Volume 28, Number 2
Volume 28, Number 2, 2023
Medvedev V. S., Zhuzhoma E. V.
Abstract
We introduce Smale Ahomeomorphisms that include regular, semichaotic, chaotic,
and superchaotic homeomorphisms of a topological $n$manifold $M^n$, $n\geqslant 2$. Smale Ahomeomorphisms
contain axiom A diffeomorphisms (in short, Adiffeomorphisms) provided that $M^n$
admits a smooth structure. Regular Ahomeomorphisms contain all Morse–Smale diffeomorphisms,
while semichaotic and chaotic Ahomeomorphisms contain Adiffeomorphisms with
trivial and nontrivial basic sets. Superchaotic Ahomeomorphisms contain Adiffeomorphisms
whose basic sets are nontrivial. The reason to consider Smale Ahomeomorphisms instead of
Adiffeomorphisms may be attributed to the fact that it is a good weakening of nonuniform
hyperbolicity and pseudohyperbolicity, a subject which has already seen an immense number
of applications.
We describe invariant sets that determine completely the dynamics of regular, semichaotic, and chaotic Smale Ahomeomorphisms. This allows us to get necessary and sufficient conditions of conjugacy for these Smale Ahomeomorphisms (in particular, for all Morse–Smale diffeomorphisms). We apply these necessary and sufficient conditions for structurally stable surface diffeomorphisms with an arbitrary number of expanding attractors. We also use these conditions to obtain a complete classification of Morse–Smale diffeomorphisms on projectivelike manifolds. 
Ronzhina M. I., Manita L. A.
Abstract
In this paper, we consider the minimum time problem for a space rocket whose
dynamics is given by a controlaffine system with drift. The admissible control set is a disc. We
study extremals in the neighbourhood of singular points of the second order. Our approach is
based on applying the method of a descending system of Poisson brackets and the
Zelikin–Borisov method for resolution of singularities to the Hamiltonian system of Pontryagin’s
maximum principle. We show that in the neighbourhood of any singular point there is a family
of spirallike solutions of the Hamiltonian system that enter the singular point in a finite time,
while the control performs an infinite number of rotations around the circle.

Zhong X., Zhao J., Yu K., Xu M.
Abstract
This paper presents the stability of resonant rotation of a symmetric gyrostat under
third and fourthorder resonances, whose center of mass moves in an elliptic orbit in a central
Newtonian gravitational field. The resonant rotation is a special planar periodic motion of
the gyrostat about its center of mass, i. e., the body performs one rotation in absolute space
during two orbital revolutions of its center of mass. The equations of motion of the gyrostat
are derived as a periodic Hamiltonian system with three degrees of freedom and a constructive
algorithm based on a symplectic map is used to calculate the coefficients of the normalized
Hamiltonian. By analyzing the Floquet multipliers of the linearized equations of perturbed
motion, the unstable region of the resonant rotation and the region of stability in the firstorder
approximation are determined in the dimensionless parameter plane. In addition, the thirdand
fourthorder resonances are obtained in the linear stability region and further nonlinear
stability analysis is performed in the third and fourthorder resonant cases.

Lopes L. M., Grácio C., Fernandes S., FournierPrunaret D.
Abstract
The chaotic behaviour of dynamical systems can be suppressed if we couple them
in some way. In order to do that, the coupling strengths must assume particular values. We
illustrate it for the situation that leads to a fixed point behaviour, using two types of couplings
corresponding either to a diffusive interaction or a migrative one. For both of them, we present
strategies that easily calculate coupling strengths that suppress the chaotic behaviour. We
analyse the particular situation of these couplings that consists in a symmetric one and we
propose a strategy that provides the suppression of the chaotic evolution of a population.

Ivanov A. V.
Abstract
We consider a skew product $F_{A} = (\sigma_{\omega}, A)$ over irrational rotation $\sigma_{\omega}(x) = x + \omega$ of a circle $\mathbb{T}^{1}$. It is supposed that the transformation $A: \mathbb{T}^{1} \to SL(2, \mathbb{R})$
which is a $C^{1}$map has the form $A(x) = R\big(\varphi(x)\big) Z\big(\lambda(x)\big)$, where $R(\varphi)$ is a rotation in $\mathbb{R}^{2}$ through the angle $\varphi$ and $Z(\lambda)= \text{diag}\{\lambda, \lambda^{1}\}$ is a diagonal matrix. Assuming that $\lambda(x) \geqslant \lambda_{0} > 1$ with a sufficiently large constant $\lambda_{0}$ and the function $\varphi$
is such that $\cos \varphi(x)$ possesses only simple zeroes, we study hyperbolic properties of
the cocycle generated by $F_{A}$. We apply the critical set method to show that, under some
additional requirements on the derivative of the function $\varphi$, the secondary collisions compensate weakening of the hyperbolicity due to primary collisions and the cocycle generated by $F_{A}$ becomes uniformly hyperbolic
in contrast to the case where secondary collisions can be partially eliminated.

Takano H.
Abstract
This paper investigates the dynamics of a toy known as the chatter ring. Specifically,
it examines the mechanism by which the small ring rotates around the large ring, the mechanism
by which the force from the large ring provides torque to the small ring, and whether the motion
of the small ring is the same as that of a hula hoop. The dynamics of a chatter ring has been
investigated in previous work [13–15]; however, a detailed analysis has not yet been performed.
Thus, to understand the mechanisms described above, the equations of motion and constraint
conditions are obtained, and an analysis of the motion is performed. To simplify the problem,
a model consisting of a straight rod and a washer ring is analyzed under the noslip condition.
The motion of a washer has two modes: the one point of contact (1PC) mode and two points
of contact (2PC) mode. The motion of the small ring of the chatter ring is similar to that of a
washer in the 2PC mode, whereas the motion of a hula hoop is similar to that of a washer in
the 1PC mode. The analysis indicates that the motion of a washer with two points of contact
is equivalent to free fall motion. However, in practice, the velocity reaches a constant value
through energy dissipation. The washer rotates around an axis that passes through the two
points of contact. The components of the forces exerted by the rod at the points of contact
that are normal to the plane of the washer provide rotational torque acting at the center of
mass. The components of the forces parallel to the horizontal plane are centripetal forces, which
induce the circular motion of the center of mass.
