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Mat. Sb., 1999, Volume 190, Number 9, Pages 127–150 (Mi msb428)  

This article is cited in 1 scientific paper (total in 2 paper)

On the bifurcations of equilibria corresponding to double eigenvalues

È. È. Shnol', E. V. Nikolaev

Institute of Mathematical Problems of Biology, Russian Academy of Sciences

Abstract: Systems of ordinary differential equations having a finite symmetry group are considered. One-parameter local bifurcations of symmetric equilibria corresponding to a double pair of purely imaginary eigenvalues are studied.
It is shown that in one case a two-dimensional torus is generated from the equilibrium. The torus contains limit cycles; their number does not depend on the values of the parameter. The trajectories of the system that do not leave a certain fixed domain may only tend to the equilibrium under study or to the 2-dimensional torus or to one of two (disjoint) limit cycles.
In all the other cases an invariant surface is generated from the equilibrium which is diffeomorphic to the three-dimensional sphere. The behaviour of the trajectories on this surface depends on the symmetry group and is not studied in this paper.
In the appendix we provide information on codimension 1 bifurcations corresponding to double zero eigenvalues.

DOI: https://doi.org/10.4213/sm428

Full text: PDF file (454 kB)
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English version:
Sbornik: Mathematics, 1999, 190:9, 1353–1376

Bibliographic databases:

UDC: 517.9
MSC: Primary 58F14, 58F21; Secondary 58F12, 34C23, 34C30
Received: 21.08.1998

Citation: È. È. Shnol', E. V. Nikolaev, “On the bifurcations of equilibria corresponding to double eigenvalues”, Mat. Sb., 190:9 (1999), 127–150; Sb. Math., 190:9 (1999), 1353–1376

Citation in format AMSBIB
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\paper On the bifurcations of equilibria corresponding to double eigenvalues
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\pages 127--150
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  • https://doi.org/10.4213/sm428
  • http://mi.mathnet.ru/eng/msb/v190/i9/p127

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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. È. È. Shnol', “Regular polyhedra and bifurcations of symmetric equilibria of ordinary differential equations”, Sb. Math., 191:8 (2000), 1243–1258  mathnet  crossref  crossref  mathscinet  zmath  isi
    2. A. I. Aptekarev, A. L. Afendikov, F. I. Ataullakhanov, N. K. Balabaev, V. N. Biktashev, I. V. Biktasheva, R. M. Borisyuk, N. D. Vvedenskaya, R. D. Dagkesamanskii, Yu. G. Zarkhin, Yu. S. Ilyashenko, V. D. Lakhno, V. Yu. Lunin, N. L. Lunina, E. V. Nikolaev, V. S. Posvyanskii, M. A. Roitberg, V. S. Ryaben'kii, L. B. Ryashko, Ya. G. Sinai, V. M. Tikhomirov, A. A. Tokarev, A. G. Urzhumtsev, A. I. Khibnik, “To the memory of Èmmanuil Èl'evich Shnol'”, Russian Math. Surveys, 72:1 (2017), 185–198  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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