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On the Isolation/Nonisolation of a Cosymmetric Equilibrium and Bifurcations in its Neighborhood

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Abstract

A dynamical system with a cosymmetry is considered. V. I. Yudovich showed that a noncosymmetric equilibrium of such a system under the conditions of the general position is a member of a one-parameter family. In this paper, it is assumed that the equilibrium is cosymmetric, and the linearization matrix of the cosymmetry is nondegenerate. It is shown that, in the case of an odd-dimensional dynamical system, the equilibrium is also nonisolated and belongs to a one-parameter family of equilibria. In the even-dimensional case, the cosymmetric equilibrium is, generally speaking, isolated. The Lyapunov – Schmidt method is used to study bifurcations in the neighborhood of the cosymmetric equilibrium when the linearization matrix has a double kernel. The dynamical system and its cosymmetry depend on a real parameter. We describe scenarios of branching for families of noncosymmetric equilibria.

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ACKNOWLEDGMENTS

We express our gratitude to the anonymous reviewer #1 for useful comments and recommendations, and to I. V. Ostrovskaya for useful discussions.

Funding

This work was carried out within the framework of the publication activity support program of the Southern Federal University.

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Correspondence to Leonid G. Kurakin or Aik V. Kurdoglyan.

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Kurakin, L.G., Kurdoglyan, A.V. On the Isolation/Nonisolation of a Cosymmetric Equilibrium and Bifurcations in its Neighborhood. Regul. Chaot. Dyn. 26, 258–270 (2021). https://doi.org/10.1134/S1560354721030047

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  • DOI: https://doi.org/10.1134/S1560354721030047

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