Volume 24, Number 1
Volume 24, Number 1, 2019
Proceedings of GDIS 2018, Moscow
Dragović V., Radnović M.
Caustics of Poncelet Polygons and Classical Extremal Polynomials
Abstract
A comprehensive analysis of periodic trajectories of billiards within ellipses in the Euclidean
plane is presented. The novelty of the approach is based on a relationship recently established
by the authors between periodic billiard trajectories and extremal polynomials on the systems
of $d$ intervals on the real line and ellipsoidal billiards in $d$dimensional space.
Even in the planar case systematically studied in the present paper, it leads to new results
in characterizing $n$ periodic trajectories vs. socalled $n$ elliptic periodic trajectories,
which are $n$periodic in elliptical coordinates. The characterizations are done both in terms
of the underlying elliptic curve and divisors on it and in terms of polynomial functional
equations, like Pell's equation. This new approach also sheds light on some classical results.
In particular, we connect the search for caustics which generate periodic trajectories with
three classical classes of extremal polynomials on two intervals, introduced by Zolotarev and Akhiezer.
The main classifying tool are winding numbers, for which we provide several interpretations, including
one in terms of numbers of points of alternance of extremal polynomials. The latter implies
important inequality between the winding numbers, which, as a consequence, provides another
proof of monotonicity of rotation numbers. A complete catalog of billiard trajectories with
small periods is provided for $n=3, 4, 5, 6$ along with an effective search for caustics.
As a byproduct, an intriguing connection between Cayleytype conditions and discriminantly
separable polynomials has been observed for all those small periods.

Ardentov A. A., Le Donne E., Sachkov Y. L.
SubFinsler Geodesics on the Cartan Group
Abstract
This paper is a continuation of the work by the same authors on the
Cartan group equipped with the subFinsler $\ell_\infty$ norm.
We start by giving a detailed presentation of the structure of bangbang extremal trajectories.
Then we prove upper bounds on the number of switchings on bangbang minimizers.
We prove that any normal extremal is either bangbang, or singular, or mixed.
Consequently, we study mixed extremals.
In particular, we prove that every two points can be connected by a piecewise smooth
minimizer, and we give a uniform bound on the number of such pieces.

Koiller J., Castilho C., Rodrigues A. R.
Vortex Pairs on the Triaxial Ellipsoid: Axis Equilibria Stability
Abstract
We consider a pair of opposite vortices moving on the surface of the triaxial ellipsoid
$\mathbb{E}(a,b,c):$ $x^2/a+y^2/b+z^2/c=1, \, a < b < c$. The equations of motion are transported to $S^2 \times S^2$ via a conformal map that combines confocal quadric coordinates for the ellipsoid and spheroconical coordinates in the sphere. The antipodal pairs form an invariant submanifold for the dynamics. We characterize the linear stability of the equilibrium pairs at the three axis endpoints.

Allilueva A. I., Shafarevich A. I.
Evolution of Lagrangian Manifolds and Asymptotic Solutions to the Linearized Equations of Gas Dynamics
Abstract
We study asymptotic solution of the Cauchy problem for linearized equations of gas dynamics with rapidly oscillating initial data. We construct the formal serie, satisfying this problem. This serie is naturally divided into three parts, corresponding to the hydrodynamic mode and two acoustic modes. The summands of the serie are expressed in terms of the Maslov canonic operator on moving Lagrangian manifolds. Evolution of the manifolds is governed by the corresponding classical Hamiltonian systems.

Kudryashov N. A.
Rational and Special Solutions for Some Painlevé Hierarchies
Abstract
A selfsimilar reduction of the Korteweg–de Vries hierarchy is considered. A linear system of equations associated with this hierarchy is presented. This Lax pair can be used to solve the Cauchy problem for the studied hierarchy. It is shown that special solutions of
the selfsimilar reduction of the KdV hierarchy are determined via the transcendents of the first Painlevé hierarchy. Rational solutions are expressed by means of the Yablonskii–Vorob’ev polynomials. The map of the solutions for the second Painlevé hierarchy into the solutions for the selfsimilar reduction of the KdV hierarchy is illustrated using the Miura transformation. Lax pairs for equations of the hierarchy for the Yablonskii–Vorob’ev polynomial are discussed. Special solutions to the hierarchy for the Yablonskii–Vorob’ev polynomials are given. Some other hierarchies with properties of the Painlevé hierarchies are presented. The list of nonlinear differential equations whose general solutions are expressed in terms of nonclassical functions is extended.

Kilin A. A., Artemova E. M.
Integrability and Chaos in Vortex Lattice Dynamics
Abstract
This paper is concerned with the problem of the interaction of vortex lattices, which
is equivalent to the problem of the motion of point vortices on a torus. It is shown that the
dynamics of a system of two vortices does not depend qualitatively on their strengths. Steadystate
configurations are found and their stability is investigated. For two vortex lattices it is
also shown that, in absolute space, vortices move along closed trajectories except for the case
of a vortex pair. The problems of the motion of three and four vortex lattices with nonzero
total strength are considered. For three vortices, a reduction to the level set of first integrals
is performed. The nonintegrability of this problem is numerically shown. It is demonstrated
that the equations of motion of four vortices on a torus admit an invariant manifold which
corresponds to centrally symmetric vortex configurations. Equations of motion of four vortices
on this invariant manifold and on a fixed level set of first integrals are obtained and their
nonintegrability is numerically proved.

Fedonyuk V., Tallapragada P.
The Dynamics of a Chaplygin Sleigh with an Elastic Internal Rotor
Abstract
In this paper the dynamics of a Chaplygin sleigh like system are investigated. The
system consists a of a Chaplygin sleigh with an internal rotor connected by a torsional spring,
which is possibly nonHookean. The problem is motivated by applications in robotics, where
the motion of a nonholonomic system is sought to be controlled by modifying or tuning the
stiffness associated with some degrees of freedom of the system. The elastic potential modifies
the dynamics of the system and produces two possible stable paths in the plane, a straight
line and a circle, each of which corresponds to fixed points in a reduced velocity space. Two
different elastic potentials are considered in this paper, a quadratic potential and a Duffing like
quartic potential. The stiffness of the elastic element, the relative inertia of the main body and
the internal rotor and the initial energy of the system are all bifurcation parameters. Through
numerics, we investigate the codimensionone bifurcations of the fixed points while holding
all the other bifurcation parameters fixed. The results show the possibility of controlling the
dynamics of the sleigh and executing different maneuvers by tuning the stiffness of the spring.
