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Volume 15, Number 4-5

Volume 15, Number 4-5, 2010
On the 60th birthday of professor V.V. Kozlov

Valery Vasilievich Kozlov. On His 60th Birthday
Abstract
Citation: Valery Vasilievich Kozlov. On His 60th Birthday , Regular and Chaotic Dynamics, 2010, vol. 15, no. 4-5, pp. 419-424
DOI:10.1134/S1560354710040015
Arnold V. I.
Are quadratic residues random?
Abstract
Keywords: arithmetical dynamics, quadratic residue, randomness
Citation: Arnold V. I., Are quadratic residues random?, Regular and Chaotic Dynamics, 2010, vol. 15, no. 4-5, pp. 425-430
DOI:10.1134/S1560354710040027
Bogoyavlenskij O. I.,  Reynolds A. P.
Criteria for existence of a Hamiltonian structure
Abstract
The necessary and sufficient conditions are derived for the existence of a Hamiltonian structure for 3-component non-diagonalizable systems of hydrodynamic type. The conditions are formulated in terms of tensor invariants defined by the metric $h_{ij}(u)$ constructed from the Haantjes (1,2)-tensor.
Keywords: Poisson brackets, conformally flat metric, covariant derivatives, Weyl–Schouten equations, Haantjes tensor
Citation: Bogoyavlenskij O. I.,  Reynolds A. P., Criteria for existence of a Hamiltonian structure, Regular and Chaotic Dynamics, 2010, vol. 15, no. 4-5, pp. 431-439
DOI:10.1134/S1560354710040039
Borisov A. V.,  Mamaev I. S.,  Ramodanov S. M.
Coupled motion of a rigid body and point vortices on a two-dimensional spherical surface
Abstract
The paper is concerned with a class of problems which involves the dynamical interaction of a rigid body with point vortices on the surface of a two-dimensional sphere. The general approach to the 2D hydrodynamics is further developed. The problem of motion of a dynamically symmetric circular body interacting with a single vortex is shown to be integrable. Mass vortices on $S^2$ are introduced and the related issues (such as equations of motion, integrability, partial solutions, etc.) are discussed. This paper is a natural progression of the author’s previous research on interaction of rigid bodies and point vortices in a plane.
Keywords: hydrodynamics on a sphere, coupled body-vortex system, mass vortex, equations of motion, integrability
Citation: Borisov A. V.,  Mamaev I. S.,  Ramodanov S. M., Coupled motion of a rigid body and point vortices on a two-dimensional spherical surface, Regular and Chaotic Dynamics, 2010, vol. 15, no. 4-5, pp. 440-461
DOI:10.1134/S1560354710040040
Gonchenko S. V.,  Gonchenko V. S.,  Shilnikov L. P.
On a homoclinic origin of Hénon-like maps
Abstract
We review bifurcations of homoclinic tangencies leading to Hénon-like maps of various kinds.
Keywords: homoclinic tangency, Hénon-like maps, saddle-focus fixed point, wild-hyperbolic attractor
Citation: Gonchenko S. V.,  Gonchenko V. S.,  Shilnikov L. P., On a homoclinic origin of Hénon-like maps, Regular and Chaotic Dynamics, 2010, vol. 15, no. 4-5, pp. 462-481
DOI:10.1134/S1560354710040052
Gutkin E.
Geometry, topology and dynamics of geodesic flows on noncompact polygonal surfaces
Abstract
We establish the background for the study of geodesics on noncompact polygonal surfaces. For illustration, we study the recurrence of geodesics on $\mathbb{Z}$-periodic polygonal surfaces. We prove, in particular, that almost all geodesics on a topologically typical $\mathbb{Z}$-periodic surface with a boundary are recurrent.
Keywords: (periodic) polygonal surface, geodesic, skew product, cross-section, displacement function, recurrence, transience, ergodicity
Citation: Gutkin E., Geometry, topology and dynamics of geodesic flows on noncompact polygonal surfaces, Regular and Chaotic Dynamics, 2010, vol. 15, no. 4-5, pp. 482-503
DOI:10.1134/S1560354710040064
Khesin B. A.,  Tabachnikov S.
Contact complete integrability
Abstract
Complete integrability in a symplectic setting means the existence of a Lagrangian foliation leaf-wise preserved by the dynamics. In the paper we describe complete integrability in a contact set-up as a more subtle structure: a flag of two foliations, Legendrian and co-Legendrian, and a holonomy-invariant transverse measure of the former in the latter. This turns out to be equivalent to the existence of a canonical $\mathbb{R} \times \mathbb{R}^{n−1}$ structure on the leaves of the co-Legendrian foliation. Further, the above structure implies the existence of n commuting contact fields preserving a special contact 1-form, thus providing the geometric framework and establishing equivalence with previously known definitions of contact integrability. We also show that contact completely integrable systems are solvable in quadratures.
