title: On the stability of the zero solution of differential equation of the second order under critical case
metadata.name: dc.contributor.author
metadata.value: Dorodenkov, Аlexander А.
metadata.lang: -
metadata.name: dc.date.accessioned
metadata.value: 2022-02-16T19:04:46Z
metadata.lang: -
metadata.name: dc.date.available
metadata.value: 2022-02-16T19:04:46Z
metadata.lang: -
metadata.name: dc.date.issued
metadata.value: 2021-12
metadata.lang: -
metadata.name: dc.identifier.citation
metadata.value: Dorodenkov А. А. On the stability of the zero solution of differential equation of the second order under critical case. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 2021, vol. 8 (66), issue 4, pp. 572–579.
metadata.lang: en_GB
metadata.name: dc.identifier.other
metadata.value: https://doi.org/10.21638/spbu01.2021.402
metadata.lang: -
metadata.name: dc.identifier.uri
metadata.value: http://hdl.handle.net/11701/35104
metadata.lang: -
metadata.name: dc.description.abstract
metadata.value: Differential equation of the form x¨+x2sgnx = Y (t, x, x˙ ), is considered, where the right-hand
  side is a small periodic perturbation of t, a sufficiently differentiable function in the origin
  neighborhood with variables x, x˙ . It is assumed that X perturbation is of order smallness
  not lower than the fifth, if x is assigned the second order, and x˙ is assigned the third order.
  Periodic functions are introduced that are the solution of the equation above with zero righthand
  side. Since differentiability of the quadratic part is bounded, then differentiability of
  the introduced functions is also bounded. These functions are used to transfer from the
  initial equation to the system of equations in coordinates similar to polar. This system with
  the help of the polynomial replacement is reduced to a system with Lyapunov constants.
  Replacement coefficients are found by the partial fraction decomposition. The conclusion
  about the nature of stability of the zero solution is made by a sign of the first non zero
  constant. Due to the bounded differentiability of the introduced functions, the degree of
  the polynomial replacement must be limited. The system of differential equations for the
  replacement coefficients is solved recursively. The number of found Lyapunov constants is
  also bounded. This article considers the case when all found constants are zero. To study
  this problem there used the method of isolating the main part of the introduced functions
  and their combinations as a result of the expantion of the latter in the Fourier series.
  The remainder of the series is assumed to be sufficiently small and it is shown that its
  presence can be neglected. The transition to the main parts instead of functions allows to
  compensate for the lack of differentiability of the introduced functions. Considering such
  systems, polynomial replacement can be again used and the Lyapunov constant for each
  main part can be found. It is shown that the sign of the constant for any main part is
  preserved. Sufficient conditions for stability and instability are indicated.
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 8 (66); Issue 4
metadata.lang: -
metadata.name: dc.subject
metadata.value: stability
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: small periodic perturbations
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: oscillator
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: Lyapunov constant
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: periodic function
metadata.lang: en_GB
metadata.name: dc.title
metadata.value: On the stability of the zero solution of differential equation of the second order under critical case
metadata.lang: en_GB
metadata.name: dc.type
metadata.value: Article
metadata.lang: en_GB
resources.filename: 572-579.pdf
resources.url: https://dspace.spbu.ru/bitstream/11701/35104/1/572-579.pdf
resources.format: Adobe PDF
resources.size: 281,77 kB