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Volume 10, Number 1

Volume 10, Number 1, 2005

Shashikanth B. N.
Poisson brackets for the dynamically interacting system of a 2D rigid cylinder and $N$ point vortices: the case of arbitrary smooth cylinder shapes
Abstract
This paper basically extends the work of Shashikanth, Marsden, Burdick and Kelly [17] by showing that the Hamiltonian (Poisson bracket) structure of the dynamically interacting system of a 2-D rigid circular cylinder and $N$ point vortices, when the vortex strengths sum to zero and the circulation around the cylinder is zero, also holds when the cylinder has arbitrary (smooth) shape. This extension is a consequence of a reciprocity relation, obtainable by an application of a classical Green's formula, that holds for this problem. Moreover, even when the vortex strengths do not sum to zero but with the circulation around the cylinder still zero, it is shown that there is a Poisson bracket for the system which differs from the previous bracket by the inclusion of a 2-cocycle term. Finally, comparisons are made to the works of Borisov, Mamaev and Ramodanov [15], [16], [5], [4].
Keywords: point vortices, rigid body, Hamiltonian, Poisson brackets, reciprocity
Citation: Shashikanth B. N., Poisson brackets for the dynamically interacting system of a 2D rigid cylinder and $N$ point vortices: the case of arbitrary smooth cylinder shapes, Regular and Chaotic Dynamics, 2005, vol. 10, no. 1, pp. 1-14
DOI:10.1070/RD2005v010n01ABEH000295
Kharlamov M. P.,  Zotev D. B.
Non-degenerate energy surfaces of rigid body in two constant fields
Abstract
The problem of motion of a rigid body in two constant fields is considered. The motion is described by the Hamiltonian system with three degrees of freedom. This system in general case does not have any explicit symmetry groups and, therefore, cannot be reduced to a family of systems with two degrees of freedom. The critical points of the energy integral are found. It appeared that the energy of the system is a Morse function and has exactly four distinct critical points with different critical values and Morse indexes 0,1,2,3. In particular, the body has four equilibria, only one of which is stable. Basing on the Morse theory the smooth type of 5-dimensional non-degenerate iso-energetic manifolds is pointed out.
Keywords: rigid body, two constant fields, iso-energetic manifolds
Citation: Kharlamov M. P.,  Zotev D. B., Non-degenerate energy surfaces of rigid body in two constant fields, Regular and Chaotic Dynamics, 2005, vol. 10, no. 1, pp. 15-19
DOI:10.1070/RD2005v010n01ABEH000296
Russmann H.
Addendum to "Invariant tori in non-degenerate nearly integrable Hamiltonian systems", RCD 6(2) 2001
Abstract
For the convenience of the reader of our above mentioned paper we prove here the theorem on the approximation of periodic holomorphic functions by trigonometric polynomials in that paper without any reference to approximation theory, especially to Akhiezer's theorem. See the footnote on p. 136 in RCD 6(2) 2001.
Keywords: good approximation of multi-periodic, analytic functions by trigonometric polynomials in a multi-dimensional strip
Citation: Russmann H., Addendum to "Invariant tori in non-degenerate nearly integrable Hamiltonian systems", RCD 6(2) 2001 , Regular and Chaotic Dynamics, 2005, vol. 10, no. 1, pp. 21-31
DOI: 10.1070/RD2005v010n01ABEH000297
Kuznetsov S. P.
Generalized dimensions of the golden-mean quasiperiodic orbit from renormalization-group functional equation
Abstract
A method is suggested for computation of the generalized dimensions for a fractal attractor associated with the quasiperiodic transition to chaos at the golden-mean rotation number. The approach is based on an eigenvalue problem formulated in terms of functional equations with coeficients expressed via the universal fixed-point function of Feigenbaum–Kadanoff–Shenker. The accuracy of the results is determined only by precision of representation of the universal function.
Keywords: circle map, golden mean, renormalization, dimension, generalized dimensions
Citation: Kuznetsov S. P., Generalized dimensions of the golden-mean quasiperiodic orbit from renormalization-group functional equation, Regular and Chaotic Dynamics, 2005, vol. 10, no. 1, pp. 33-38
DOI:10.1070/RD2005v010n01ABEH000298
Laurent-Polz F.
Point vortices on a rotating sphere
Abstract
We study the dynamics of $N$ point vortices on a rotating sphere. The Hamiltonian system becomes infinite dimensional due to the non-uniform background vorticity coming from the Coriolis force. We prove that a relative equilibrium formed of latitudinal rings of identical vortices for the non-rotating sphere persists to be a relative equilibrium when the sphere rotates. We study the nonlinear stability of a polygon of planar point vortices on a rotating plane in order to approximate the corresponding relative equilibrium on the rotating sphere when the ring is close to the pole. We then perform the same study for geostrophic vortices. To end, we compare our results to the observations on the southern hemisphere atmospheric circulation.
