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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.
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
Linking options:
https://www.mathnet.ru/eng/sm428https://doi.org/10.1070/sm1999v190n09ABEH000428 https://www.mathnet.ru/eng/sm/v190/i9/p127
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Abstract page: | 629 | Russian version PDF: | 233 | English version PDF: | 7 | References: | 75 | First page: | 2 |
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