Volume 27, Number 6
Volume 27, Number 6, 2022
Alexey Borisov Memorial Volume
Bizyaev I. A., Mamaev I. S.
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.

Grines V. Z., Medvedev V. S., Zhuzhoma E. V.
Abstract
Let $M^n$, $n\geqslant 3$, be a closed orientable $n$manifold and $\mathbb{G}(M^n)$ the set of Adiffeomorp\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 onedimensional 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 $\mathbb S^{n1}\times \mathbb S^1$ 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 $n1$, $0\leqslant l_f\leqslant \lambda_f$, $\lambda_f\geqslant 0$ is the number of all periodic points whose Morse index does not belong to the set $\{0,1,n1,n\}$ of diffeomorphism $f$. Similar statements hold for gradientlike 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.

Pires L.
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
infinitedimensional dynamics of the Chafee – Infante equation. As a consequence,
we obtain geometric properties of $\mathcal{A}^\lambda$ using the Morse – Smale inequalities.

Gonchenko S. V., Safonov K. A., Zelentsov N. G.
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 twodimensional 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 period2 points, among which there are both standard bifurcations (parabolic, perioddoubling and pitchfork) and singular ones (during transition through $c=0$).

Markeev A. P.
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 areapreserving maps. None of the parameters of
the Duffing equation is assumed to be small.

Kudryashov N. A.
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.

Kuptsov P. V.
Abstract
A spintransfer oscillator is a nanoscale device demonstrating selfsustained precession
of its magnetization vector whose length is preserved. Thus, the phase space of this
dynamical system is limited by a threedimensional 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 higherorder synchronization tongues except
the main one are observed.

Artemova E. M., Kilin A. A.
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.

Zhang D., Qu R.
Abstract
In this paper, we focus on the persistence of degenerate lowerdimensional 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 lowerdimensional 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 lowerdimensional invariant tori with prescribed frequencies.
