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# Volume 27, Number 6, 2022 Alexey Borisov Memorial Volume

 Bizyaev I. A.,  Mamaev I. S. Permanent Rotations in Nonholonomic Mechanics. Omnirotational Ellipsoid Abstract This paper is concerned with the study of permanent rotations of a rigid body rolling without slipping on a horizontal plane (i. e., the velocity of the point of contact of the ellipsoid with the plane is zero). By permanent rotations we will mean motions of a rigid body on a horizontal plane such that the angular velocity of the body remains constant and the point of contact does not change its position. A more detailed analysis is made of permanent rotations of an omnirotational ellipsoid whose characteristic feature is the possibility of permanent rotations about any point of its surface. Keywords: nonholonomic mechanics, poincare map, stability, permanent rotations Citation: Bizyaev I. A.,  Mamaev I. S., Permanent Rotations in Nonholonomic Mechanics. Omnirotational Ellipsoid, Regular and Chaotic Dynamics, 2022, vol. 27, no. 6, pp. 587-612 DOI:10.1134/S1560354722060016
 Grines V. Z.,  Medvedev V. S.,  Zhuzhoma E. V. On the Topological Structure of Manifolds Supporting Axiom A Systems Abstract Let $M^n$, $n\geqslant 3$, be a closed orientable $n$-manifold and $\mathbb{G}(M^n)$ the set of A-diffeomorp\-hisms $f: M^n\to M^n$ whose nonwandering set satisfies the following conditions: $(1)$ each nontrivial basic set of the nonwandering set is either an orientable codimension one expanding attractor or an orientable codimension one contracting repeller; $(2)$ the invariant manifolds of isolated saddle periodic points intersect transversally and codimension one separatrices of such points can intersect only one-dimensional separatrices of other isolated periodic orbits. We prove that the ambient manifold $M^n$ is homeomorphic to either the sphere $\mathbb S^n$ or the connected sum of $k_f \geqslant 0$ copies of the torus $\mathbb T^n$, $\eta_f\geqslant 0$ copies of $\mathbb S^{n-1}\times \mathbb S^1$ and $l_f\geqslant 0$ simply connected manifolds $N^n_1, \dots, N^n_{l_f}$ which are not homeomorphic to the sphere. Here $k_f\geqslant 0$ is the number of connected components of all nontrivial basic sets, $\eta_{f}=\frac{\kappa_f}{2} -k_f+\frac{\nu_f - \mu_f +2}{2},$ $\kappa_f\geqslant 0$ is the number of bunches of all nontrivial basic sets, $\mu_f\geqslant 0$ is the number of sinks and sources, $\nu_f\geqslant 0$ is the number of isolated saddle periodic points with Morse index $1$ or $n-1$, $0\leqslant l_f\leqslant \lambda_f$, $\lambda_f\geqslant 0$ is the number of all periodic points whose Morse index does not belong to the set $\{0,1,n-1,n\}$ of diffeomorphism $f$. Similar statements hold for gradient-like flows on $M^n$. In this case there are no nontrivial basic sets in the nonwandering set of a flow. As an application, we get sufficient conditions for the existence of heteroclinic intersections and periodic trajectories for Morse – Smale flows. Keywords: Decomposition of manifolds, axiom A systems, Morse – Smale systems, heteroclinic intersections Citation: Grines V. Z.,  Medvedev V. S.,  Zhuzhoma E. V., On the Topological Structure of Manifolds Supporting Axiom A Systems, Regular and Chaotic Dynamics, 2022, vol. 27, no. 6, pp. 613-628 DOI:10.1134/S1560354722060028
 Pires L. Morse – Smale Inequalities and Chafee – Infante Attractors Abstract In this paper, we are concerned with the shape of the attractor $\mathcal{A}^\lambda$ of the scalar Chafee – Infante equation. We construct a Morse – Smale vector field in the disk $\mathbb{D}^k$ topologically equivalent to infinite-dimensional dynamics of the Chafee – Infante equation. As a consequence, we obtain geometric properties of $\mathcal{A}^\lambda$ using the Morse – Smale inequalities. Keywords: Morse – Smale systems, Chafee – Infante equation, Morse inequalities Citation: Pires L., Morse – Smale Inequalities and Chafee – Infante Attractors, Regular and Chaotic Dynamics, 2022, vol. 27, no. 6, pp. 629-646 DOI:10.1134/S156035472206003X
