- EDITOR-IN-CHIEF
- DEPUTY EDITOR-IN-CHIEF
- Editorial board
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- Passed Away
Volume 30, Number 1
Special Issue: In Honor of the 90th Anniversary of L. P. Shilnikov: Part I (Guest Editors: Sergey Gonchenko, Andrey Shilnikov, Dmitry Turaev, and Alexey Kazakov)
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Gonchenko S. V., Malkin M. I., Turaev D. V.
Abstract
This issue of RCD is dedicated to the outstanding Russian mathematician Leonid Pavlovich
Shilnikov, a leading figure in the theory of dynamical systems, one of the founders of the
mathematical theory of dynamical chaos.
The impact of the Shilnikov work on nonlinear dynamics and its applications is indeed enormous.
We mention only a few of his groundbreaking scientific achievements: the theory of global
bifurcations of multidimensional dynamical systems, the discovery of spiral chaos, the theory of
Lorenz-like attractors, the mathematical theory of transition from synchronization to chaos, the
theory of homoclinic chaos, among many others.
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Gonchenko S. V., Gordeeva O. V.
Abstract
We consider a one-parameter family $f_\mu$ of multidimensional diffeomorphisms such that for $\mu=0$ the diffeomorphism $f_0$ has a transversal homoclinic orbit to a nonhyperbolic fixed point of arbitrary finite order $n\geqslant 1$ of degeneracy, and for $\mu>0$ the fixed point becomes a hyperbolic saddle. In the paper, we give a complete description of the structure of the set $N_\mu$ of all orbits entirely lying in a sufficiently small fixed neighborhood of the homoclinic orbit. Moreover, we show that for $\mu\geqslant 0$ the set $N_\mu$ is hyperbolic (for $\mu=0$ it is nonuniformly hyperbolic) and the dynamical system $f_\mu\bigl|_{N_\mu}$ (the restriction of $f_\mu$ to $N_\mu$) is topologically conjugate to a certain nontrivial subsystem of the topological Bernoulli scheme of two symbols.
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Malkin M. I., Safonov K. A.
Abstract
This paper deals with the problem of smoothness of the stable invariant foliation
for a homoclinic bifurcation with a neutral saddle in symmetric systems of differential equations.
We give an improved sufficient condition for the existence of an invariant smooth foliation on a
cross-section transversal to the stable manifold of the saddle. It is shown that the smoothness
of the invariant foliation depends on the gap between the leading stable eigenvalue of the saddle
and other stable eigenvalues. We also obtain an equation to describe the one-dimensional factor
map, and we study the renormalization properties of this map. The improved information on
the smoothness of the foliation and the factor map allows one to extend Shilnikov’s results on
the birth of Lorenz attractors under the bifurcation considered.
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Glyzin S. D., Kolesov A. Y.
Abstract
An arbitrary diffeomorphism $f$ of class $C^1$ acting from an open subset $U$ of Riemannian manifold $M$ of dimension $m,$ $m\ge 2,$ into $f(U)\subset M$ is considered.
Let $A$ be a compact subset of $U$ invariant for $f,$ i.e. $f(A)=A.$
Various sufficient conditions are proposed under which $A$ is a hyperbolic set of the diffeomorphism $f.$
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Emelianova A. A., Nekorkin V. I.
Abstract
This paper provides an overview of the results obtained from the study of adaptive
dynamical networks of Kuramoto oscillators with higher-order interactions. The main focus
is on results in the field of synchronization and collective chaotic dynamics. Identifying the
dynamical mechanisms underlying the synchronization of oscillator ensembles with higher-order
interactions may contribute to further advances in understanding the work of some complex
systems such as the neural networks of the brain.
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Bolotin S. V.
Abstract
We study the dynamics of a multidimensional slow-fast Hamiltonian system in a
neighborhood of the slow manifold under the assumption that the frozen system has a hyperbolic
equilibrium with complex simple leading eigenvalues and there exists a transverse homoclinic
orbit. We obtain formulas for the corresponding Shilnikov separatrix map and prove the
existence of trajectories in a neighborhood of the homoclinic set with a prescribed evolution of
the slow variables. An application to the 3 body problem is given.
