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# Volume 24, Number 3, 2019

 Kholostova O. V. On the Motions of One Near-Autonomous Hamiltonian System at a 1:1:1 Resonance Abstract We consider the motion of a $2 \pi$-periodic in time two-degree-of-freedom Hamiltonian system in a neighborhood of the equilibrium position. It is assumed that the system depends on a small parameter $e$ and other parameters and is autonomous at $e = 0$. It is also assumed that in the autonomous case there is a set of parameter values for which a 1:1 resonance occurs, and the matrix of the linearized equations of perturbed motion is reduced to a diagonal form. The study is carried out using an example of the problem of the motion of a dynamically symmetric rigid body (satellite) relative to its center of mass in a central Newtonian gravitational field on an elliptical orbit with small eccentricity in the neighborhood of the cylindrical precession. The character of the motions of the reduced two-degree-of-freedom system in the vicinity of the resonance point in the three-dimensional  parameter space  is studied. Stability regions of the unperturbed motion (the cylindrical precession) and two types of parametric resonance regions corresponding to the case of zero frequency and the case of equal frequencies in the transformed approximate system of the linearized equations of perturbed motion are considered. The problem of the existence, number and stability of $2 \pi$-periodic motions of the satellite is solved, and conclusions on the existence of two- and three-frequency conditionally periodic motions are obtained. Keywords: Hamiltonian system, resonance, stability, cylindrical precession of a satellite, periodic motion, conditionally periodic motion Citation: Kholostova O. V., On the Motions of One Near-Autonomous Hamiltonian System at a 1:1:1 Resonance, Regular and Chaotic Dynamics, 2019, vol. 24, no. 3, pp. 235-265 DOI:10.1134/S1560354719030018
 Bolsinov A. V.,  Bao J. A Note about Integrable Systems on Low-dimensional Lie Groups and Lie Algebras Abstract The goal of the paper is to explain why any left-invariant Hamiltonian system on (the cotangent bundle of) a 3-dimensonal Lie group $G$ is Liouville integrable. We derive this property from the fact that the coadjoint orbits of $G$ are two-dimensional so that the integrability of left-invariant systems is a common property of all such groups regardless their dimension. We also give normal forms for left-invariant Riemannian and sub-Riemannian metrics on 3-dimensional Lie groups focusing on the case of solvable groups, as the cases of $SO(3)$ and $SL(2)$ have been already extensively studied. Our description is explicit and is given in global coordinates on $G$ which allows one to easily obtain parametric equations of geodesics in quadratures. Keywords: Integrable systems, Lie groups, geodesic flow, left-invariant metric, sub-Riemannian structure Citation: Bolsinov A. V.,  Bao J., A Note about Integrable Systems on Low-dimensional Lie Groups and Lie Algebras, Regular and Chaotic Dynamics, 2019, vol. 24, no. 3, pp. 266-280 DOI:10.1134/S156035471903002X
 Polekhin I. Y. Precession of the Kovalevskaya and Goryachev – Chaplygin Tops Abstract The change of the precession angle is studied analytically and numerically for two classical integrable tops: the Kovalevskaya top and the Goryachev – Chaplygin top. Based on the known results on the topology of Liouville foliations for these systems, we find initial conditions for which the average change of the precession angle is zero or can be estimated asymptotically. Some more difficult cases are studied numerically. In particular, we show that the average change of the precession angle for the Kovalevskaya top can be non-zero even in the case of zero area integral. Keywords: mean motion, Kovalevskaya top, Goryachev – Chaplygin top, integrable system, precession Citation: Polekhin I. Y., Precession of the Kovalevskaya and Goryachev – Chaplygin Tops, Regular and Chaotic Dynamics, 2019, vol. 24, no. 3, pp. 281-297 DOI:10.1134/S1560354719030031
