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Volume 15, Number 6

Volume 15, Number 6, 2010

Ratiu T.
Remembering Jerry Marsden
Abstract
Citation: Ratiu T., Remembering Jerry Marsden, Regular and Chaotic Dynamics, 2010, vol. 15, no. 6, pp. 635-636
DOI:10.1134/S1560354710060018
Butta P.,  Negrini P.
On the stability problem of stationary solutions for the Euler equation on a 2-dimensional torus
Abstract
We study the linear stability problem of the stationary solution $\psi^* = −\cos y$ for the Euler equation on a 2-dimensional flat torus of sides $2\pi L$ and $2\pi$. We show that $\psi^*$ is stable if $L \in (0, 1)$ and that exponentially unstable modes occur in a right neighborhood of $L = n$ for any integer $n$. As a corollary, we gain exponentially instability for any $L$ large enough and an unbounded growth of the number of unstable modes as $L$ diverges.
Keywords: Euler equation, shear flows, linear stability
Citation: Butta P.,  Negrini P., On the stability problem of stationary solutions for the Euler equation on a 2-dimensional torus, Regular and Chaotic Dynamics, 2010, vol. 15, no. 6, pp. 637-645
DOI:10.1134/S1560354710510143
Massart D.
Two remarks about Mañé's conjecture
Abstract
We consider Mañé's conjectures and prove that the one he made in [1] is stronger than the one he made in [2]. Then we prove that the most straightforward approach to prove the strong conjecture doesn’t work in the $C^4$ topology.
Keywords: Lagrangian dynamics, minimizing measures
Citation: Massart D., Two remarks about Mañé's conjecture, Regular and Chaotic Dynamics, 2010, vol. 15, no. 6, pp. 646-651
DOI:10.1134/S1560354710510155
Marikhin V. G.,  Sokolov V. V.
Transformation of a pair of commuting Hamiltonians quadratic in momenta to canonical form and real partial separation of variables for the Clebsch top
Abstract
In the case of two degrees of freedom the simultaneous diagonalization of pairs of Hamiltonians quadratic on momenta that commute with respect to the standard Poisson bracket is considered. A general scheme of partial separation of variables for such pairs is discussed. As an example the Clebsch top is considered.
Keywords: separation of variables, the Clebsch top
Citation: Marikhin V. G.,  Sokolov V. V., Transformation of a pair of commuting Hamiltonians quadratic in momenta to canonical form and real partial separation of variables for the Clebsch top, Regular and Chaotic Dynamics, 2010, vol. 15, no. 6, pp. 652-658
DOI:10.1134/S1560354710510167
Tsiganov A. V.
New variables of separation for particular case of the Kowalevski top
Abstract
We discuss the polynomial bi-Hamiltonian structures for the Kowalevski top in special case of zero square integral. An explicit procedure to find variables of separation and separation relations is considered in detail.
Keywords: Kowalevski top, separation of variables, bi-Hamiltonian geometry, differential geometry, algebraic curves
Citation: Tsiganov A. V., New variables of separation for particular case of the Kowalevski top, Regular and Chaotic Dynamics, 2010, vol. 15, no. 6, pp. 659-669
DOI:10.1134/S156035471006002X
Vershilov A. V.
On the bi-Hamiltonian structure of Bogoyavlensky system on $so(4)$
Abstract
We discuss bi-Hamiltonian structure for the Bogoyavlensky system on $so(4)$ with an additional integral of fourth order in momenta. An explicit procedure to find the variables of separation and the separation relations is considered in detail.
Keywords: integrable systems, separation of variables, bi-Hamiltonian geometry, Lie algebras
Citation: Vershilov A. V., On the bi-Hamiltonian structure of Bogoyavlensky system on $so(4)$, Regular and Chaotic Dynamics, 2010, vol. 15, no. 6, pp. 670-676
DOI:10.1134/S1560354710060031
Basak I.
Bifurcation analysis of the Zhukovskii–Volterra system via bi-Hamiltonian approach
Abstract
The main goal of this paper consists of bifurcation analysis of classical integrable Zhukovskii–Volterra system. We use the fact that the ZV system is bi-Hamiltonian and apply new techniques [1] for analysis of singularities of bi-Hamiltonian systems, which can be formulated as follows: the structure of singularities of a bi-Hamiltonian system is determined by that of the corresponding compatible Poisson brackets.
