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# Volume 19, Number 3, 2014

 Zhang P. Algebraic Properties of Compatible Poisson Brackets Abstract We discuss algebraic properties of a pencil generated by two compatible Poisson tensors $\mathcal{A}(x)$ and $\mathcal{B}(x)$. From the algebraic viewpoint this amounts to studying the properties of a pair of skew-symmetric bilinear forms $\mathcal{A}$ and $\mathcal{B}$ defined on a finite-dimensional vector space. We describe the Lie group $G_\mathcal{P}$ of linear automorphisms of the pencil $\mathcal{P}={\mathcal{A}+\lambda \mathcal{B}}$. In particular, we obtain an explicit formula for the dimension of $G_\mathcal{P}$ and discuss some other algebraic properties such as solvability and Levi–Malcev decomposition. Keywords: compatible Poisson brackets, Jordan–Kronecker decomposition, pencils of skew symmetric matrices, bi-Hamiltonian systems Citation: Zhang P., Algebraic Properties of Compatible Poisson Brackets, Regular and Chaotic Dynamics, 2014, vol. 19, no. 3, pp. 267-288 DOI:10.1134/S1560354714030010
 Chenciner A.,  Leclerc B. Between Two Moments Abstract In this short note, we draw attention to a relation between two Horn polytopes which is proved in [3] as a result, on the one hand, of a deep combinatorial result in [5] and, on the other hand, of a simple computation involving complex structures. This suggests an inequality between Littlewood–Richardson coefficients, which we prove using the symmetric characterization of these coefficients given in [1]. Keywords: Littlewood–Richardson coefficients, Horn polytopes, moment maps Citation: Chenciner A.,  Leclerc B., Between Two Moments, Regular and Chaotic Dynamics, 2014, vol. 19, no. 3, pp. 289-295 DOI:10.1134/S1560354714030022
 Burlakov D.,  Treschev D. V. A Rigid Body on a Surface with Random Roughness Abstract Consider an interval on a horizontal line with random roughness. With probability one it is supported at two points: one on the left, and another on the right from its center. We compute the probability distribution of the support points provided the roughness is fine grained. We also solve an analogous problem where a circle or a disk lies on a rough plane. Some applications in static are given. Keywords: rigid body, support with random roughness Citation: Burlakov D.,  Treschev D. V., A Rigid Body on a Surface with Random Roughness, Regular and Chaotic Dynamics, 2014, vol. 19, no. 3, pp. 296-309 DOI:10.1134/S1560354714030034
 Kulczycki M. A Unified Approach to Theories of Shadowing Abstract This paper introduces the notion of a general approximation property, which encompasses many existing types of shadowing. It is proven that there exists a metric space $X$ such that the sets of maps with many types of general approximation properties (including the classic shadowing, the $\mathcal{L}_p$-shadowing, limit shadowing, and the $s$-limit shadowing) are not dense in $C(X)$, $S(X)$, and $H(X)$ (the space of continuous self-maps of $X$, continuous surjections of $X$ onto itself, and self-homeomorphisms of $X$) and that there exists a manifold M such that the sets of maps with general approximation properties of nonlocal type (including the average shadowing property and the asymptotic average shadowing property) are not dense in $C(M)$, $S(M)$, and $H(M)$. Furthermore, it is proven that the sets of maps with a wide range of general approximation properties (including the classic shadowing, the $\mathcal{L}_p$-shadowing, and the $s$-limit shadowing) are dense in the space of continuous self-maps of the Cantor set. A condition is given that guarantees transfer of general approximation property from a map on $X$ to the map induced by it on the hyperspace of $X$. It is also proven that the transfer in the opposite direction always takes place. Keywords: shadowing, average shadowing, limit shadowing, pseudo-orbit, chain-transitivity Citation: Kulczycki M., A Unified Approach to Theories of Shadowing, Regular and Chaotic Dynamics, 2014, vol. 19, no. 3, pp. 310-317 DOI:10.1134/S1560354714030046
