University proceedings. Volga region. Physical and mathematical sciences
RUS  ENG    JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB  
General information
Latest issue
Archive

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



University proceedings. Volga region. Physical and mathematical sciences:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


University proceedings. Volga region. Physical and mathematical sciences, 2022, Issue 1, Pages 45–55
DOI: https://doi.org/10.21685/2072-3040-2022-1-5
(Mi ivpnz5)
 

Mathematics

On generic homogeneous vector fields

V. Sh. Roitenberg

Yaroslavl State Technical University, Yaroslavl, Russia
References:
Abstract: Background. The study of dynamical systems with symmetry is of both theoretical and applied interest. Phase portraits of dynamical systems defined by homogeneous vector fields are invariant with respect to the phase space dilation group. First of all, it is natural to describe generic homogeneous vector fields. The structure of the phase portraits of generic homogeneous vector fields on the projective plane is known. The purpose of this work is to obtain some generic properties of homogeneous vector fields in a phase space of three or more dimensions. Materials and methods. Methods of the qualitative theory of differential equations and functional analysis are applied. Results. We consider homogeneous vector fields in an $m$-dimensional arithmetic space of degree $n > 0$, which have continuous derivatives everywhere, except perhaps at the origin. The trajectories of homogeneous vector fields extend naturally to the space compactification, the Poincar'e ball. It is shown that vector fields that have only hyperbolic singular points and closed trajectories in the Poincaré ball with a punctured origin are generic: they form a massive set in the Banach space of homogeneous vector fields. The concept of a weakly structurally stable homogeneous vector field is introduced. Necessary and sufficient conditions for weak structural stability at $m = 3$ are obtained. It is shown that weakly structurally stable vector fields in the Banach space of homogeneous vector fields form an open everywhere dense set. Conclusions. Weak structural stability of homogeneous vector fields in three-dimensional space is a generic property.
Keywords: homogeneous vector field, compactification of the phase space, hyperbolic singular points and closed trajectories, generality, weak structural stability.
Document Type: Article
UDC: 517.925
Language: Russian
Citation: V. Sh. Roitenberg, “On generic homogeneous vector fields”, University proceedings. Volga region. Physical and mathematical sciences, 2022, no. 1, 45–55
Citation in format AMSBIB
\Bibitem{Roi22}
\by V.~Sh.~Roitenberg
\paper On generic homogeneous vector fields
\jour University proceedings. Volga region. Physical and mathematical sciences
\yr 2022
\issue 1
\pages 45--55
\mathnet{http://mi.mathnet.ru/ivpnz5}
\crossref{https://doi.org/10.21685/2072-3040-2022-1-5}
Linking options:
  • https://www.mathnet.ru/eng/ivpnz5
  • https://www.mathnet.ru/eng/ivpnz/y2022/i1/p45
  • Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    University proceedings. Volga region. Physical and mathematical sciences
    Statistics & downloads:
    Abstract page:44
    Full-text PDF :14
    References:17
     
      Contact us:
     Terms of Use  Registration to the website  Logotypes © Steklov Mathematical Institute RAS, 2024