 EDITORINCHIEF
 Honorary Editor
 Editorial board

 Passed away
Condolence note
Professor Alexey V. Borisov, EditorinChief of the journal RJND, suddenly passed away at the age of 55 on January 24, 2021.

Vol. 16, no. 4
Kozhevnikov I. F.
Abstract
We investigate a special case of vibrations of a loaded tire rolling at constant speed. A previously
proposed analytical model of a radial tire is considered. The surface of the tire is a flexible
tread combined with elastic sidewalls. In the undeformed state, the tread is a circular cylinder.
The tread is reinforced with inextensible cords. The tread is the part of the tire that makes
actual contact with the ground plane. In the undeformed state, the sidewalls are represented by
parts of two tori and consist of incompressible rubber described by the Mooney –Rivlin model.
The previously obtained partial differential equation which describes the tire radial inplane vibrations
about steadystate regime of rolling is investigated. Analyzing the discriminant of the
quartic polynomial, which is the function of the frequency of the tenth degree and the function of
the angular velocity of the sixth degree, the rare case of a root of multiplicity three is discovered.
The angular velocity of rotation, the tire speed and the natural frequency, corresponding to
this case, are determined analytically. The mode shape of vibration in the neighborhood of the
singular point is determined analytically.

Lim C. C.
Abstract
A new energyenstrophy model for the equilibrium statistical mechanics of barotropic flow
on a sphere is introduced and solved exactly for phase transitions to quadrupolar vortices when
the kinetic energy level is high. Unlike the Kraichnan theory, which is a Gaussian model, we
substitute a microcanonical enstrophy constraint for the usual canonical one, a step which is based
on sound physical principles. This yields a spherical model with zero total circulation, a microcanonical
enstrophy constraint and a canonical constraint on energy, with angular momentum fixed
to zero. A closedform solution of this spherical model, obtained by the Kac – Berlin method
of steepest descent, provides critical temperatures and amplitudes of the symmetrybreaking
quadrupolar vortices. This model and its results differ from previous solvable models for related
phenomena in the sense that they are not based on a meanfield assumption.

Rashkovskiy S. A.
Abstract
It is believed that thermodynamic laws are associated with random processes occurring
in the system and, therefore, deterministic mechanical systems cannot be described within the
framework of the thermodynamic approach. In this paper, we show that thermodynamics (or,
more precisely, a thermodynamicallylike description) can be constructed even for deterministic
Hamiltonian systems, for example, systems with only one degree of freedom. We show that
for such systems it is possible to introduce analogs of thermal energy, temperature, entropy,
Helmholtz free energy, etc., which are related to each other by the usual thermodynamic relations.
For the Hamiltonian systems considered, the first and second laws of thermodynamics are
rigorously derived, which have the same form as in ordinary (molecular) thermodynamics. It is
shown that for Hamiltonian systems it is possible to introduce the concepts of a thermodynamic
state, a thermodynamic process, and thermodynamic cycles, in particular, the Carnot cycle,
which are described by the same relations as their usual thermodynamic analogs.

Bardin B. S.
Abstract
A method is presented of constructing a nonlinear canonical change of variables which
makes it possible to introduce local coordinates in a neighborhood of periodic motions of an
autonomous Hamiltonian system with two degrees of freedom. The problem of the orbital stability
of pendulumlike oscillations of a heavy rigid body with a fixed point in the Bobylev – Steklov
case is discussed as an application. The nonlinear analysis of orbital stability is carried out
including terms through degree six in the expansion of the Hamiltonian function in a neighborhood
of the unperturbed periodic motion. This makes it possible to draw rigorous conclusions
on orbital stability for the parameter values corresponding to degeneracy of terms of degree four
in the normal form of the Hamiltonian function of equations of perturbed motion.