Keywords: integrable Hamiltonian sistems, compatible Poisson structures, bifurcations, semisimple Lie algebras
Citation: Basak I., Bifurcation analysis of the Zhukovskii–Volterra system via bi-Hamiltonian approach, Regular and Chaotic Dynamics, 2010, vol. 15, no. 6, pp. 677-684
DOI:10.1134/S1560354710060043
Avrutin V.,  Schanz M.,  Gardini L.
Self-similarity of the bandcount adding structures: calculation by map replacement
Abstract
Recently it has been demonstrated that the domain of robust chaos close to the periodic domain, which is organized by the period-adding structure, contains an infinite number of interior crisis bifurcation curves. These curves form the so-called bandcount adding scenario, which determines the occurrence of multi-band chaotic attractors. The analytical calculation of the interior crisis bifurcations represents usually a quite sophisticated and cumbersome task. In this work we demonstrate that, using the map replacement approach, the bifurcation curves can be calculated much easier. Moreover, using this approach recursively, we confirm the hypothesis regarding the self-similarity of the bandcount adding structure.
Keywords: piecewise-linear maps, crisis bifurcations, chaotic attractors, bandcount adding and doubling, self-similarityand renormalization
Citation: Avrutin V.,  Schanz M.,  Gardini L., Self-similarity of the bandcount adding structures: calculation by map replacement, Regular and Chaotic Dynamics, 2010, vol. 15, no. 6, pp. 685-703
DOI:10.1134/S1560354710060055
Bardin B. S.
On the orbital stability of pendulum-like motions of a rigid body in the Bobylev–Steklov case
Abstract
We deal with the problem of orbital stability of pendulum-like periodic motions of a heavy rigid body with a fixed point. We suppose that the geometry of the mass of the body corresponds to the Bobylev–Steklov case. Unperturbed motion represents oscillations or rotations of the body around a principal axis, occupying a fixed horizontal position. The problem of the orbital stability is considered on the basis of a nonlinear analysis.
In the case of oscillations with small amplitudes as well as in the case of rotations with high angular velocities we study the problem analytically. In the general case we reduce the problem to the stability study of a fixed point of the symplectic map generated by equations of perturbed motion. We calculate coefficients of the symplectic map numerically. By analyzing the abovementioned coefficients we establish the orbital stability or instability of the unperturbed motion. The results of the study are represented in the form of a stability diagram.
Keywords: Hamiltonian system, periodic orbits, normal form, resonance, action-angel variables, KAM theory
Citation: Bardin B. S., On the orbital stability of pendulum-like motions of a rigid body in the Bobylev–Steklov case, Regular and Chaotic Dynamics, 2010, vol. 15, no. 6, pp. 704-716
DOI:10.1134/S1560354710060067
Novokshenov V. Y.
Poles of tritronquée solution to the Painlevé I equation and cubic anharmonic oscillator
Abstract
The distribution of poles of zero-parameter solution to Painlevé I, specified by P. Boutroux as intégrale tritronquée, is studied in the complex plane. This solution has regular asymptotics $−\sqrt{z/6}+ O(1)$ as $z \to \infty$, $|\arg z| < 4\pi/5$. At the sector $|\arg z| > 4\pi/5$ it is a meromorphic function with regular asymptotic distribution of poles at infinity. This fact together with numeric simulations for $|z| < const$ allowed B. Dubrovin to make a conjecture that all poles of the intégrale tritronquée belong to this sector. As a step to prove this conjecture, we study the Riemann–Hilbert (RH) problem related to the specified solution of the Painlevé I equation. It is "undressed" to a similar RH problem for the Schrödinger equation with cubic potential. The latter determines all coordinates of poles for the intégrale tritronquée via a Bohr–Sommerfeld quantization conditions.
Citation: Novokshenov V. Y., Poles of tritronquée solution to the Painlevé I equation and cubic anharmonic oscillator, Regular and Chaotic Dynamics, 2010, vol. 15, no. 6, pp. 390-403
DOI:10.1134/S1560354710020243

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