RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
General information
Latest issue
Archive
Impact factor

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Vestn. Tomsk. Gos. Univ. Mat. Mekh.:
Year:
Volume:
Issue:
Page:
Find






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


Vestn. Tomsk. Gos. Univ. Mat. Mekh., 2018, Number 54, Pages 46–57 (Mi vtgu659)  

MATHEMATICS

Asymptotics of the Cauchy problem solution in the case of instability of a stationary point in the plane of "rapid motions"

D. A. Tursunov

Osh State University, Kyrgyzstan

Abstract: In this paper, the Cauchy problem for a normal system of two linear inhomogeneous ordinary differential equations with a small parameter at the derivative is considered. The coefficient matrix of the linear part of the system has complex conjugate eigenvalues. The real parts of the complex conjugate eigenvalues in the considered interval change signs from negative to positive ones. A singularly perturbed Cauchy problem is investigated in the case of instability, i.e., when the asymptotic stability condition is violated. Moreover, the singularly perturbed Cauchy problem has an additional singularity, namely, the corresponding unperturbed equation has a non-smooth solution in the investigated extended domain. More exactly, the solution of the corresponding unperturbed equation has poles in the complex plane. Therefore, the Cauchy problem under consideration can be called bisingular in the terminology introduced by Academician A.M. Il'in.
The aim of the research is to construct the principal term of the asymptotic behavior of the Cauchy problem solution when the asymptotic stability condition is violated.
In the study, methods of the stationary phase, saddle point, successive approximations, and L.S. Pontryagin's idea-the transition to a complex plane-are applied.
An asymptotic estimate is obtained for the solution of a bisingularly perturbed Cauchy problem in the case of a change in the stability of a stationary point in the plane of "rapid motions" is violated. The principal term of the asymptotic expansion of the solution is constructed. It has a negative fractional power with respect to a small parameter, which is characteristic of bisingularly perturbed equations or equations with turning points.
The obtained results can find applications in chemical kinetics, in the study of Ziegler's pendulum, etc.

Keywords: asymptotic expansion of the solution, bisingular problem, singular perturbation, Cauchy problem, small parameter, stationary phase method, system of ordinary differential equations with a small parameter for the derivative.

DOI: https://doi.org/10.17223/19988621/54/4

Full text: PDF file (486 kB)
References: PDF file   HTML file

Bibliographic databases:

UDC: 517.928
MSC: 34M60, 34E10, 34A12
Received: 21.02.2018

Citation: D. A. Tursunov, “Asymptotics of the Cauchy problem solution in the case of instability of a stationary point in the plane of "rapid motions"”, Vestn. Tomsk. Gos. Univ. Mat. Mekh., 2018, no. 54, 46–57

Citation in format AMSBIB
\Bibitem{Tur18}
\by D.~A.~Tursunov
\paper Asymptotics of the Cauchy problem solution in the case of instability of a stationary point in the plane of "rapid motions"
\jour Vestn. Tomsk. Gos. Univ. Mat. Mekh.
\yr 2018
\issue 54
\pages 46--57
\mathnet{http://mi.mathnet.ru/vtgu659}
\crossref{https://doi.org/10.17223/19988621/54/4}
\elib{http://elibrary.ru/item.asp?id=35424225}


Linking options:
  • http://mi.mathnet.ru/eng/vtgu659
  • http://mi.mathnet.ru/eng/vtgu/y2018/i54/p46

    SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru


    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles
  • Вестник Томского государственного университета. Математика и механика
    Number of views:
    This page:81
    Full text:18
    References:3

     
    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2020