Keywords: point vortices, rotating sphere, relative equilibria, nonlinear stability, planar vortices, geostrophic vortices, Southern Hemisphere Circulation
Citation: Laurent-Polz F., Point vortices on a rotating sphere , Regular and Chaotic Dynamics, 2005, vol. 10, no. 1, pp. 39-58
DOI: 10.1070/RD2005v010n01ABEH000299
Marikhin V. G.,  Sokolov V. V.
Separation of variables on a non-hyperelliptic curve
Abstract
In the paper we consider several dynamical systems that admit a separation of variables on the algebraic curve of genus 4. The main result of the paper is an explicit formula for the action of these systems. It is obtained directly from the Hamilton–Jacobi equation. We find the action and a separation of variables for the Clebsch and the $so(4)$ Schottky–Manakov spinning tops.
Keywords: integrable tops, separation of variables, Hamilton–Jacobi equation
Citation: Marikhin V. G.,  Sokolov V. V., Separation of variables on a non-hyperelliptic curve , Regular and Chaotic Dynamics, 2005, vol. 10, no. 1, pp. 59-70
DOI: 10.1070/RD2005v010n01ABEH000300
Guru Prem Prasad M.
Chaotic burst in the dynamics of $f_\lambda (z) = \lambda \frac{\sinh (z)}{z}$
Abstract
In this paper, a one-parameter family of non-critically finite entire functions $\mathscr{F} \equiv \{f_\lambda(z)=\lambda f(z): \lambda \in \mathbb{R} \setminus \{0\}\}$ with $f(z) = \frac{\sinh z}{z}$ is considered and the dynamics of the entire transcendental functions $f_\lambda \in \mathscr{F}$ is studied in detail. It is shown that there exists a parameter value $\lambda^* > 0$ such that the Julia set of $f_\lambda (z)$ is nowhere dense subset for $0 < |\lambda| \leqslant \lambda^* (\approx 1.104)$. For $|\lambda| > \lambda^*$ the set explodes and becomes equal to the extended complex plane. This phenomenon is referred to as a chaotic burst in the dynamics of the functions $f_\lambda$ in the one-parameter family $\mathscr{F}$.
Keywords: Fatou sets, Julia sets and Chaotic Burst
Citation: Guru Prem Prasad M., Chaotic burst in the dynamics of $f_\lambda (z) = \lambda \frac{\sinh (z)}{z}$ , Regular and Chaotic Dynamics, 2005, vol. 10, no. 1, pp. 71-80
DOI:10.1070/RD2005v010n01ABEH000301
Markeev A. P.
On the Steklov case in rigid body dynamics
Abstract
We study the motion of a heavy rigid body with a fixed point. The center of mass is located on mean or minor axis of the ellipsoid of inertia, with the moments of inertia satisfying the conditions $B>A>2C$ or $2B>A>B>C$, $A>2C$ as well as the usual triangle inequalities. Under these circumstances the Euler–Poisson equations have the particular periodic solutions mentioned by V. A. Steklov. We examine the problem of the orbital stability of the periodic motions of a rigid body, which correspond to the Steklov solutions.
Keywords: rigid body dynamics, Euler–Poisson equations, Steklov solutions, orbital stability of the periodic motions
Citation: Markeev A. P., On the Steklov case in rigid body dynamics , Regular and Chaotic Dynamics, 2005, vol. 10, no. 1, pp. 81-93
DOI:On the Steklov case in rigid body dynamics
Vidal C.,  Dos Santos F.
Stability of equilibrium positions of periodic Hamiltonian systems under third and fourth order resonances
Abstract
The problem of the stability of an equilibrium position of a nonautonomous 2$\pi$-periodic Hamiltonian system with $n$ degrees of freedom ($n \geqslant 2$), in a nonlinear setting, is studied in the presence of a single third and fourth order resonance. We give conditions of instability in the sense of Lyapunov and formal stability of the equilibrium position, depending on the coefficients of the Hamiltonian function.
Keywords: periodic Hamiltonian system, Lyapunov stability, formal stability, resonance, normal form
Citation: Vidal C.,  Dos Santos F., Stability of equilibrium positions of periodic Hamiltonian systems under third and fourth order resonances , Regular and Chaotic Dynamics, 2005, vol. 10, no. 1, pp. 95-111
DOI: 10.1070/RD2005v010n01ABEH000303
Wang H.,  Xiong J.
Sweep out and chaos
Abstract
Let $X$ be a compact metric space and let $\mathscr{B}$ be a $\sigma$-algebra of all Borel subsets of $X$. Let $m$ be a probability outer measure on $X$ with the properties that each non-empty open set has non-zero $m$-measure and every open set is $m$-measurable. And for every subset $Y$ of $X$ there is a Borel set $B$ of $X$ such that $Y \subset B$ and $m(Y)=m(B)$. We prove that $f : (X, \mathscr{B}, m) \to (X, \mathscr{B}, m)$ sweeps out if and only if for any increasing sequence $J$ of positive integers, there is a finitely chaotic set $C$ for $f$ with respect to $J$ such that $m(C)=1$.
Keywords: sweep out, chaos, measure
Citation: Wang H.,  Xiong J., Sweep out and chaos , Regular and Chaotic Dynamics, 2005, vol. 10, no. 1, pp. 113-118
DOI:10.1070/RD2005v010n01ABEH000304

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