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2013
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Volume 28, Number 1
Proceeding of GDIS 2022

The Eighth International Conference “GEOMETRY, DYNAMICS, INTEGRABLE SYSTEMS — GDIS 2022”
Abstract
Citation: The Eighth International Conference “GEOMETRY, DYNAMICS, INTEGRABLE SYSTEMS — GDIS 2022”, Regular and Chaotic Dynamics, 2023, vol. 28, no. 1, pp. 1-4
DOI:10.1134/S156035472301001X
Bolotin S. V.,  Treschev D. V.
Quasiperiodic Version of Gordon’s Theorem
Abstract
We consider Hamiltonian systems possessing families of nonresonant invariant tori whose frequencies are all collinear. Then under certain conditions the frequencies depend on energy only. This is a generalization of the well-known Gordon’s theorem about periodic solutions of Hamiltonian systems. While the proof of Gordon’s theorem uses Hamilton’s principle, our result is based on Percival’s variational principle. This work was motivated by the problem of isochronicity in Hamiltonian systems.
Keywords: isochronicity, superintegrability, Hamiltonian systems, variational pronciples
Citation: Bolotin S. V.,  Treschev D. V., Quasiperiodic Version of Gordon’s Theorem, Regular and Chaotic Dynamics, 2023, vol. 28, no. 1, pp. 5-13
DOI:10.1134/S1560354723010021
Dragović V.,  Radnović M.
Billiards Within Ellipsoids in the 4-Dimensional Pseudo-Euclidean Spaces
Abstract
We study billiard systems within an ellipsoid in the 4-dimensional pseudo-Euclidean spaces. We provide an analysis and description of periodic and weak periodic trajectories in algebro-geometric and functional-polynomial terms.
Keywords: ellipsoidal billiards, caustics, confocal families of quadrics, extremal polynomials, periodic trajectories, Poncelet porism
Citation: Dragović V.,  Radnović M., Billiards Within Ellipsoids in the 4-Dimensional Pseudo-Euclidean Spaces, Regular and Chaotic Dynamics, 2023, vol. 28, no. 1, pp. 14-43
DOI:10.1134/S1560354723010033
Jovanović B.,  Šukilović  T.,  Vukmirović S.
Integrable Systems Associated to the Filtrations of Lie Algebras
Abstract
In 1983 Bogoyavlenski conjectured that, if the Euler equations on a Lie algebra $\mathfrak{g}_0$ are integrable, then their certain extensions to semisimple lie algebras $\mathfrak{g}$ related to the filtrations of Lie algebras $\mathfrak{g}_0\subset\mathfrak{g}_1\subset\mathfrak{g}_2\dots\subset\mathfrak{g}_{n-1}\subset \mathfrak{g}_n=\mathfrak{g}$ are integrable as well. In particular, by taking $\mathfrak{g}_0=\{0\}$ and natural filtrations of ${\mathfrak{so}}(n)$ and $\mathfrak{u}(n)$, we have Gel’fand – Cetlin integrable systems. We prove the conjecture for filtrations of compact Lie algebras $\mathfrak{g}$: the system is integrable in a noncommutative sense by means of polynomial integrals. Various constructions of complete commutative polynomial integrals for the system are also given.
Keywords: noncommutative integrability, invariant polynomials, Gel’fand – Cetlin systems
Citation: Jovanović B.,  Šukilović  T.,  Vukmirović S., Integrable Systems Associated to the Filtrations of Lie Algebras, Regular and Chaotic Dynamics, 2023, vol. 28, no. 1, pp. 44-61
DOI:10.1134/S1560354723010045
Dragović V.,  Gajić B.,  Jovanović B.
Spherical and Planar Ball Bearings — a Study of Integrable Cases
Abstract
We consider the nonholonomic systems of $n$ homogeneous balls $\mathbf B_1,\dots,\mathbf B_n$ with the same radius $r$ that are rolling without slipping about a fixed sphere $\mathbf S_0$ with center $O$ and radius $R$. In addition, it is assumed that a dynamically nonsymmetric sphere $\mathbf S$ with the center that coincides with the center $O$ of the fixed sphere $\mathbf S_0$ rolls without slipping in contact with the moving balls $\mathbf B_1,\dots,\mathbf B_n$. The problem is considered in four different configurations, three of which are new. We derive the equations of motion and find an invariant measure for these systems. As the main result, for $n=1$ we find two cases that are integrable by quadratures according to the Euler – Jacobi theorem. The obtained integrable nonholonomic models are natural extensions of the well-known Chaplygin ball integrable problems. Further, we explicitly integrate the planar problem consisting of $n$ homogeneous balls of the same radius, but with different masses, which roll without slipping over a fixed plane $\Sigma_0$ with a plane $\Sigma$ that moves without slipping over these balls.
Keywords: nonholonimic dynamics, rolling without slipping, invariant measure, integrability
Citation: Dragović V.,  Gajić B.,  Jovanović B., Spherical and Planar Ball Bearings — a Study of Integrable Cases, Regular and Chaotic Dynamics, 2023, vol. 28, no. 1, pp. 62-77
DOI:10.1134/S1560354723010057
Kilin A. A.,  Pivovarova E. N.
Dynamics of an Unbalanced Disk with a Single Nonholonomic Constraint
Abstract
The problem of the rolling of a disk on a plane is considered under the assumption that there is no slipping in the direction parallel to the horizontal diameter of the disk and that the center of mass does not move in the horizontal direction. This problem is reduced to investigating a system of three first-order differential equations. It is shown that the reduced system is reversible relative to involution of codimension one and admits a two-parameter family of fixed points. The linear stability of these fixed points is analyzed. Using numerical simulation, the nonintegrability of the problem is shown. It is proved that the reduced system admits, even in the nonintegrable case, a two-parameter family of periodic solutions. A number of dynamical effects due to the existence of involution of codimension one and to the degeneracy of the fixed points of the reduced system are found.
Keywords: nonholonomic constraint, unbalanced disk, omnidisk, permanent rotations, periodic solutions, stability, integrability, chaos, invariant manifolds, manifolds of fall
Citation: Kilin A. A.,  Pivovarova E. N., Dynamics of an Unbalanced Disk with a Single Nonholonomic Constraint, Regular and Chaotic Dynamics, 2023, vol. 28, no. 1, pp. 78-106
DOI:10.1134/S1560354723010069
Bizyaev I. A.,  Mamaev I. S.
Roller Racer with Varying Gyrostatic Momentum: Acceleration Criterion and Strange Attractors
Abstract
In this paper we investigate a nonholonomic system with parametric excitation, a Roller Racer with variable gyrostatic momentum. We examine in detail the problem of the existence of regimes with unbounded growth of energy (nonconservative Fermi acceleration). We find a criterion for the existence of trajectories for which one of the velocity components increases withound bound and has asymptotics $t^{1/3}$. In addition, we show that the problem under consideration reduces to analysis of a three-dimensional Poincaré map. This map exhibits both regular attractors (a fixed point, a limit cycle and a torus) and strange attractors.
Keywords: nonholonomic mechanics, Roller Racer, Andronov – Hopf bifurcation, stability, central manifold, unbounded speedup, Poincaré map, limit cycle, strange attractor
Citation: Bizyaev I. A.,  Mamaev I. S., Roller Racer with Varying Gyrostatic Momentum: Acceleration Criterion and Strange Attractors, Regular and Chaotic Dynamics, 2023, vol. 28, no. 1, pp. 107-130
DOI:10.1134/S1560354723010070