Evgeniy Zhuzhoma

    ul. Bolshaya Pecherskaya 25/12, Nizhni Novgorod, 603005, Russia
    National research University “Higher school of Economics”


    Medvedev V. S., Zhuzhoma E. V.
    We prove that, given any $n\geqslant 3$ and $2\leqslant q\leqslant n-1$, there is a closed $n$-manifold $M^n$ admitting a chaotic lamination of codimension $q$ whose support has the topological dimension ${n-q+1}$. For $n=3$ and $q=2$, such chaotic laminations can be represented as nontrivial 2-dimensional basic sets of axiom A flows on 3-manifolds. We show that there are two types of compactification (called casings) for a basin of a nonmixing 2-dimensional basic set by a finite family of isolated periodic trajectories. It is proved that an axiom A flow on every casing has repeller-attractor dynamics. For the first type of casing, the isolated periodic trajectories form a fibered link. The second type of casing is a locally trivial fiber bundle over a circle. In the latter case, we classify (up to neighborhood equivalence) such nonmixing basic sets on their casings with solvable fundamental groups. To be precise, we reduce the classification of basic sets to the classification (up to neighborhood conjugacy) of surface diffeomorphisms with one-dimensional basic sets obtained previously by V. Grines, R. Plykin and Yu. Zhirov [16, 28, 31].
    Keywords: chaotic lamination, basic set, axiom A flow
    Citation: Medvedev V. S., Zhuzhoma E. V.,  On a Classification of Chaotic Laminations which are Nontrivial Basic Sets of Axiom A Flows, Rus. J. Nonlin. Dyn., 2023, Vol. 19, no. 2, pp.  227-237
    Grines V. Z., Zhuzhoma E. V.
    Cantor Type Basic Sets of Surface $A$-endomorphisms
    2021, Vol. 17, no. 3, pp.  335-345
    The paper is devoted to an investigation of the genus of an orientable closed surface $M^2$ which admits $A$-endomorphisms whose nonwandering set contains a one-dimensional strictly invariant contracting repeller $\Lambda_r^{}$ with a uniquely defined unstable bundle and with an admissible boundary of finite type. First, we prove that, if $M^2$ is a torus or a sphere, then $M^2$ admits such an endomorphism. We also show that, if $ \Omega$ is a basic set with a uniquely defined unstable bundle of the endomorphism $f\colon M^2\to M^2$ of a closed orientable surface $M^2$ and $f$ is not a diffeomorphism, then $ \Omega$ cannot be a Cantor type expanding attractor. At last, we prove that, if $f\colon M^2\to M^2$ is an $A$-endomorphism whose nonwandering set consists of a finite number of isolated periodic sink orbits and a one-dimensional strictly invariant contracting repeller of Cantor type $\Omega_r^{}$ with a uniquely defined unstable bundle and such that the lamination consisting of stable manifolds of $\Omega_r^{}$ is regular, then $M^2$ is a two-dimensional torus $\mathbb{T}^2$ or a two-dimensional sphere $\mathbb{S}^2$.
    Keywords: $A$-endomorphism, regular lamination, attractor, repeller, strictly invariant set
    Citation: Grines V. Z., Zhuzhoma E. V.,  Cantor Type Basic Sets of Surface $A$-endomorphisms, Rus. J. Nonlin. Dyn., 2021, Vol. 17, no. 3, pp.  335-345
    Zhuzhoma E. V., Medvedev V. S., Isaenkova N. V.
    In this paper, using methods of Morse – Smale dynamical systems, we consider the topological structure of the magnetic field of regions of the photosphere for a point-charge model. For an arbitrary number of charges (regardless of their location), without assuming a potentiality of the field $\boldsymbol{\vec B}$ (and hence without applying specific formulas), we give estimates that connect the numbers of charges of a certain type with the numbers of null-points. For the boundary estimates, we describe the topological structure of the magnetic field. We present a bifurcation of the birth of a large number of separators.
    Keywords: dynamical Morse–Smale system, null-points, separator
    Citation: Zhuzhoma E. V., Medvedev V. S., Isaenkova N. V.,  On the topological structure of the magnetic field of regions of the photosphere, Rus. J. Nonlin. Dyn., 2017, Vol. 13, No. 3, pp.  399-412
    Grines V. Z., Gurevich E. Y., Zhuzhoma E. V., Zinina S. K.
    We obtain properties of three-dimensional phase space and dynamics of Morse–Smale diffeomorphism that led to existence of at least one heteroclinical curve in non-wandering set of the diffeomorphism. We apply this result to solve a problem of existence of separators in magnetic field of plasma.
    Keywords: Morse – Smale cascades, heteroclinic curves, mapping torus, locally trivial bundle, separators of magnetic field
    Citation: Grines V. Z., Gurevich E. Y., Zhuzhoma E. V., Zinina S. K.,  Heteroclinic curves of Morse – Smale cascades and separators in magnetic field of plasma, Rus. J. Nonlin. Dyn., 2014, Vol. 10, No. 4, pp.  427-438

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