We present an example of contact complete integrability: the billiard system inside an ellipsoid in pseudo-Euclidean space, restricted to the space of oriented null geodesics. We describe a surprising acceleration mechanism for closed light-like billiard trajectories.
Keywords: complete integrability, contact structure, Legendrian foliation, pseudo-Euclidean geometry, billiard map
Citation: Khesin B. A.,  Tabachnikov S., Contact complete integrability, Regular and Chaotic Dynamics, 2010, vol. 15, no. 4-5, pp. 504-520
DOI:10.1134/S1560354710040076
Li D.,  Sinai Y. G.
Blowups of complex-valued solutions for some hydrodynamic models
Abstract
We study complex-valued blowups of solutions for several hydrodynamic models. For complex-valued initial conditions, smooth local solutions can have finite-time singularities since the energy inequality does not hold. By using some version of the renormalization group method, we derive the equations for corresponding fixed points and analyze the spectrum of the linearized operator. We describe the open set of initial conditions for which blowups at finite time can occur.
Keywords: blowup, renormalization group method
Citation: Li D.,  Sinai Y. G., Blowups of complex-valued solutions for some hydrodynamic models, Regular and Chaotic Dynamics, 2010, vol. 15, no. 4-5, pp. 521-531
DOI:10.1134/S1560354710040088
Mari Beffa  G.,  Olver P. J.
Poisson structures for geometric curve flows in semi-simple homogeneous spaces
Abstract
We apply the equivariant method of moving frames to investigate the existence of Poisson structures for geometric curve flows in semi-simple homogeneous spaces. We derive explicit compatibility conditions that ensure that a geometric flow induces a Hamiltonian evolution of the associated differential invariants. Our results are illustrated by several examples of geometric interest.
Keywords: moving frame, Poisson structure, homogeneous space, invariant curve flow, differential invariant, invariant variational bicomplex
Citation: Mari Beffa  G.,  Olver P. J., Poisson structures for geometric curve flows in semi-simple homogeneous spaces, Regular and Chaotic Dynamics, 2010, vol. 15, no. 4-5, pp. 532-550
DOI:10.1134/S156035471004009X
Maciejewski A. J.,  Przybylska M.
Partial integrability of Hamiltonian systems with homogeneous potential
Abstract
In this paper we consider systems with n degrees of freedom given by the natural Hamiltonian function of the form
$H = \frac{1}{2} {\bf p}^T {\bf Mp} + V({\bf q})$,
where ${\bf q} = (q_1, \ldots, q_n) \in \mathbb{C}^n$, ${\bf p}= (p_1, \ldots, p_n) \in \mathbb{C}^n$, are the canonical coordinates and momenta, $\bf M$ is a symmetric non-singular matrix, and $V({\bf q})$ is a homogeneous function of degree $k \in Z^*$. We assume that the system admits $1 \leqslant m < n$ independent and commuting first integrals $F_1, \ldots F_m$. Our main results give easily computable and effective necessary conditions for the existence of one more additional first integral $F_{m+1}$ such that all integrals $F_1, \ldots F_{m+1}$ are independent and pairwise commute. These conditions are derived from an analysis of the differential Galois group of variational equations along a particular solution of the system. We apply our result analysing the partial integrability of a certain $n$ body problem on a line and the planar three body problem.
Keywords: integrability, non-integrability criteria, monodromy group, differential Galois group, hypergeometric equation, Hamiltonian equations
Citation: Maciejewski A. J.,  Przybylska M., Partial integrability of Hamiltonian systems with homogeneous potential, Regular and Chaotic Dynamics, 2010, vol. 15, no. 4-5, pp. 551-563
DOI:10.1134/S1560354710040106
Neishtadt A. I.,  Artemyev A. V.,  Zelenyi L. M.
Regular and chaotic charged particle dynamics in low frequency waves and role of separatrix crossings
Abstract
We consider interaction of charged particles with an electromagnetic (electrostatic) low frequency wave propagating perpendicular to a uniform background magnetic field. The effects of particle trapping by the wave and further acceleration of a surfatron type are discussed in details. Method for this analysis based on the adiabatic theory of separatrix crossing is used. It is shown that particle can unlimitedly accelerate in the trapping in electromagnetic waves and energy of particle does not increase for the system with electrostatic wave.
