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Volume 25, Number 1

Volume 25, Number 1, 2020
Special issue: In honor of Valery Kozlov for his 70th birthday

Valery V. Kozlov. On the Occasion of his 70th Birthday
Abstract
Citation: Valery V. Kozlov. On the Occasion of his 70th Birthday, Regular and Chaotic Dynamics, 2020, vol. 25, no. 1, pp. 1
DOI:10.1134/S1560354720010013
Artemyev A. V.,  Neishtadt A. I.,  Vasiliev A. A.
A Map for Systems with Resonant Trappings and Scatterings
Abstract
Slow-fast dynamics and resonant phenomena can be found in a wide range of physical systems, including problems of celestial mechanics, fluid mechanics, and charged particle dynamics. Important resonant effects that control transport in the phase space in such systems are resonant scatterings and trappings. For systems with weak diffusive scatterings the transport properties can be described with the Chirikov standard map, and the map parameters control the transition between stochastic and regular dynamics. In this paper we put forward the map for resonant systems with strong scatterings that result in nondiffusive drift in the phase space, and trappings that produce fast jumps in the phase space. We demonstrate that this map describes the transition between stochastic and regular dynamics and find the critical parameter values for this transition.
Keywords: scattering on resonance, capture into resonance
Citation: Artemyev A. V.,  Neishtadt A. I.,  Vasiliev A. A., A Map for Systems with Resonant Trappings and Scatterings, Regular and Chaotic Dynamics, 2020, vol. 25, no. 1, pp. 2-10
DOI:10.1134/S1560354720010025
Tabachnikov S.
Two Variations on the Periscope Theorem
Abstract
A (multidimensional) spherical periscope is a system of two ideal mirrors that reflect every ray of light emanating from some point back to this point. A spherical periscope defines a local diffeomorphism of the space of rays through this point, and we describe such diffeomorphisms. We also solve a similar problem for (multidimensional) reversed periscopes, the systems of two mirrors that reverse the direction of a parallel beam of light.
Keywords: periscope, optical reflection, projectively gradient vector field
Citation: Tabachnikov S., Two Variations on the Periscope Theorem, Regular and Chaotic Dynamics, 2020, vol. 25, no. 1, pp. 11-17
DOI:10.1134/S1560354720010037
Borisov A. V.,  Tsiganov A. V.
On the Nonholonomic Routh Sphere in a Magnetic Field
Abstract
This paper is concerned with the motion of an unbalanced dynamically symmetric sphere rolling without slipping on a plane in the presence of an external magnetic field. It is assumed that the sphere can consist completely or partially of dielectric, ferromagnetic, superconducting and crystalline materials. According to the existing phenomenological theory, the analysis of the sphere’s dynamics requires in this case taking into account the Lorentz torque, the Barnett – London effect and the Einstein – de Haas effect. Using this mathematical model, we have obtained conditions for the existence of integrals of motion which allow one to reduce integration of the equations of motion to a quadrature similar to the Lagrange quadrature for a heavy rigid body.
Keywords: nonholonomic systems, integrable systems, magnetic field, Barnett – London effect, Einstein – de Haas effect
Citation: Borisov A. V.,  Tsiganov A. V., On the Nonholonomic Routh Sphere in a Magnetic Field, Regular and Chaotic Dynamics, 2020, vol. 25, no. 1, pp. 18-32
DOI:10.1134/S1560354720010049
Sachkov Y. L.
Periodic Controls in Step 2 Strictly Convex Sub-Finsler Problems
Abstract
We consider control-linear left-invariant time-optimal problems on step 2 Carnot groups with a strictly convex set of control parameters (in particular, sub-Finsler problems). We describe all Casimirs linear in momenta on the dual of the Lie algebra.
In the case of rank 3 Lie groups we describe the symplectic foliation on the dual of the Lie algebra. On this basis we show that extremal controls are either constant or periodic. Some related results for other Carnot groups are presented.
