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# Vol. 16, no. 4, 2020

 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 in-plane vibrations about steady-state 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. Keywords: radial tire, analytical model, rolling, modal analysis, vibrations, multiple roots Citation: Kozhevnikov I. F., Another Special Case of Vibrations of a Rolling Tire, Rus. J. Nonlin. Dyn., 2020, Vol. 16, no. 4, pp. 531-542 DOI:10.20537/nd200401
 Lim C. C. Abstract A new energy-enstrophy 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 closed-form solution of this spherical model, obtained by the Kac – Berlin method of steepest descent, provides critical temperatures and amplitudes of the symmetry-breaking 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 mean-field assumption. Keywords: energy-enstrophy theory, long-range spherical model, phase transition, rotating atmospheres Citation: Lim C. C., Phase Transition to Quadrupolar Vortices in a Spherical Model of the Energy-Enstrophy Theory — Exact Solution, Rus. J. Nonlin. Dyn., 2020, Vol. 16, no. 4, pp. 543-555 DOI:10.20537/nd200402
 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 thermodynamically-like 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. Keywords: Hamiltonian system, adiabatic invariants, thermodynamics, temperature, heat, entropy, thermodynamic processes, the first and second laws of thermodynamics Citation: Rashkovskiy S. A., Hamiltonian Thermodynamics, Rus. J. Nonlin. Dyn., 2020, Vol. 16, no. 4, pp. 557-580 DOI:10.20537/nd200403
 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 pendulum-like 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. Keywords: rigid body, rotations, oscillations, orbital stability, Hamiltonian system, local coordinates, normal form Citation: Bardin B. S., On a Method of Introducing Local Coordinates in the Problem of the Orbital Stability of Planar Periodic Motions of a Rigid Body, Rus. J. Nonlin. Dyn., 2020, Vol. 16, no. 4, pp. 581-594 DOI:10.20537/nd200404
 Grines V. Z.,  Kruglov E. V.,  Pochinka O. V. Abstract This paper is devoted to the topological classification of structurally stable diffeomorphisms of the two-dimensional torus whose nonwandering set consists of an orientable one-dimensional 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 one-dimensional 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 algebra-geometric invariants. In this paper, we distinguish a class of diffeomorphisms that admit purely algebraic differentiating invariants. Keywords: A-diffeomorphisms of a torus, topological classification, orientable attractor Citation: Grines V. Z.,  Kruglov E. V.,  Pochinka O. V., The Topological Classification of Diffeomorphisms of the Two-Dimensional Torus with an Orientable Attractor, Rus. J. Nonlin. Dyn., 2020, Vol. 16, no. 4, pp. 595-606 DOI:10.20537/nd200405
 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). Keywords: nonlinear oscillations, resonance, stability, canonical transformations Citation: Markeev A. P.,  Chekhovskaya T. N., On Nonlinear Oscillations and Stability of Coupled Pendulums in the Case of a Multiple Resonance, Rus. J. Nonlin. Dyn., 2020, Vol. 16, no. 4, pp. 607-623 DOI:10.20537/nd200406
 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 Kupka-Smale 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 well-known results from Aubry – Mather theory. Keywords: Tonelli Lagrangian system, Aubry – Mather theory, static classes Citation: Damasceno J. G.,  Miranda J. A.,   Perona L. G., A Note on Tonelli Lagrangian Systems on $\mathbb{T}^2$ with Positive Topological Entropy on a High Energy Level, Rus. J. Nonlin. Dyn., 2020, Vol. 16, no. 4, pp. 625-635 DOI:10.20537/nd200407
 Guha P.,  Garai S.,  Choudhury A. G. Abstract Recently Sinelshchikov et al. [1] formulated a Lax representation for a family of nonautonomous second-order differential equations. In this paper we extend their result and obtain the Lax pair and the associated first integral of a non-autonomous 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 non-autonomous equations Nos. XV and XVI in Ince’s book. Keywords: Lax representation, Liénard type equations, Painlevé – Gambier equations, first integrals Citation: Guha P.,  Garai S.,  Choudhury A. G., Lax Pairs and First Integrals for Autonomous and Non-Autonomous Differential Equations Belonging to the Painlevé – Gambier List, Rus. J. Nonlin. Dyn., 2020, Vol. 16, no. 4, pp. 637-650 DOI:10.20537/nd200408
 Ndawa Tangue B. Abstract We consider order-preserving $C^3$ 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. Keywords: circle map, irrational rotation number, flat piece, critical exponent, geometry, Hausdorff dimension Citation: Ndawa Tangue B., Cherry Maps with Different Critical Exponents: Bifurcation of Geometry, Rus. J. Nonlin. Dyn., 2020, Vol. 16, no. 4, pp. 651-672 DOI:10.20537/nd200409
 Abstract Citation: In memory of Professor Vadim S. Anishchenko, Rus. J. Nonlin. Dyn., 2020, Vol. 16, no. 4, pp. 673-674 DOI:10.20537/nd200410

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