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Volume 9, Number 2

Volume 9, Number 2, 2004

Tsiganov A. V.
On the Steklov–Lyapunov case of the rigid body motion
Abstract
We construct a Poisson map between manifolds with linear Poisson brackets corresponding to the two samples of Lie algebra $e(3)$. Using this map we establish equivalence of the Steklov–Lyapunov system and the motion of a particle on the surface of the sphere under the influence of the fourth order potential. To study separation of variables for the Steklov case on the Lie algebra $so(4)$ we use the twisted Poisson map between the bi-Hamiltonian manifolds $e(3)$ and $so(4)$.
Citation: Tsiganov A. V., On the Steklov–Lyapunov case of the rigid body motion, Regular and Chaotic Dynamics, 2004, vol. 9, no. 2, pp. 77-89
DOI:10.1070/RD2004v009n02ABEH000267
Kozlov V. V.
Billiards, invariant measures, and equilibrium thermodynamics. II
Abstract
The kinetics of collisionless continuous medium is studied in a bounded region on a curved manifold. We have assumed that in statistical equilibrium, the probability distribution density depends only on the total energy. It is shown that in this case, all the fundamental relations for a multi-dimensional ideal gas in thermal equilibrium hold true.
Citation: Kozlov V. V., Billiards, invariant measures, and equilibrium thermodynamics. II, Regular and Chaotic Dynamics, 2004, vol. 9, no. 2, pp. 91-100
DOI:10.1070/RD2004v009n02ABEH000268
Borisov A. V.,  Mamaev I. S.,  Kilin A. A.
Absolute and relative choreographies in the problem of point vortices moving on a plane
Abstract
We obtained new periodic solutions in the problems of three and four point vortices moving on a plane. In the case of three vortices, the system is reduced to a Hamiltonian system with one degree of freedom, and it is integrable. In the case of four vortices, the order is reduced to two degrees of freedom, and the system is not integrable. We present relative and absolute choreographies of three and four vortices of the same intensity which are periodic motions of vortices in some rotating and fixed frame of reference, where all the vortices move along the same closed curve. Similar choreographies have been recently obtained by C. Moore, A. Chenciner, and C. Simo for the $n$-body problem in celestial mechanics [6, 7, 17]. Nevertheless, the choreographies that appear in vortex dynamics have a number of distinct features.
Citation: Borisov A. V.,  Mamaev I. S.,  Kilin A. A., Absolute and relative choreographies in the problem of point vortices moving on a plane, Regular and Chaotic Dynamics, 2004, vol. 9, no. 2, pp. 101-111
DOI:10.1070/RD2004v009n02ABEH000269
Cordani B.
Global study of the 2d secular 3-body problem
Abstract
Following the same central idea of Féjoz [9] [10] [8], we study the planar averaged $3$-body problem without making use of series developments, as is usual, but instead we perform a global geometric analysis: the space of the orbits for a fixed energy is reduced under the rotational symmetry to a $2$-dimensional symplectic manifold, where the motion is described by the level curves of the reduced Hamiltonian. The number and location of the critical points are investigated both analytically and numerically, confirming a conjecture of Féjoz.
Citation: Cordani B., Global study of the 2d secular 3-body problem, Regular and Chaotic Dynamics, 2004, vol. 9, no. 2, pp. 113-128
DOI:10.1070/RD2004v009n02ABEH000270
Rozanova O. S.
Note on the typhoon eye trajectory
Abstract
We consider a model of typhoon based on the three-dimensional baroclinic compressible equations of atmosphere dynamics averaged over hight and describe a qualitative behavior of the vortex and possible trajectories of the typhoon eye.
Citation: Rozanova O. S., Note on the typhoon eye trajectory, Regular and Chaotic Dynamics, 2004, vol. 9, no. 2, pp. 129-142
DOI:10.1070/RD2004v009n02ABEH000271
Sajid M.,  Kapoor G. P.
Dynamics of a family of non-critically finite even transcendental meromorphic functions
Abstract
