Volume 25, Number 1
Volume 25, Number 1, 2020
Special issue: In honor of Valery Kozlov for his 70th birthday
Artemyev A. V., Neishtadt A. I., Vasiliev A. A.
A Map for Systems with Resonant Trappings and Scatterings
Abstract
Slowfast 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.

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.

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.

Sachkov Y. L.
Periodic Controls in Step 2 Strictly Convex SubFinsler Problems
Abstract
We consider controllinear leftinvariant timeoptimal problems on step 2 Carnot
groups with a strictly convex set of control parameters (in particular, subFinsler 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. 
RauchWojciechowski 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.

Kudryashov N. A.
Lax Pairs and Special Polynomials Associated with Selfsimilar Reductions of Sawada – Kotera and Kupershmidt Equations
Abstract
Selfsimilar 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 selfsimilar reductions for the Sawada – Kotera and Kupershmidt equations are
given.

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 twobody 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 $N >2$. 
Markeev A. P.
On Periodic Poincaré Motions in the Case of Degeneracy of an Unperturbed System
Abstract
This paper is concerned with a onedegreeoffreedom 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 smallamplitude oscillations along the vertical. Families of
subharmonic oscillations of the pendulum are shown and the problem of their Lyapunov stability
is solved.

Burov A., Guerman A., Nikonov V.
Asymptotic Invariant Surfaces for NonAutonomous PendulumType 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 socalled Hess case in the mechanics of a
heavy rigid body with a fixed point.
We consider here the motion of a nonautonomous mechanical pendulumlike system with one degree of freedom. The conditions of existence for invariant surfaces of such a system corresponding to nonsplit 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. 