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Abstract
This issue of RJND is dedicated to the 80th birthdays of the remarkable Russian mathematicians
Lev Mikhailovich Lerman and Albert Dmitrievich Morozov.
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Gonchenko S. V., Morozov K. E.
Abstract
We give a review of scientific results of the remarkable Russian mathematician Albert
Dmitrievich Morozov who is the world’s recognized leader in the theory of nearly Hamiltonian
systems and one of the founders of the modern mathematical theory of synchronization in oscillatory
systems. This review was prepared in connection with the 80th birthday of A.D. Morozov
and the authors wish him all the best, good health and creative success.
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Lerman L. M.
Abstract
This text presents a loose journey on my activity over the scientific life on the background of
the rapid development of the theory of dynamical systems during the last 50 years. I intentionally
do not explain nor go into mathematical details, otherwise this would require too many pages.
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Morozov K. E., Morozov A. D.
Abstract
We consider nonconservative quasi-periodic perturbations of two-dimensional Hamiltonian
systems with nonmonotonic rotation. The peculiarity of these systems is that degenerate resonances
can take place. Special focus is on those systems for which the corresponding autonomous
perturbed system has a structurally stable limit cycle. If the cycle appears in the neighborhood
of a nonresonance phase curve, then it corresponds to an invariant torus in the initial system.
In this regard, the problem of synchronization arises when the invariant torus passes through
a resonance zone. In this paper, we distinguish a class of perturbations (the so-called parametric
perturbations) under which synchronization can be violated. Also, we introduce the concept of
generalized synchronization and give conditions for this type of synchronization to occur. As an
example, we study a Duffing-like equation with an asymmetric potential function.
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Glyzin S. D., Kashchenko S. A., Kosterin D. S.
Abstract
Spatially distributed integro-differential systems of equations with periodic boundary conditions
are considered. In applications, such systems arise as limiting ones for some nonlinear
fully coupled ensembles. The simplest critical cases of zero and purely imaginary eigenvalues in
the problem of stability of the zero equilibrium state are considered.
In these two situations, quasinormal forms are constructed, for which the question of the
existence of piecewise constant solutions is studied. In the case of a simple zero root, the
conditions for the stability of these solutions are determined. The existence of piecewise constant
solutions with more than one discontinuity point is shown. An algorithm for calculating solutions
of the corresponding boundary value problem by numerical methods is presented. A numerical
experiment is performed, confirming the analytical constructions.
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Bardakov V. G., Kozlovskaya T. A., Pochinka O. V.
Abstract
Knots naturally appear in continuous dynamical systems as flow periodic trajectories. However, discrete dynamical systems are also closely connected with the theory of knots and links. For example, for Pixton diffeomorphisms, the equivalence class of the Hopf knot, which is the orbit space of the unstable saddle separatrix in the manifold $\mathbb{S}^2\times \mathbb{S}^1$, is a complete invariant of the topological conjugacy of the system. In this paper we distinguish a class of three-dimensional Morse – Smale diffeomorphisms for which the complete invariant of topological conjugacy is the equivalence class of a link in $\mathbb{S}^2\times \mathbb{S}^1$.
We prove that, if $M$ is a link complement in $\mathbb{S}^3$, or a handlebody $H_g^{}$ of genus $g\geqslant 0$, or a closed, connected, orientable 3-manifold, then the set of equivalence classes of tame links in $M$ is countable. As a corollary, we prove that there exists a countable number of equivalence classes of tame links in $\mathbb{S}^2\times \mathbb{S}^1$. It is proved that any essential link can be realized by a diffeomorphism of the class under consideration.
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Dedaev R. A., Zhukova N. I.
Abstract
In this work, by a dynamical system we mean a pair $(S, \,X)$, where $S$ is either a pseudogroup of local diffeomorphisms, or a transformation group, or a smooth foliation of the manifold $X$. The groups of transformations can be both discrete and nondiscrete. We define the concepts of attractor and global attractor of the dynamical system $(S, \,X)$ and investigate the properties of attractors and the problem of the existence of attractors of dynamical systems $(S, \,X)$. Compactness of attractors and ambient manifolds is not assumed. A property of the dynamical system is called transverse if it can be expressed in terms of the orbit space or the leaf space (in the case of foliations). It is shown that the existence of an attractor of a dynamical system is a transverse property. This property is applied by us in proving two subsequent criteria for the existence of an attractor (and global attractor): for foliations of codimension $q$ on an $n$-dimensional manifold, $0 < q < n$, and for foliations covered by fibrations. A criterion for the existence of an attractor that is a minimal set for an arbitrary dynamical system is also proven. Various examples of both regular attractors and attractors of transformation groups that are fractals are constructed.
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Chilina E. E.
Abstract
The present paper is devoted to the study of the dynamics of mappings commuting with pseudo-Anosov surface homeomorphisms. It is proved that the centralizer of a pseudo-Anosov homeomorphism $P$ consists of pairwise nonhomotopic mappings, each of which is a composition of a power of the pseudo-Anosov mapping and a periodic homeomorphism. For periodic mappings commuting with $P$, it is proved that their number is finite and does not exceed the number $N_P^{}$, which is equal to the minimum among the number of all separatrices related to saddles of the same valency of $P$-invariant foliations. For a periodic homeomorphism $J$ lying in the centralizer of $P$, it is also shown that, if the period of a point is equal to half the period of the homeomorphism $J$, then this point is located in the complement of the separatrices of saddle singularities. If the period of the point is less than half the period of $J$, then this point is contained in the set of saddle singularities. In addition, it is proved that there exists a monomorphism from the group of periodic maps commuting with a pseudo-Anosov homeomorphism to the symmetric group of degree $N_P^{}$. Each permutation from the image of the monomorphism is represented as a product of disjoint cycles of the same length. Furthermore, it is deduced that a pseudo-Anosov homeomorphism with the trivial centralizer exists on each orientable closed surface of genus greater than $2$. As an application of the results related to the structure of the centralizer of pseudo-Anosov homeomorphisms to their topological classification, it is proved that there are no pairwise distinct homotopic conjugating mappings for topologically conjugated pseudo-Anosov homeomorphisms.
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