Keywords: surfatron acceleration, separatrix crossings, adiabatic invariant
Citation: Neishtadt A. I.,  Artemyev A. V.,  Zelenyi L. M., Regular and chaotic charged particle dynamics in low frequency waves and role of separatrix crossings, Regular and Chaotic Dynamics, 2010, vol. 15, no. 4-5, pp. 564-574
DOI:10.1134/S1560354710040118
Piftankin G. N.,  Treschev D. V.
Coarse-grained entropy in dynamical systems
Abstract
Let $M$ be the phase space of a physical system. Consider the dynamics, determined by the invertible map $T: M \to M$, preserving the measure $\mu$ on $M$. Let $\nu$ be another measure on $M$, $d\nu = \rho d\mu$. Gibbs introduced the quantity $s(\rho) = − \int \rho \log \rho d \mu$ as an analog of the thermodynamical entropy. We consider a modification of the Gibbs (fine-grained) entropy the so called coarse-grained entropy.
First we obtain a formula for the difference between the coarse-grained and Gibbs entropy. The main term of the difference is expressed by a functional usually referenced to as the Fisher information.
Then we consider the behavior of the coarse-grained entropy as a function of time. The dynamics transforms $\nu$ in the following way: $\nu \mapsto \nu_n$, $d\nu_n = \rho \circ T^{-n} d\mu$. Hence, we obtain the sequence of densities $\rho_n = \rho \circ T^{-n}$ and the corresponding values of the Gibbs and the coarse-grained entropy. We show that while the Gibbs entropy remains constant, the coarse-grained entropy has a tendency to a growth and this growth is determined by dynamical properties of the map $T$. Finally, we give numerical calculation of the coarse-grained entropy as a function of time for systems with various dynamical properties: integrable, chaotic and with mixed dynamics and compare these calculation with theoretical statements.
Keywords: Gibbs entropy, nonequilibrium thermodynamics, Lyapunov exponents, Gibbs ensemble
Citation: Piftankin G. N.,  Treschev D. V., Coarse-grained entropy in dynamical systems, Regular and Chaotic Dynamics, 2010, vol. 15, no. 4-5, pp. 575-597
DOI:10.1134/S156035471004012X
Taimanov I. A.
Periodic magnetic geodesics on almost every energy level via variational methods
Abstract
For strong exact magnetic fields the action functional (i.e., the length plus the linear magnetic term) is not bounded from below on the space of closed contractible curves and the lower estimates for critical levels are derived by using the principle of throwing out cycles. It is proved that for almost every energy level the principle of throwing out cycles gives periodic magnetic geodesics on the critical levels defined by the "thrown out" cycles.
Keywords: magnetic geodesics, closed extremals, calculus of variations in the large
Citation: Taimanov I. A., Periodic magnetic geodesics on almost every energy level via variational methods, Regular and Chaotic Dynamics, 2010, vol. 15, no. 4-5, pp. 598-605
DOI:10.1134/S1560354710040131
Vankerschaver J.,  Kanso E.,  Marsden J. E.
The dynamics of a rigid body in potential flow with circulation
Abstract
We consider the motion of a two-dimensional body of arbitrary shape in a planar irrotational, incompressible fluid with a given amount of circulation around the body. We derive the equations of motion for this system by performing symplectic reduction with respect to the group of volume-preserving diffeomorphisms and obtain the relevant Poisson structures after a further Poisson reduction with respect to the group of translations and rotations. In this way, we recover the equations of motion given for this system by Chaplygin and Lamb, and we give a geometric interpretation for the Kutta–Zhukowski force as a curvature-related effect. In addition, we show that the motion of a rigid body with circulation can be understood as a geodesic flow on a central extension of the special Euclidian group $SE(2)$, and we relate the cocycle in the description of this central extension to a certain curvature tensor.
Keywords: fluid-structure interactions, potential flow, circulation, symplectic reduction, diffeomorphism groups, oscillator group
Citation: Vankerschaver J.,  Kanso E.,  Marsden J. E., The dynamics of a rigid body in potential flow with circulation, Regular and Chaotic Dynamics, 2010, vol. 15, no. 4-5, pp. 606-629
DOI:10.1134/S1560354710040143
Ziglin S. L.
On the absence of an additional meromorphic first integral in the Riemann problem on the motion of a homogeneous liquid ellipsoid
Abstract
We prove the absence of an additional meromorphic first integral in the Riemann problem on the motion of a homogeneous liquid ellipsoid with zero angular and vortex momenta in the case of zero self-gravitation.
Keywords: Riemann problem, liquid ellipsoid, meromorphic first
Citation: Ziglin S. L., On the absence of an additional meromorphic first integral in the Riemann problem on the motion of a homogeneous liquid ellipsoid, Regular and Chaotic Dynamics, 2010, vol. 15, no. 4-5, pp. 630-633
DOI:10.1134/S1560354710040155

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