Periodic solutions to symmetric Newtonian systems in neighborhoods of orbits of equilibria

Anna Gołȩbiewska; Marta Kowalczyk; Sławomir Rybicki; Piotr Stefaniak; Anna Gołȩbiewska; Marta Kowalczyk; Sławomir Rybicki; Piotr Stefaniak

doi:10.3934/era.2022085

Electronic Research Archive

2022, Volume 30, Issue 5: 1691-1707. doi: 10.3934/era.2022085

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Periodic solutions to symmetric Newtonian systems in neighborhoods of orbits of equilibria

Faculty of Mathematics and Computer Science, Nicolaus Copernicus University in Toruń, PL-87-100 Toruń, ul. Chopina 12/18, Poland

Received: 13 August 2021 Revised: 15 January 2022 Accepted: 15 January 2022 Published: 25 March 2022

The aim of this paper is to prove the existence of periodic solutions to symmetric Newtonian systems in any neighborhood of an isolated orbit of equilibria. Applying equivariant bifurcation techniques we obtain a generalization of the classical Lyapunov center theorem to the case of symmetric potentials with orbits of non-isolated critical points. Our tool is an equivariant version of the Conley index. To compare the indices we compute cohomological dimensions of some orbit spaces.
- Lyapunov center theorem,
- symmetric Newtonian systems,
- equivariant Conley index,
- periodic solutions,
- symmetric bifurcations
Citation: Anna Gołȩbiewska, Marta Kowalczyk, Sławomir Rybicki, Piotr Stefaniak. Periodic solutions to symmetric Newtonian systems in neighborhoods of orbits of equilibria[J]. Electronic Research Archive, 2022, 30(5): 1691-1707. doi: 10.3934/era.2022085

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Abstract

The aim of this paper is to prove the existence of periodic solutions to symmetric Newtonian systems in any neighborhood of an isolated orbit of equilibria. Applying equivariant bifurcation techniques we obtain a generalization of the classical Lyapunov center theorem to the case of symmetric potentials with orbits of non-isolated critical points. Our tool is an equivariant version of the Conley index. To compare the indices we compute cohomological dimensions of some orbit spaces.

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