title: Systems that generate solutions with small period
metadata.name: dc.contributor.author
metadata.value: Pilyugin, Sergey Yu.
metadata.lang: -
metadata.name: dc.contributor.author
metadata.value: Rodionova, Anastasiya A.
metadata.lang: -
metadata.name: dc.date.accessioned
metadata.value: 2016-09-28T15:58:37Z
metadata.lang: -
metadata.name: dc.date.available
metadata.value: 2016-09-28T15:58:37Z
metadata.lang: -
metadata.name: dc.date.issued
metadata.value: 2016-09
metadata.lang: -
metadata.name: dc.identifier.citation
metadata.value: Pilyugin S.Yu., Rodionova A.A. Systems that generate solutions with small period. Vestnik of Saint Petersburg University. Series 1. Mathematics. Mechanics. Astronomy, 2016, vol. 3 (61), issue 3, pp. 424–428.
metadata.lang: en_GB
metadata.name: dc.identifier.other
metadata.value: 10.21638/11701/spbu01.2016.310
metadata.lang: -
metadata.name: dc.identifier.uri
metadata.value: http://hdl.handle.net/11701/3912
metadata.lang: -
metadata.name: dc.description.abstract
metadata.value: The problem of existence of periodic solutions is one of the traditional problems of the theory of differential
  equations. In this short note, we select a class of systems of differential equations for which conditions of
  the existence of periodic solutions have an extremely simple form. In particular, this class includes systems
  that correspond to periodically perturbed equations of oscillations without friction,
  ¨x + f(x) = h(t);
  for such systems, this problem was intensively studied.
  Let (j1, . . . , jn) be a permutation of the set (1, . . . ,n). We consider a system of differential equations
  x˙ i = fi(xji ), i= 1, . . . ,n,
  in which any function fi is continuous on R.
  This system has the property of generation of solutions with a small period if for any numberM >0
  there exists a number ω0 = ω0(M) > 0 such that if 0 < ω ≤ ω0 and hi(t, x1, . . . , xn) are continuous on
  R × Rn, ω-periodic in t functions that satisfy the inequalities |hi| ≤ M, then the system
  x˙ i = fi(xji) + hi(t, x1, . . . ,xn), i= 1, . . . ,n,
  has an ω-periodic solution.
  We show that a system has the property of generation of solutions with a small period if and only
  if the following equalities hold:
  fi(R) = R, i= 1, . . . ,n.
  It is also shown that the condition of smallness of the period is essential. Refs 5.
metadata.lang: en_GB
metadata.name: dc.description.sponsorship
metadata.value: Работа выполнена при финансовой поддержке РФФИ (грант 15-01-03797а) и программы СПбГУ
  (шифр ИАС 6.38.223.2014).
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. Series 1. Mathematics. Mechanics. Astronomy;Vol. 3 (61); Issue 3
metadata.lang: -
metadata.name: dc.subject
metadata.value: systems of differential equations
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: periodic solutions
metadata.lang: en_GB
metadata.name: dc.title
metadata.value: Systems that generate solutions with small period
metadata.lang: en_GB
metadata.name: dc.type
metadata.value: Article
metadata.lang: en_GB
resources.filename: Pilyugin.pdf
resources.url: https://dspace.spbu.ru/bitstream/11701/3912/1/Pilyugin.pdf
resources.format: Adobe PDF
resources.size: 213,91 kB