Volume 15, Number 6
Volume 15, Number 6, 2010
Ratiu T.
Remembering Jerry Marsden
Abstract

Butta P., Negrini P.
On the stability problem of stationary solutions for the Euler equation on a 2dimensional torus
Abstract
We study the linear stability problem of the stationary solution $\psi^* = −\cos y$ for the Euler equation on a 2dimensional 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.

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.

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.

Tsiganov A. V.
New variables of separation for particular case of the Kowalevski top
Abstract
We discuss the polynomial biHamiltonian 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.

Vershilov A. V.
On the biHamiltonian structure of Bogoyavlensky system on $so(4)$
Abstract
We discuss biHamiltonian 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.

Basak I.
Bifurcation analysis of the Zhukovskii–Volterra system via biHamiltonian 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 biHamiltonian and apply new techniques [1] for analysis of singularities of biHamiltonian systems, which can be formulated as follows: the structure of singularities of a biHamiltonian system is determined by that of the corresponding compatible Poisson brackets.

Avrutin V., Schanz M., Gardini L.
Selfsimilarity 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 periodadding structure, contains an infinite number of interior crisis bifurcation curves. These curves form the socalled bandcount adding scenario, which determines the occurrence of multiband 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 selfsimilarity of the bandcount adding structure.

Bardin B. S.
On the orbital stability of pendulumlike motions of a rigid body in the Bobylev–Steklov case
Abstract
We deal with the problem of orbital stability of pendulumlike 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. 
Novokshenov V. Y.
Poles of tritronquée solution to the Painlevé I equation and cubic anharmonic oscillator
Abstract
The distribution of poles of zeroparameter 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.