 Pepa R. Y.,  Popelensky T. Y. Combinatorial Ricci Flow for Degenerate Circle Packing Metrics Abstract Chow and Luo [3] showed in 2003 that the combinatorial analogue of the Hamilton Ricci flow on surfaces converges under certain conditions to Thruston’s circle packing metric of constant curvature. The combinatorial setting includes weights defined for edges of a triangulation. A crucial assumption in [3] was that the weights are nonnegative. Recently we have shown that the same statement on convergence can be proved under a weaker condition: some weights can be negative and should satisfy certain inequalities [4]. On the other hand, for weights not satisfying conditions of Chow – Luo’s theorem we observed in numerical simulation a degeneration of the metric with certain regular behaviour patterns [5]. In this note we introduce degenerate circle packing metrics, and under weakened conditions on weights we prove that under certain assumptions for any initial metric an analogue of the combinatorial Ricci flow has a unique limit metric with a constant curvature outside of singularities. Keywords: combinatorial Ricci flow, degenerate circle packing metric Citation: Pepa R. Y.,  Popelensky T. Y., Combinatorial Ricci Flow for Degenerate Circle Packing Metrics, Regular and Chaotic Dynamics, 2019, vol. 24, no. 3, pp. 298-311 DOI:10.1134/S1560354719030043
 Ardentov A. A.,  Karavaev Y. L.,  Yefremov K. S. Euler Elasticas for Optimal Control of the Motion of Mobile Wheeled Robots: the Problem of Experimental Realization Abstract This paper is concerned with the problem of optimal path planning for a mobile wheeled robot. Euler elasticas, which ensure minimization of control actions, are considered as optimal trajectories. An algorithm for constructing controls that realizes the motion along the trajectory in the form of an Euler elastica is presented. Problems and special features of the application of this algorithm in practice are discussed. In particular, analysis is made of speedup and deceleration along the elastica, and of the influence of the errors made in manufacturing the mobile robot on the precision with which the prescribed trajectory is followed. Special attention is also given to the problem of forming optimal trajectories of motion along Euler elasticas to a preset point at different angles of orientation. Results of experimental investigations are presented. Keywords: mobile wheeled robot, Euler’s elastica, optimal control, experimental investigations Citation: Ardentov A. A.,  Karavaev Y. L.,  Yefremov K. S., Euler Elasticas for Optimal Control of the Motion of Mobile Wheeled Robots: the Problem of Experimental Realization, Regular and Chaotic Dynamics, 2019, vol. 24, no. 3, pp. 312-328 DOI:10.1134/S1560354719030055
 Borisov A. V.,  Kilin A. A.,  Mamaev I. S. A Parabolic Chaplygin Pendulum and a Paul Trap: Nonintegrability, Stability, and Boundedness Abstract This paper is a small review devoted to the dynamics of a point on a paraboloid. Specifically, it is concerned with the motion both under the action of a gravitational field and without it. It is assumed that the paraboloid can rotate about a vertical axis with constant angular velocity. The paper includes both well-known results and a number of new results. We consider the two most widespread friction (resistance) models: dry (Coulomb) friction and viscous friction. It is shown that the addition of external damping (air drag) can lead to stability of equilibrium at the saddle point and hence to preservation of the region of bounded motion in a neighborhood of the saddle point. Analysis of three-dimensional Poincaré sections shows that limit cycles can arise in this case in the neighborhood of the saddle point. Keywords: parabolic pendulum, Paul trap, rotating paraboloid, internal damping, external damping, friction, resistance, linear stability, Hill's region, bifurcational diagram, Poincaré section, bounded trajectory, chaos, integrability, nonintegrability, separatrix Citation: Borisov A. V.,  Kilin A. A.,  Mamaev I. S., A Parabolic Chaplygin Pendulum and a Paul Trap: Nonintegrability, Stability, and Boundedness, Regular and Chaotic Dynamics, 2019, vol. 24, no. 3, pp. 329-352 DOI:10.1134/S1560354719030067

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