The dynamics of one parameter family of non-critically finite even transcendental meromorphic function $\xi_{\lambda}(z)=\lambda \dfrac{\sinh^{2} z}{z^4}$, $\lambda>0$ is investigated in the present paper. It is shown that bifurcations in the dynamics of the function $\xi_{\lambda}(x)$ for $x\in{\mathbb{R}}\setminus\{0\}$ occur at two critical parameter values $\lambda=\dfrac{x_{1}^{5}}{\sinh^{2} x_{1}}\; (\approx 1.26333)$ and $\lambda=\dfrac{\tilde{x}^5}{\sinh^{2}\tilde{x}}\; (\approx 2.7.715)$, where $x_{1}$ and $\tilde{x}$ are the unique positive real roots of the equations $\tanh x=\dfrac{2x}{3}$ and $\tanh x=\dfrac{2x}{5}$ respectively. For certain ranges of parameter values of $\lambda$, it is proved that the Julia set of the function $\xi_{\lambda}(z)$ contains both real and imaginary axes. The images of the Julia sets of $\xi_{\lambda}(z)$ are computer generated by using the characterization of the Julia set of $\xi_{\lambda}(z)$ as the closure of the set of points whose orbits escape to infinity under iterations. Finally, our results are compared with the recent results on dynamics of (i) critically finite transcendental meromorphic functions $\lambda \tan z$ having polynomial Schwarzian Derivative [10,15,19] and (ii) non-critically finite transcendental entire functions $\lambda \dfrac{e^{z}-1}{z}$ [14].
Citation: Sajid M.,  Kapoor G. P., Dynamics of a family of non-critically finite even transcendental meromorphic functions, Regular and Chaotic Dynamics, 2004, vol. 9, no. 2, pp. 143-162
DOI:10.1070/RD2004v009n02ABEH000272
Khushalani B.
Equilibrium configurations based on Platonic geometries
Abstract
The problem of the stable configurations of N electrons on a sphere minimizing the potential energy of the system is related to the mathematical problem of the extremal configurations in the distance geometry and to the problem of the densest lattice packing of the congruent closed spheres. The arrangement of the points on a sphere in three-space leading to the equilibrium solutions has been of interest since 1904 when J. J. Thomson tried to obtain the stable equilibrium patterns of electrons moving on a sphere and subject to the electrostatic force inversely proportional to the square of the distance between them. Utilizing the theory of the point vortex motion on a sphere, Platonic polyhedral extremal configurations are obtained in this paper using numerical methods.
Citation: Khushalani B., Equilibrium configurations based on Platonic geometries, Regular and Chaotic Dynamics, 2004, vol. 9, no. 2, pp. 163-167
DOI:10.1070/RD2004v009n02ABEH000273
Komarov I. V.,  Tsiganov A. V.
On integration of the Kowalevski gyrostat and the Clebsch problems
Abstract
For the Kowalevski gyrostat change of variables similar to that of the Kowalevski top is done. We establish one to one correspondence between the Kowalevski gyrostat and the Clebsch system and demonstrate that Kowalevski variables for the gyrostat practically coincide with elliptic coordinates on sphere for the Clebsch case. Equivalence of considered integrable systems allows to construct two Lax matrices for the gyrostat using known rational and elliptic Lax matrices for the Clebsch model. Associated with these matrices solutions of the Clebsch system and, therefore, of the Kowalevski gyrostat problem are discussed. The Kötter solution of the Clebsch system in modern notation is presented in detail.
Citation: Komarov I. V.,  Tsiganov A. V., On integration of the Kowalevski gyrostat and the Clebsch problems, Regular and Chaotic Dynamics, 2004, vol. 9, no. 2, pp. 169-187
DOI:10.1070/RD2004v009n02ABEH000274
Khushalani B.
The families of periodic orbits bifurcating from the fixed equilibria in a $48$-dimensional system
Abstract
When inscribed inside a sphere of radius $R$, each of the five Platonic solids with a vortex of strength $\Gamma$ placed at each vertex gives rise to an equilibrium solution of the point vortex equations. In this paper, it will be shown how these equilibria can be used to generate families of periodic orbits on the sphere. These orbits are centered either around these equilibria or around more exotic equilibria, such as staggered ring configurations. Focussing on the cube as a generic case, four distinct families of periodic orbits made up of $24$ vortices (a $48$-dimensional system) are generated. These orbits bifurcate from the cube as each vertex is opened up with a splitting parameter $\theta$. The bifurcation from one orbit family to another is tracked by following the Floquet multipliers around the unit circle as the splitting parameter is varied.
Citation: Khushalani B., The families of periodic orbits bifurcating from the fixed equilibria in a $48$-dimensional system, Regular and Chaotic Dynamics, 2004, vol. 9, no. 2, pp. 189-198
DOI:10.1070/RD2004v009n02ABEH000275

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