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 Rus. J. Nonlin. Dyn., 2019, Volume 15, Number 4, Pages 525–531 (Mi nd679)

Semi-Invariant Form of Equilibrium Stability Criteria for Systems with One Cosymmetry

L. G. Kurakinab, A. V. Kurdoglyanc

a Southern Mathematical Institute, Vladikavkaz Scienific Center of RAS, ul. Markusa 22, Vladikavkaz, 362027 Russia
b Water Problems Institute of RAS, ul. Gubkina 3, Moscow, 119333 Russia
c Southern Federal University, ul. Milchakova 8a, Rostov-on-Don, 344090 Russia

Abstract: The systems of differential equations with one cosymmetry are considered [1]. The ordinary object for such systems is a one-dimensional continuous family of equilibria. The stability spectrum changes along this family, but it necessarily contains zero. We consider the nondegeneracy condition, thus the boundary equilibria separate the family on linearly stable and instable areas. The stability of the boundary equilibria depends on nonlinear terms of the system.
The stability problem for the systems with one cosymmetry is studied in [2]. The general problem is to apply the stability criteria one needs to compute coefficients of the model system. It is especially difficult if the system has a large dimension, while a number of critical variables may be small. A method for calculating coefficients is proposed in [3].
In this work the expressions for the known stability criteria are proposed in a form convenient for calculation. The explicit formulas of the coefficients of the model system are given in semi-invariant form. They are expressed using the generalized eigenvectors of the linear matrix and its conjugate matrix.

Keywords: stability, critical case, neutral manifold, cosymmetry, semi-invariant form

 Funding Agency Grant Number Ministry of Education and Science of the Russian Federation 1.5169.2017/8.9 This work was supported by the Ministry of Education and Science of the Russian Federation, Southern Federal University (Projects No. 1.5169.2017/8.9).

DOI: https://doi.org/10.20537/nd190411

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MSC: 34D20, 70K20
Accepted:05.11.2019
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Citation: L. G. Kurakin, A. V. Kurdoglyan, “Semi-Invariant Form of Equilibrium Stability Criteria for Systems with One Cosymmetry”, Rus. J. Nonlin. Dyn., 15:4 (2019), 525–531

Citation in format AMSBIB
\Bibitem{KurKur19} \by L. G. Kurakin, A. V. Kurdoglyan \paper Semi-Invariant Form of Equilibrium Stability Criteria for Systems with One Cosymmetry \jour Rus. J. Nonlin. Dyn. \yr 2019 \vol 15 \issue 4 \pages 525--531 \mathnet{http://mi.mathnet.ru/nd679} \crossref{https://doi.org/10.20537/nd190411} \elib{https://elibrary.ru/item.asp?id=43280422} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85084319411}