Keywords: optimal control, sub-Finsler geometry, Lie groups, Pontryagin maximum principle
Citation: Sachkov Y. L., Periodic Controls in Step 2 Strictly Convex Sub-Finsler Problems, Regular and Chaotic Dynamics, 2020, vol. 25, no. 1, pp. 33-39
DOI:10.1134/S1560354720010050
Rauch-Wojciechowski S.,  Przybylska M.
On Dynamics of Jellet's Egg. Asymptotic Solutions Revisited
Abstract
We study here the asymptotic condition $\dot E=-\mu g_n {\boldsymbol v}_A^2=0$ for an eccentric rolling and sliding ellipsoid with axes of principal moments of inertia directed along geometric axes of the ellipsoid, a rigid body which we call here Jellett's egg (JE). It is shown by using dynamic equations expressed in terms of Euler angles that the asymptotic condition is satisfied by stationary solutions. There are 4 types of stationary solutions: tumbling, spinning, inclined rolling and rotating on the side solutions. In the generic situation of tumbling solutions concise explicit formulas for stationary angular velocities $\dot\varphi_{\mathrm{JE}}(\cos\theta)$, $\omega_{3\mathrm{JE}}(\cos\theta)$ as functions of JE parameters $\widetilde{\alpha},\alpha,\gamma$ are given. We distinguish the case $1-\widetilde{\alpha}<\alpha^2<1+\widetilde{\alpha}$, $1-\widetilde{\alpha}<\alpha^2\gamma<1+\widetilde{\alpha}$ when velocities $\dot\varphi_{\mathrm{JE}}$, $\omega_{3\mathrm{JE}}$ are defined for the whole range of inclination angles $\theta\in(0,\pi)$. Numerical simulations illustrate how, for a JE launched almost vertically with $\theta(0)=\tfrac{1}{100},\tfrac{1}{10}$, the inversion of the JE depends on relations between parameters.
Keywords: rigid body, nonholonomic mechanics, Jellett egg, tippe top
Citation: Rauch-Wojciechowski S.,  Przybylska M., On Dynamics of Jellet's Egg. Asymptotic Solutions Revisited, Regular and Chaotic Dynamics, 2020, vol. 25, no. 1, pp. 40-58
DOI:10.1134/S1560354720010062
Kudryashov N. A.
Lax Pairs and Special Polynomials Associated with Self-similar Reductions of Sawada – Kotera and Kupershmidt Equations
Abstract
Self-similar reductions of the Sawada – Kotera and Kupershmidt equations are studied. Results of Painlevé's test for these equations are given. Lax pairs for solving the Cauchy problems to these nonlinear ordinary differential equations are found. Special solutions of the Sawada – Kotera and Kupershmidt equations expressed via the first Painlevé equation are presented. Exact solutions of the Sawada – Kotera and Kupershmidt equations by means of general solution for the first member of $K_2$ hierarchy are given. Special polynomials for expressions of rational solutions for the equations considered are introduced. The differentialdifference equations for finding special polynomials corresponding to the Sawada – Kotera and Kupershmidt equations are found. Nonlinear differential equations of sixth order for special polynomials associated with the Sawada – Kotera and Kupershmidt equations are obtained. Lax pairs for nonlinear differential equations with special polynomials are presented. Rational solutions of the self-similar reductions for the Sawada – Kotera and Kupershmidt equations are given.
Keywords: higher-order Painlevé equation, Sawada – Kotera equation, Kupershmidt equation, self-similar reduction, special polynomial, exact solution
Citation: Kudryashov N. A., Lax Pairs and Special Polynomials Associated with Self-similar Reductions of Sawada – Kotera and Kupershmidt Equations, Regular and Chaotic Dynamics, 2020, vol. 25, no. 1, pp. 59-77
DOI:10.1134/S1560354720010074
Andrade J.,  Boatto S.,  Combot T.,  Duarte G.,  Stuchi T. J.
