title: On the stability of the zero solution of a differential equation of the second order under periodic perturbation of a center
metadata.name: dc.contributor.author
metadata.value: Dorodenkov, Аlexander А.
metadata.lang: -
metadata.name: dc.date.accessioned
metadata.value: 2018-04-26T09:21:58Z
metadata.lang: -
metadata.name: dc.date.available
metadata.value: 2018-04-26T09:21:58Z
metadata.lang: -
metadata.name: dc.date.issued
metadata.value: 2018-03
metadata.lang: -
metadata.name: dc.identifier.citation
metadata.value: Dorodenkov А.А. On the stability of the zero solution of a differential equation of the second order under periodic perturbation of a center. Vestnik SPbSU. Mathematics. Mechanics. Astronomy, 2018, vol. 5 (63), issue 1, pp. 44–50.
metadata.lang: en_GB
metadata.name: dc.identifier.other
metadata.value: 10.21638/11701/spbu01.2018.105
metadata.lang: -
metadata.name: dc.identifier.uri
metadata.value: http://hdl.handle.net/11701/9497
metadata.lang: -
metadata.name: dc.description.abstract
metadata.value: Small periodic perturbations of the oscillator ¨x+x2n−1 = 0 are considered, where 1 < n is a
  natural number, and the right-hand side is an analytic function in the origin neighborhood
  with variables x˙ , x. The equilibrium position of the given equation on stability is investigated.
  As a result, sufficient conditions for asymptotic stability and instability are formulated. In
  this work new periodic functions of the Lyapunov type are introduced. With their help, a
  transition to the system of equations is performed, similar to the transition to a system in
  polar coordinates. A system of two differential equations is obtained, the unknown functions
  of which are the amplitude and the “angular” variable. Then, polynomial change of variables
  in powers of the amplitude is made. The coefficients are periodic in time and “angular”
  variable functions. This replacement leads to a system of differential equations with a
  Lyapunov constant which in general is non-zero. Its sign determines the stability of the zero
  solution of the original equation. It is important that cases of even and odd n differ from
  each other. For an even n a non-zero Lyapunov constant can be found from one equation,
  and for an odd one it can be found from a system of three equations. The system is solved
  recurrently.
metadata.lang: en_GB
metadata.name: dc.language.iso
metadata.value: ru
metadata.lang: en_GB
metadata.name: dc.publisher
metadata.value: St Petersburg State University
metadata.lang: en_GB
metadata.name: dc.relation.ispartofseries
metadata.value: Vestnik of St Petersburg University. Mathematics. Mechanics. Astronomy;Volume 5(63); Issue 1
metadata.lang: -
metadata.name: dc.subject
metadata.value: asymptotic stability
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: small periodic perturbation
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: oscillator
metadata.lang: en_GB
metadata.name: dc.title
metadata.value: On the stability of the zero solution of a differential equation of the second order under periodic perturbation of a center
metadata.lang: en_GB
metadata.name: dc.type
metadata.value: Article
metadata.lang: en_GB
resources.filename: 05-Dorodenkov.pdf
resources.url: https://dspace.spbu.ru/bitstream/11701/9497/1/05-Dorodenkov.pdf
resources.format: Adobe PDF
resources.size: 286,07 kB