 Gonchenko S. V.,  Safonov K. A.,  Zelentsov N. G. Antisymmetric Diffeomorphisms and Bifurcations of a Double Conservative Hénon Map Abstract We propose a new method for constructing multidimensional reversible maps by only two input data: a diffeomorphism $T_1$ and an involution $h$, i.e., a map (diffeomorphism) such that $h^2 = Id$. We construct the desired reversible map $T$ in the form $T = T_1\circ T_2$, where $T_2 = h\circ T_1^{-1}\circ h$. We also discuss how this method can be used to construct normal forms of Poincaré maps near mutually symmetric pairs of orbits of homoclinic or heteroclinic tangencies in reversible maps. One of such normal forms, as we show, is a two-dimensional double conservative Hénon map $H$ of the form $\bar x = M + cx - y^2; \ y = M + c\bar y - \bar x^2$. We construct this map by the proposed method for the case when $T_1$ is the standard Hénon map and the involution $h$ is $h: (x,y) \to (y,x)$. For the map $H$, we study bifurcations of fixed and period-2 points, among which there are both standard bifurcations (parabolic, period-doubling and pitchfork) and singular ones (during transition through $c=0$). Keywords: reversible diffeomorphism, parabolic bifurcation, period-doubling bifurcation Citation: Gonchenko S. V.,  Safonov K. A.,  Zelentsov N. G., Antisymmetric Diffeomorphisms and Bifurcations of a Double Conservative Hénon Map, Regular and Chaotic Dynamics, 2022, vol. 27, no. 6, pp. 647-667 DOI:10.1134/S1560354722060041
 Markeev A. P. On the Stability of Exact Subharmonic Solutions of the Duffing Equation Abstract This paper is concerned with the classical Duffing equation which describes the motion of a nonlinear oscillator with an elastic force that is odd with respect to the value of deviation from its equilibrium position, and in the presence of an external periodic force. The equation depends on three dimensionless parameters. When they satisfy some relation, the equation admits exact periodic solutions with a period that is a multiple of the period of external forcing. These solutions can be written in explicit form without using series. The paper studies the nonlinear problem of the stability of these periodic solutions. The study is based on the classical Lyapunov methods, methods of KAM theory for Hamiltonian systems and the computer algorithms for analysis of area-preserving maps. None of the parameters of the Duffing equation is assumed to be small. Keywords: nonlinear oscillations, stability, Duffing equation Citation: Markeev A. P., On the Stability of Exact Subharmonic Solutions of the Duffing Equation, Regular and Chaotic Dynamics, 2022, vol. 27, no. 6, pp. 668-679 DOI:10.1134/S1560354722060053
 Kudryashov N. A. Embedded Solitons of the Generalized Nonlinear Schrödinger Equation with High Dispersion Abstract The family of generalized Schrödinger equations is considered with the Kerr nonlinearity. The partial differential equations are not integrable by the inverse scattering transform and new solutions of this family are sought taking into account the traveling wave reduction. The compatibility of the overdetermined system of equations is analyzed and constraints for parameters of equations are obtained. A modification of the simplest equation method for finding embedded solitons is presented. A block diagram for finding a solution to the nonlinear ordinary differential equation is given. The theorem on the existence of bright solitons for differential equations of any order with Kerr nonlinearity of the family considered is proved. Exact solutions of embedded solitons described by fourth-, sixth-, eighth and tenthorder equations are found using the modified algorithm of the simplest equation method. New solutions for embedded solitons of generalized nonlinear Schrödinger equations with several extremes are obtained. Keywords: generalized Schrödinger equation, optical soliton, embedded soliton, simplest equation method, exact solution Citation: Kudryashov N. A., Embedded Solitons of the Generalized Nonlinear Schrödinger Equation with High Dispersion, Regular and Chaotic Dynamics, 2022, vol. 27, no. 6, pp. 680-696 DOI:10.1134/S1560354722060065