$N$-body Dynamics on an Infinite Cylinder: the Topological Signature in the Dynamics
Abstract
The formulation of the dynamics of $N$-bodies on the surface of an infinite cylinder is considered. We have chosen such a surface to be able to study the impact of the surface’s topology in the particle’s dynamics. For this purpose we need to make a choice of how to generalize the notion of gravitational potential on a general manifold. Following Boatto, Dritschel and Schaefer [5], we define a gravitational potential as an attractive central force which obeys Maxwell’s like formulas.
As a result of our theoretical differential Galois theory and numerical study — Poincaré sections, we prove that the two-body dynamics is not integrable. Moreover, for very low energies, when the bodies are restricted to a small region, the topological signature of the cylinder is still present in the dynamics. A perturbative expansion is derived for the force between the two bodies. Such a force can be viewed as the planar limit plus the topological perturbation. Finally, a polygonal configuration of identical masses (identical charges or identical vortices) is proved to be an unstable relative equilibrium for all $N >2$.
Keywords: $N$-body problem, Hodge decomposition, central forces on manifolds, topology and integrability, differential Galois theory, Poincaré sections, stability of relative equilibria
Citation: Andrade J.,  Boatto S.,  Combot T.,  Duarte G.,  Stuchi T. J., $N$-body Dynamics on an Infinite Cylinder: the Topological Signature in the Dynamics, Regular and Chaotic Dynamics, 2020, vol. 25, no. 1, pp. 78-110
DOI:10.1134/S1560354720010086
Markeev A. P.
On Periodic Poincaré Motions in the Case of Degeneracy of an Unperturbed System
Abstract
This paper is concerned with a one-degree-of-freedom system close to an integrable system. It is assumed that the Hamiltonian function of the system is analytic in all its arguments, its perturbing part is periodic in time, and the unperturbed Hamiltonian function is degenerate. The existence of periodic motions with a period divisible by the period of perturbation is shown by the Poincaré methods. An algorithm is presented for constructing them in the form of series (fractional degrees of a small parameter), which is implemented using classical perturbation theory based on the theory of canonical transformations of Hamiltonian systems. The problem of the stability of periodic motions is solved using the Lyapunov methods and KAM theory. The results obtained are applied to the problem of subharmonic oscillations of a pendulum placed on a moving platform in a homogeneous gravitational field. The platform rotates with constant angular velocity about a vertical passing through the suspension point of the pendulum, and simultaneously executes harmonic small-amplitude oscillations along the vertical. Families of subharmonic oscillations of the pendulum are shown and the problem of their Lyapunov stability is solved.
Keywords: Hamiltonian system, degeneracy, periodic motion, stability
Citation: Markeev A. P., On Periodic Poincaré Motions in the Case of Degeneracy of an Unperturbed System, Regular and Chaotic Dynamics, 2020, vol. 25, no. 1, pp. 111-120
DOI:10.1134/S1560354720010098
Burov A.,  Guerman A.,  Nikonov V.
Asymptotic Invariant Surfaces for Non-Autonomous Pendulum-Type Systems
Abstract
Invariant surfaces play a crucial role in the dynamics of mechanical systems separating regions filled with chaotic behavior. Cases where such surfaces can be found are rare enough. Perhaps the most famous of these is the so-called Hess case in the mechanics of a heavy rigid body with a fixed point.
We consider here the motion of a non-autonomous mechanical pendulum-like system with one degree of freedom. The conditions of existence for invariant surfaces of such a system corresponding to non-split separatrices are investigated. In the case where an invariant surface exists, combination of regular and chaotic behavior is studied analytically via the Poincaré – Mel'nikov separatrix splitting method, and numerically using the Poincaré maps.
Keywords: separatrices splitting, chaotic dynamics, invariant surface
Citation: Burov A.,  Guerman A.,  Nikonov V., Asymptotic Invariant Surfaces for Non-Autonomous Pendulum-Type Systems, Regular and Chaotic Dynamics, 2020, vol. 25, no. 1, pp. 121-130
DOI:10.1134/S1560354720010104

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