 Aquilanti V.,  Lombardi A.,  Sevryuk M. B. Statistics of Energy Partitions for Many-Particle Systems in Arbitrary Dimension Abstract In some previous articles, we defined several partitions of the total kinetic energy $T$ of a system of $N$ classical particles in $\mathbb{R}^d$ into components corresponding to various modes of motion. In the present paper, we propose formulas for the mean values of these components in the normalization $T=1$ (for any $d$ and $N$) under the assumption that the masses of all the particles are equal. These formulas are proven at the “physical level” of rigor and numerically confirmed for planar systems $(d=2)$ at $3\leqslant N \leqslant 100$. The case where the masses of the particles are chosen at random is also considered. The paper complements our article of 2008 [Russian J. Phys. Chem. B, 2(6):947–963] where similar numerical experiments were carried out for spatial systems $(d=3)$ at $3\leqslant N \leqslant 100$. Keywords: multidimensional systems of classical particles, instantaneous phase-space invariants, kinetic energy partitions, formulas for the mean values, hyperangular momenta Citation: Aquilanti V.,  Lombardi A.,  Sevryuk M. B., Statistics of Energy Partitions for Many-Particle Systems in Arbitrary Dimension, Regular and Chaotic Dynamics, 2014, vol. 19, no. 3, pp. 318-347 DOI:10.1134/S1560354714030058
 Carvalho A.,  Cabral H. E. Lyapunov Orbits in the $n$-Vortex Problem Abstract In the reduced phase space by rotation, we prove the existence of periodic orbits of the $n$-vortex problem emanating from a relative equilibrium formed by $n$ unit vortices at the vertices of a regular polygon, both in the plane and at a fixed latitude when the ideal fluid moves on the surface of a sphere. In the case of a plane we also prove the existence of such periodic orbits in the $(n+1)$-vortex problem, where an additional central vortex of intensity κ is added to the ring of the polygonal configuration. Keywords: point vortices, relative equilibria, periodic orbits, Lyapunov center theorem Citation: Carvalho A.,  Cabral H. E., Lyapunov Orbits in the $n$-Vortex Problem, Regular and Chaotic Dynamics, 2014, vol. 19, no. 3, pp. 348-362 DOI:10.1134/S156035471403006X
 Fortunati A.,  Wiggins S. Normal Form and Nekhoroshev Stability for Nearly Integrable Hamiltonian Systems with Unconditionally Slow Aperiodic Time Dependence Abstract The aim of this paper is to extend the result of Giorgilli and Zehnder for aperiodic time dependent systems to a case of nearly integrable convex analytic Hamiltonians. The existence of a normal form and then a stability result are shown in the case of a slow aperiodic time dependence that, under some smallness conditions, is independent of the size of the perturbation. Keywords: Hamiltonian systems, Nekhoroshev theorem, aperiodic time dependence Citation: Fortunati A.,  Wiggins S., Normal Form and Nekhoroshev Stability for Nearly Integrable Hamiltonian Systems with Unconditionally Slow Aperiodic Time Dependence, Regular and Chaotic Dynamics, 2014, vol. 19, no. 3, pp. 363-373 DOI:10.1134/S1560354714030071
 Labrousse C.,  Marco J. Polynomial Entropies for Bott Integrable Hamiltonian Systems Abstract In this paper, we study the entropy of a Hamiltonian flow in restriction to an energy level where it admits a first integral which is nondegenerate in the sense of Bott. It is easy to see that for such a flow, the topological entropy vanishes. We focus on the polynomial and the weak polynomial entropies $h_{pol}$ and $h^*_{pol}$. We show that, under natural conditions on the critical levels of the Bott first integral and on the Hamiltonian function $H, h^*_{pol} \in {0,1}$ and $h_{pol} \in {0,1,2}$. To prove this result, our main tool is a semi-global desingularization of the Hamiltonian system in the neighborhood of a polycycle. Keywords: dynamical complexity, entropy, integrability, Bott integrable Hamiltonians Citation: Labrousse C.,  Marco J., Polynomial Entropies for Bott Integrable Hamiltonian Systems, Regular and Chaotic Dynamics, 2014, vol. 19, no. 3, pp. 374-414 DOI:10.1134/S1560354714030083
 Bizyaev I. A.,  Borisov A. V.,  Mamaev I. S. Superintegrable Generalizations of the Kepler and Hook Problems Abstract In this paper we consider superintegrable systems which are an immediate generalization of the Kepler and Hook problems, both in two-dimensional spaces — the plane $\mathbb{R}^2$ and the sphere $S^2$ — and in three-dimensional spaces $\mathbb{R}^3$ and $S^3$. Using the central projection and the reduction procedure proposed in [21], we show an interrelation between the superintegrable systems found previously and show new ones. In all cases the superintegrals are presented in explicit form. Keywords: superintegrable systems, Kepler and Hook problems, isomorphism, central projection, reduction, highest degree polynomial superintegrals Citation: Bizyaev I. A.,  Borisov A. V.,  Mamaev I. S., Superintegrable Generalizations of the Kepler and Hook Problems, Regular and Chaotic Dynamics, 2014, vol. 19, no. 3, pp. 415-434 DOI:10.1134/S1560354714030095

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