the chaotization of one-dimensional gas consisting of interactive particles, Rus. J. Nonlin. Dyn., 2007, Vol. 3, No. 4, pp. 393-399
DOI:10.20537/nd0704002
Shaimuratov R. F.
Abstract
For sets with finite measure, the frequency of recurrence is shown to be positive. In the event of infinite measure, an example with zero frequency of recurrence is presented. The paper provides a sufficient condition under which the frequency of recurrence is not separated from zero.
Keywords: ergodic theory, Poincare' recurrence theorem, frequency of recurrence
Citation: Shaimuratov R. F., On the frequency of recurrence, Rus. J. Nonlin. Dyn., 2007, Vol. 3, No. 4, pp. 401-410
DOI:10.20537/nd0704003
Borisov A. V.,  Mamaev I. S.,  Ramodanov S. M.
Abstract
The paper deals with the derivation of the equations of motion for two spheres in an unbounded volume of ideal and incompressible fluid in 3D Euclidean space. Reduction of order, based on the use of new variables that form a Lie algebra, is offered. A trivial case of integrability is indicated.
Keywords: motion of two spheres, ideal fluid, reduction, integrability
Citation: Borisov A. V.,  Mamaev I. S.,  Ramodanov S. M., Motion of two spheres in ideal fluid. I. Equations o motions in the Euclidean space. First integrals and reduction, Rus. J. Nonlin. Dyn., 2007, Vol. 3, No. 4, pp. 411-422
DOI:10.20537/nd0704004
Gonchenko S. V.,  Gonchenko A. S.
Abstract
We consider the problem of classification of Smale horseshoes from point of view of the local topological conjugacy of two-dimensionalmaps which generate the horseshoes.We show that there are 10 different types of linear horseshoes. As it was established in the recent paper [4], there are infinitelymany different types of nonlinear horseshoes. All of them belong to the class of the so-called half-orientable horseshoes and can be realized for endomorphisms (not one-to-one maps) of disk or for diffeomorphisms of non-orientable two-dimensional manifolds. We give also a short review of related results from [4].
Keywords: Smale horseshoe, local topological conjugacy, hyperbolic set, standard and generalized Henon maps
Citation: Gonchenko S. V.,  Gonchenko A. S., Towards a classification of linear and nonlinear Smale horseshoes, Rus. J. Nonlin. Dyn., 2007, Vol. 3, No. 4, pp. 423-443
DOI:10.20537/nd0704005
Smale S.
Abstract
Citation: Smale S., A Structurally Stable Differentiable Homeomorphism with an Infinite Number of Periodic Points, Rus. J. Nonlin. Dyn., 2007, Vol. 3, No. 4, pp. 445-446
DOI:10.20537/nd0704006

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