title: On the stability of the zero solution of a periodic reversible secondorder differential equation
metadata.name: dc.contributor.author
metadata.value: Bibikov, Yuri N.
metadata.lang: -
metadata.name: dc.date.accessioned
metadata.value: 2022-10-20T13:40:03Z
metadata.lang: -
metadata.name: dc.date.available
metadata.value: 2022-10-20T13:40:03Z
metadata.lang: -
metadata.name: dc.date.issued
metadata.value: 2022-09
metadata.lang: -
metadata.name: dc.identifier.citation
metadata.value: Bibikov Yu.N. On the stability of the zero solution of a periodic reversible secondorder differential equation. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 2022, vol. 9 (67), issue 3, pp. 474–479. https://doi.org/10.21638/spbu01.2022.308 (In Russian)
metadata.lang: en_GB
metadata.name: dc.identifier.other
metadata.value: https://doi.org/10.21638/spbu01.2022.308
metadata.lang: -
metadata.name: dc.identifier.uri
metadata.value: http://hdl.handle.net/11701/38302
metadata.lang: -
metadata.name: dc.description.abstract
metadata.value: The problem of the stability of the zero solution of the second-order differential equation
  describing the periodic perturbations of an oscillator with a nonlinear reducing force is
  studied. The problem in the autonomous case was solved by A.M. Lyapunov. The so called
  transcendental case when all members of the decomposition of the right part of the differential
  equation into series are to be taken into account, is considered. This case takes place
  for reversible differential equations, i. e. equations that do not change when time is replaced by the opposite value. The problem is solved by the methods of the KAM theory, according
  to which in any neighborhood of the equilibrium position at the origin of the phase
  plane there are periodic invariant two-dimensional tori that separate the three-dimensional
  configuration space. These tori are considered as two-dimensional periodic invariant surfaces
  covering the time axis from where the stability (non-asymptotic) of the zero solution
  followed. The problem to be solved is characterized by the fact that the unperturbed part
  of the equation contains a dissipative term (a term dependent on velocity) which has the
  same order of smallness as the restoring force. It is established that if a dissipative part
  of the perturbation is small enough then the unperturbed movement is stable according to
  Lyapunov.
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 9 (67); Issue 3
metadata.lang: -
metadata.name: dc.subject
metadata.value: second-order differential equations
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: periodic perturbations
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: oscillator
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: reversibility
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: transcendence
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: stability
metadata.lang: en_GB
metadata.name: dc.title
metadata.value: On the stability of the zero solution of a periodic reversible secondorder differential equation
metadata.lang: en_GB
metadata.name: dc.type
metadata.value: Article
metadata.lang: en_GB
resources.filename: 474-479.pdf
resources.url: https://dspace.spbu.ru/bitstream/11701/38302/1/474-479.pdf
resources.format: Adobe PDF
resources.size: 265,31 kB