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Medvedev V. S., Zhuzhoma E. V.
Abstract
We prove that an $n$-sphere $\mathbb{S}^n$, $n\geqslant 2$, admits structurally stable diffeomorphisms $\mathbb{S}^n\to\mathbb{S}^n$ with nonorientable expanding attractors of any topological dimension $d\in\{1,\ldots,[\frac{n}{2}]\}$ where $[x]$ is the integer part of $x$. In addition, any $n$-sphere $\mathbb{S}^n$, $n\geqslant 3$, admits axiom A diffeomorphisms $\mathbb{S}^n\to\mathbb{S}^n$ with orientable expanding attractors of any topological dimension $d\in\{1,\ldots,[\frac{n}{3}]\}$. We prove that an $n$-torus $\mathbb{T}^n$, $n\geqslant 2$, admits structurally stable diffeomorphisms $\mathbb{T}^n\to\mathbb{T}^n$ with orientable expanding attractors of any topological dimension $d\in\{1,\ldots,n-1\}$. We also prove that, given any closed $n$-manifold $M^n$, $n\geqslant 2$, and any $d\in\{1,\ldots,[\frac{n}{2}]\}$, there is an axiom A diffeomorphism $f: M^n\to M^n$ with a $d$-dimensional nonorientable expanding attractor. Similar statements hold for axiom A flows.
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Kashchenko A. A., Kashchenko S. A.
Abstract
The purpose of this work is to study small oscillations and oscillations with an
asymptotically large amplitude in nonlinear systems of two equations with delay, regularly
depending on a small parameter. We assume that the nonlinearity is compactly supported,
i. e., its action is carried out only in a certain finite region of phase space. Local oscillations
are studied by classical methods of bifurcation theory, and the study of nonlocal dynamics
is based on a special large-parameter method, which makes it possible to reduce the original
problem to the analysis of a specially constructed finite-dimensional mapping. In all cases,
algorithms for constructing the asymptotic behavior of solutions are developed. In the case of
local analysis, normal forms are constructed that determine the dynamics of the original system
in a neighborhood of the zero equilibrium state, the asymptotic behavior of the periodic solution
is constructed, and the question of its stability is answered. In studying nonlocal solutions,
one-dimensional mappings are constructed that make it possible to determine the behavior of
solutions with an asymptotically large amplitude. Conditions for the existence of a periodic
solution are found and its stability is investigated.
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Efremova L. S., Novozhilov D. A.
Abstract
In this paper we prove criteria of a $C^0$- $\Omega$-blowup in $C^1$-smooth skew products with a
closed set of periodic points on multidimensional cells and give examples of maps that admit such a $\Omega$-blowup.
Our method is based on the study of the properties of the set of chain-recurrent points. We also
prove that the set of weakly nonwandering points of maps under consideration coincides with
the chain-recurrent set, investigate the approximation (in the $C^0$-norm) and entropy properties
of $C^1$-smooth skew products with a closed set of periodic points.
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Meshcheryakov M. V., Zhukova N. I.
Abstract
Continuous actions of topological semigroups on products $X$ of an arbitrary family of topological spaces $X_i$, $i\in J,$ are studied. The relationship between the dynamical properties of semigroups acting on the factors $X_i$ and the same properties of the product of semigroups on the product $X$ of these spaces is investigated. We consider the following dynamical properties: topological transitivity, existence of a dense orbit, density of a union of minimal sets, and density of the set of points with closed orbits. The sensitive dependence on initial conditions is investigated for countable products of metric spaces. Various examples are constructed. In particular, on an infinite-dimen\-sio\-nal torus we have constructed a continual
family of chaotic semi\-group dynamical systems
that are pairwise topologi\-cal\-ly not conjugate by homeomorphisms preserving the structure of the
product of this torus.
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