 Kuptsov P. V. Synchronization and Bistability of Two Uniaxial Spin-Transfer Oscillators with Field Coupling Abstract A spin-transfer oscillator is a nanoscale device demonstrating self-sustained precession of its magnetization vector whose length is preserved. Thus, the phase space of this dynamical system is limited by a three-dimensional sphere. A generic oscillator is described by the Landau – Lifshitz – Gilbert – Slonczewski equation, and we consider a particular case of uniaxial symmetry when the equation yet experimentally relevant is reduced to a dramatically simple form. The established regime of a single oscillator is a purely sinusoidal limit cycle coinciding with a circle of sphere latitude (assuming that points where the symmetry axis passes through the sphere are the poles). On the limit cycle the governing equations become linear in two oscillating magnetization vector components orthogonal to the axis, while the third one along the axis remains constant. In this paper we analyze how this effective linearity manifests itself when two such oscillators are mutually coupled via their magnetic fields. Using the phase approximation approach, we reveal that the system can exhibit bistability between synchronized and nonsynchronized oscillations. For the synchronized one the Adler equation is derived, and the estimates for the boundaries of the bistability area are obtained. The twodimensional slices of the basins of attraction of the two coexisting solutions are considered. They are found to be embedded in each other, forming a series of parallel stripes. Charts of regimes and charts of Lyapunov exponents are computed numerically. Due to the effective linearity the overall structure of the charts is very simple; no higher-order synchronization tongues except the main one are observed. Keywords: uniaxial spin-transfer oscillators, mutual synchronization, bistability Citation: Kuptsov P. V., Synchronization and Bistability of Two Uniaxial Spin-Transfer Oscillators with Field Coupling, Regular and Chaotic Dynamics, 2022, vol. 27, no. 6, pp. 697-712 DOI:10.1134/S1560354722060077
 Artemova E. M.,  Kilin A. A. Dynamics of Two Vortex Rings in a Bose – Einstein Condensate Abstract In this paper, we consider the dynamics of two interacting point vortex rings in a Bose – Einstein condensate. The existence of an invariant manifold corresponding to vortex rings is proved. Equations of motion on this invariant manifold are obtained for an arbitrary number of rings from an arbitrary number of vortices. A detailed analysis is made of the case of two vortex rings each of which consists of two point vortices where all vortices have same topological charge. For this case, partial solutions are found and a complete bifurcation analysis is carried out. It is shown that, depending on the parameters of the Bose – Einstein condensate, there are three different types of bifurcation diagrams. For each type, typical phase portraits are presented. Keywords: Bose – Einstein condensate, point vortices, vortex rings, bifurcation analysis Citation: Artemova E. M.,  Kilin A. A., Dynamics of Two Vortex Rings in a Bose – Einstein Condensate, Regular and Chaotic Dynamics, 2022, vol. 27, no. 6, pp. 713-732 DOI:10.1134/S1560354722060089
 Zhang D.,  Qu R. Persistence of Multiscale Degenerate Invariant Tori for Reversible Systems with Multiscale Degenerate Equilibrium Points Abstract In this paper, we focus on the persistence of degenerate lower-dimensional invariant tori with a normal degenerate equilibrium point in reversible systems. Based on the Herman method and the topological degree theory, it is proved that if the frequency mapping has nonzero topological degree and the frequency $\omega_0$ satisfies the Diophantine condition, then the lower-dimensional invariant torus with the frequency $\omega_0$ persists under sufficiently small perturbations. Moreover, the above result can also be obtained when the reversible system is Gevrey smooth. As some applications, we apply our theorem to some specific examples to study the persistence of multiscale degenerate lower-dimensional invariant tori with prescribed frequencies. Keywords: Reversible systems, KAM iteration, topological degree, degenerate lower-dimensional tori, degenerate equilibrium points Citation: Zhang D.,  Qu R., Persistence of Multiscale Degenerate Invariant Tori for Reversible Systems with Multiscale Degenerate Equilibrium Points, Regular and Chaotic Dynamics, 2022, vol. 27, no. 6, pp. 733-756 DOI:10.1134/S1560354722060090

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