title: On the stability of the zero solution of an essentially non-linear differential equation of the second order
metadata.name: dc.contributor.author
metadata.value: Bibikov, Yury N.
metadata.lang: -
metadata.name: dc.contributor.author
metadata.value: Pliss, Victor A.
metadata.lang: -
metadata.name: dc.contributor.author
metadata.value: Trushina, Natalia V.
metadata.lang: -
metadata.name: dc.date.accessioned
metadata.value: 2017-09-20T13:34:41Z
metadata.lang: -
metadata.name: dc.date.available
metadata.value: 2017-09-20T13:34:41Z
metadata.lang: -
metadata.name: dc.date.issued
metadata.value: 2017-09
metadata.lang: -
metadata.name: dc.identifier.citation
metadata.value: Bibikov Yu.N., Pliss V.A., Trushina N.V. On the stability of the zero solution of an essentially non-linear differential equation of the second order. Vestnik SPbSU. Mathematics. Mechanics. Astronomy, 2017, vol. 4 (62), issue 3, pp. 394–401.
metadata.lang: en_GB
metadata.name: dc.identifier.other
metadata.value: 10.21638/11701/spbu01.2017.304
metadata.lang: -
metadata.name: dc.identifier.uri
metadata.value: http://hdl.handle.net/11701/7144
metadata.lang: -
metadata.name: dc.description.abstract
metadata.value: Small periodic in time perturbations of an essentially non-linear differential equation of the second order,
  are studied. It is supposed that the restoring force of the unperturbed equation consists of both conservative
  and dissipative parts, the first one being a root function. It is also supposed that all solutions of the
  unperturbed equation are periodic. Thus, the unperturbed equation is an oscillator. Peculiarity of the
  problem under consideration is that the frequency of oscillations is an infinitely small function of the
  amplitude.
  A problem of zero-solution stability is considered. Lyapunov investigated the case of autonomous
  perturbations. He showed that the asymptotic stability or the instability depends on the sign of a certain
  constant and presented a method of its calculation. Liapunov’s approach can not be applied to nonautonomous,
  in particular to periodic perturbations because it is based on the possibility to exclude the
  time variable from the system.
  Through modification of Lyapunov’s method the following results are obtained.
  “Action  angle” variables are introduced. A polynomial transformation of the action variable that
  permits to calculate Lyapunov’s constant is presented. In general case the structure of the polynomial
  transformation is studied. It appears that the “length” of the polynomial is a periodic function of the
  exponent of a root function in the conservative part of the restoring force in unperturbed equation. The
  least period is equal to 4. Refs 6.
metadata.lang: en_GB
metadata.name: dc.description.sponsorship
metadata.value: Работа выполнена при частичной финансовой поддержке РФФИ (грант №16-01-00452).
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 4 (62); Issue 3
metadata.lang: -
metadata.name: dc.subject
metadata.value: stability
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: periodic perturbation
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: oscillator
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: essentially nonlinear differential equations
metadata.lang: en_GB
metadata.name: dc.title
metadata.value: On the stability of the zero solution of an essentially non-linear differential equation of the second order
metadata.lang: en_GB
metadata.name: dc.type
metadata.value: Article
metadata.lang: en_GB
resources.filename: 04-Bibikov.pdf
resources.url: https://dspace.spbu.ru/bitstream/11701/7144/1/04-Bibikov.pdf
resources.format: Adobe PDF
resources.size: 250,74 kB