0
2013
Impact Factor

    Evgeniy Zhuzhoma

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

    Publications:

    Grines V. Z., Zhuzhoma E. V.
    Cantor Type Basic Sets of Surface $A$-endomorphisms
    2021, Vol. 17, no. 3, pp.  335-345
    Abstract
    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
    DOI:10.20537/nd210307
    Zhuzhoma E. V., Medvedev V. S., Isaenkova N. V.
    Abstract
    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
    DOI:10.20537/nd1703007
    Grines V. Z., Gurevich E. Y., Zhuzhoma E. V., Zinina S. K.
    Abstract
    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
    DOI:10.20537/nd1404003

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