title: On the stability of the zero solution with respect to a part of variables in linear approximation
metadata.name: dc.contributor.author
metadata.value: Shamanaev, Pavel A.
metadata.lang: -
metadata.name: dc.date.accessioned
metadata.value: 2023-10-23T18:11:24Z
metadata.lang: -
metadata.name: dc.date.available
metadata.value: 2023-10-23T18:11:24Z
metadata.lang: -
metadata.name: dc.date.issued
metadata.value: 2023-09
metadata.lang: -
metadata.name: dc.identifier.citation
metadata.value: Shamanaev P. A. On the stability of the zero solution with respect to a part of variables in linear approximation. Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes, 2023, vol. 19, iss. 3, pp. 374–390. https://doi.org/10.21638/11701/spbu10.2023.306 (In Russian)
metadata.lang: en_GB
metadata.name: dc.identifier.other
metadata.value: https://doi.org/10.21638/11701/spbu10.2023.306
metadata.lang: -
metadata.name: dc.identifier.uri
metadata.value: http://hdl.handle.net/11701/44291
metadata.lang: -
metadata.name: dc.description.abstract
metadata.value: The article presents the sufficient conditions for stability and asymptotic stability with respect
  to a part of the variables of the zero solution of a nonlinear system in the linear
  approximation. the case is considered when the matrix of the linear approximation may
  contain eigenvalues with zero real parts and the algebraic and geometric multiplicities of
  these eigenvalues may not coincide. The approach is based on establishing some correspondence
  between the solutions of the investigated system and its linear approximation. The
  solutions of such systems starting in a sufficiently small zero neighborhood and the systems
  themselves possess the same componentwise asymptotic properties in this case. Such solutions’
  properties are stability and asymptotic stability with respect to some variables, and for
  systems componentwise local asymptotic equivalence and componentwise local asymptotic
  equilibrium. Considering the correspondence between the solutions of systems as an operator
  defined in a Banach space, there is proved that it has at least one fixed point according
  to the Schauder’s principle. The operator allows to construct a mapping that establishes
  the relationship between the initial points of the investigated system and its linear approximation.
  Further, a conclusion about the componentwise asymptotic properties of solutions of
  the nonlinear system is made on the basis of estimates of the fundamental matrix of the linear
  approximation rows’ entries. There is given an example of the investigation of stability
  and asymptotic stability with respect to a part of the variables of the zero solution of a nonlinear
  system is given, when the linear approximation matrix contains one negative and one
  zero eigenvalues, and the algebraic and geometric multiplicities of the zero eigenvalue do not
  coincide.
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. Applied Mathematics. Computer Science. Control Processes;Volume 19; Issue 3
metadata.lang: -
metadata.name: dc.subject
metadata.value: ordinary differential equations
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: partial stability
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: local componentwise asymptotic equivalence
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: Schauder principle
metadata.lang: en_GB
metadata.name: dc.title
metadata.value: On the stability of the zero solution with respect to a part of variables in linear approximation
metadata.lang: en_GB
metadata.name: dc.type
metadata.value: Article
metadata.lang: en_GB
resources.filename: 06.pdf
resources.url: https://dspace.spbu.ru/bitstream/11701/44291/1/06.pdf
resources.format: Adobe PDF
resources.size: 259,91 kB