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Volume 13, Number 1

Volume 13, Number 1, 2008

O'Neil K. A.
Equilibrium Configurations of Point Vortices on a Sphere
Abstract
Configurations of point vortices on the sphere are considered in which all vortex velocities are zero. A sharp upper bound for the number of equilibria lying on a great circle is found, valid for generic circulations, and some unusual equilibrium configurations with a free real parameter are described. Equilibria of rings (vortices evenly spaced along circles of latitude) are also discussed. All equilibrium configurations of four vortices are determined.
Keywords: point vortices, sphere, equilibrium
Citation: O'Neil K. A., Equilibrium Configurations of Point Vortices on a Sphere, Regular and Chaotic Dynamics, 2008, vol. 13, no. 1, pp. 1-8
DOI:10.1134/S1560354708010012
Kuznetsov A. P.,  Sataev I. R.,  Sedova Y. V.
Dynamics of Coupled Non-Identical Systems with Period-Doubling Cascade
Abstract
We discuss the structure of bifurcation diagram in the plane of parameters controlling period-doublings for the system of coupled logistic maps. The analysis is carried out by computing the charts of dynamical regimes and charts of Lyapunov exponents giving showy and effective illustrations. The critical point of codimension two at the border of chaos is found. It is a terminal point for the Feigenbaum critical line. The bifurcation analysis in the vicinity of this point is presented.
Keywords: criticality, universality, transition to chaos, coupled maps, bifurcation, terminal point
Citation: Kuznetsov A. P.,  Sataev I. R.,  Sedova Y. V., Dynamics of Coupled Non-Identical Systems with Period-Doubling Cascade, Regular and Chaotic Dynamics, 2008, vol. 13, no. 1, pp. 9-18
DOI:10.1134/S1560354708010024
Dittrich J.,  Inozemtsev V. I.
The Commutativity of Integrals of Motion for Quantum Spin Chains and Elliptic Functions Identities
Abstract
We prove the commutativity of the first two nontrivial integrals of motion for quantum spin chains with elliptic form of the exchange interaction. We also show their liner independence for the number of spins larger than 4. As a byproduct, we obtained several identities between elliptic Weierstrass functions of three and four arguments.
Keywords: quantum elliptic spin chains, transpositions, integrability
Citation: Dittrich J.,  Inozemtsev V. I., The Commutativity of Integrals of Motion for Quantum Spin Chains and Elliptic Functions Identities, Regular and Chaotic Dynamics, 2008, vol. 13, no. 1, pp. 19-26
DOI:10.1134/S1560354708010036
Morozov A. D.
On Degenerate Resonances and "Vortex Pairs"
Abstract
Hamiltonian systems with 3/2 degrees of freedom close to non-linear autonomous are studied. For unperturbed equations with a nonlinearity in the form of a polynomial of the fourth or fifth degree, their coefficients are specified for which the period on closed phase curves is not a monotone function of the energy and has extreme values of the maximal order. When the perturbation is periodic in time, this non-monotonicity leads to the existence of degenerate resonances. The numerical study of the Poincaré map was carried out and bifurcations related to the formation of the vortex pairs within the resonance zones were found. For systems of a general form at arbitrarily small perturbations the absence of vortex pairs is proved. An explanation of the appearance of these structures for the Poincaré map is presented.
Keywords: Resonances, degenerate resonances, Hamiltonian systems, averaged systems, separatrix, vortex pairs, Poincaré map
Citation: Morozov A. D., On Degenerate Resonances and "Vortex Pairs", Regular and Chaotic Dynamics, 2008, vol. 13, no. 1, pp. 27-36
DOI:10.1134/S1560354708010048
Kostko A. L.,  Tsiganov A. V.
On the Bi-Hamiltonian Structures for the Goryachev–Chaplygin Top
Abstract
The bi-hamiltonian structures for the Goryachev–Chaplygin top are constructed by using the Chaplygin variables and the Sklyanin bracket.
Keywords: integrable and bi-hamiltonian system, the Goryachev–Chaplygin top
Citation: Kostko A. L.,  Tsiganov A. V., On the Bi-Hamiltonian Structures for the Goryachev–Chaplygin Top, Regular and Chaotic Dynamics, 2008, vol. 13, no. 1, pp. 37-44
DOI:10.1134/S156035470801005X
Cordani B.
Arnold’s Web and Diffusion in the Stark–Quadratic–Zeeman Problem
Abstract
The Arnold web and the Arnold diffusion arise when an integrable Hamiltonian system is slightly perturbed: the first concerns the peculiar topology characterizing the set of the resonance lines in phase space, the latter the extremaly slow motion (if any) along these lines. While Arnold has proved the possibility of diffusion, it is still unknown if the phenomenon is generic in realistic physical systems. The system we consider is the Hydrogen atom (or Kepler problem) subject to the combined action of a constant electric and magnetic field, which is known as Stark–Zeeman problem. We describe the results of numerical experiments: the Arnold web is clearly highlighted and, looking at the behaviour of the KAM frequencies on orbits of 108 revolutions, evidence for the diffusion existence is reached.
Keywords: Arnold’s diffusion, Arnold’s web, Perturbation theory, Stark–Quadratic–Zeeman problem
Citation: Cordani B., Arnold’s Web and Diffusion in the Stark–Quadratic–Zeeman Problem, Regular and Chaotic Dynamics, 2008, vol. 13, no. 1, pp. 45-55
DOI:10.1134/S1560354708010061
Yehia H. M.,  Elmandouh A. A.
New Integrable Systems with a Quartic Integral and New Generalizations of Kovalevskaya’s and Goriatchev’s Cases
Abstract
In his paper [1], one of us has introduced a method for constructing integrable conservative two-dimensional mechanical systems, on Riemannian 2D spaces, whose second integral is a polynomial in the velocities. This method was applied successfully in [2] to construction of systems admitting a cubic integral and in [3, 4] and [5] to cases of a quartic integral. The present work is devoted to construction of new integrable systems with a quartic integral. The potential is assumed to have the structure
$V = u(y) + v(y)(a \cos x + b \sin x) + w(y)(c \cos 2x + d \sin 2x)$.
This is inspired by the structure of potential in the famous generalization of Kovalevskaya’s case in rigid body dynamics introduced by Goriatchev. The resulting differential equations were completely solved only for time reversible systems. A 10-parameter family of systems of the searched type is obtained. Four parameters determine the structure of the line element of the configuration manifold and the others contribute only to the potential function. In the case of time-irreversible systems the governing equations were solved in the three cases when the metric is identical to that of reduced rigid body motion. Those lead to three new several-parameter generalizations of known cases, including the classical cases of Kovalevskaya, Chaplygin and Goriatchev.
Keywords: integrability, quartic integral
Citation: Yehia H. M.,  Elmandouh A. A., New Integrable Systems with a Quartic Integral and New Generalizations of Kovalevskaya’s and Goriatchev’s Cases, Regular and Chaotic Dynamics, 2008, vol. 13, no. 1, pp. 56-69
DOI:10.1134/S1560354708010073

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