Grines V. Z., Kruglov E. V., Pochinka O. V.
Abstract
This paper is devoted to the topological classification of structurally stable diffeomorphisms
of the twodimensional torus whose nonwandering set consists of an orientable onedimensional
attractor and finitely many isolated source and saddle periodic points, under the assumption
that the closure of the union of the stable manifolds of isolated periodic points consists of simple
pairwise nonintersecting arcs. The classification of onedimensional basis sets on surfaces has
been exhaustively obtained in papers by V. Grines. He also obtained a classification of some
classes of structurally stable diffeomorphisms of surfaces using combined algebrageometric invariants.
In this paper, we distinguish a class of diffeomorphisms that admit purely algebraic
differentiating invariants.

Markeev A. P., Chekhovskaya T. N.
Abstract
The points of suspension of two identical pendulums moving in a homogeneous gravitational
field are located on a horizontal beam performing harmonic oscillations of small amplitude along
a fixed horizontal straight line passing through the points of suspension of the pendulums. The
pendulums are connected to each other by a spring of low stiffness. It is assumed that the partial
frequency of small oscillations of each pendulum is exactly equal to the frequency of horizontal
oscillations of the beam. This implies that a multiple resonance occurs in this problem, when the
frequency of external periodic action on the system is equal simultaneously to two its frequencies
of small (linear) natural oscillations. This paper solves the nonlinear problem of the existence and
stability of periodic motions of pendulums with a period equal to the period of oscillations of the
beam. The study uses the classical methods due to Lyapunov and Poincaré, KAM (Kolmogorov,
Arnold and Moser) theory, and algorithms of computer algebra. The existence and uniqueness of the periodic motion of pendulums are shown, its analytic representation as a series is obtained, and its stability is investigated. For sufficiently small oscillation amplitudes of the beam, depending on the value of the dimensionless parameter which characterizes the stiffness of the spring connecting the pendulums, the found periodic motion is either Lyapunov unstable or stable for most (in the sense of Lebesgue measure) initial conditions or formally stable (stable in an arbitrarily large, but finite, nonlinear approximation). 
Damasceno J. G., Miranda J. A., Perona L. G.
Abstract
In this work we study the dynamical behavior of Tonelli Lagrangian systems defined on
the tangent bundle of the torus $\mathbb{T}^2=\mathbb{R}^2/\mathbb{Z}^2$. We prove that the Lagrangian flow restricted to
a high energy level $E_{L}^{1}(c)$ (i.e., $c > c_0(L)$) has positive topological entropy if the flow satisfies the
KupkaSmale property in $E_{L}^{1}(c)$ (i.e., all closed orbits with energy c are hyperbolic or elliptic and
all heteroclinic intersections are transverse on $E_{L}^{1}(c)$). The proof requires the use of wellknown
results from Aubry – Mather theory.

Guha P., Garai S., Choudhury A. G.
Abstract
Recently Sinelshchikov et al. [1] formulated a Lax representation for a family of nonautonomous
secondorder differential equations. In this paper we extend their result and obtain
the Lax pair and the associated first integral of a nonautonomous version of the Levinson –
Smith equation. In addition, we have obtained Lax pairs and first integrals for several equations
of the Painlevé – Gambier list, namely, the autonomous equations numbered XII, XVII, XVIII,
XIX, XXI, XXII, XXIII, XXIX, XXXII, XXXVII, XLI, XLIII, as well as the nonautonomous
equations Nos. XV and XVI in Ince’s book.

Ndawa Tangue B.
Abstract
We consider orderpreserving $C^3$ circle maps with a flat piece, irrational rotation number
and critical exponents $(l_1, l_2)$. We detect a change in the geometry of the system. For $(l_1, l_2) \in [1, 2]^2$ the geometry is degenerate and becomes bounded for $(l_1, l_2) \in [2, \infty)^2 \backslash \{(2, 2)\}$. When the rotation number is of the form $[abab \ldots]$; for some $a, b \in \mathbb{N}^*$, the geometry is bounded for $(l_1, l_2)$ belonging above a curve defined on $]1, +\infty[^2$. As a consequence, we estimate the Hausdorff dimension of the nonwandering set $K_f=\mathcal{S}^1\backslash \bigcup^\infty_{i=0}f^{i}(U)$. Precisely, the Hausdorff dimension of this set is equal to zero when the geometry is degenerate and it is strictly positive when the geometry is bounded. 