Advances in Differential Equations

Symmetric Lyapunov center theorem for orbit with nontrivial isotropy group

Marta Kowalczyk, Ernesto Pérez-Chavela, and Sławomir Rybicki

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In this article, we prove two versions of the Lyapunov center theorem for symmetric potentials. We consider a second order autonomous system $$ \ddot q(t)=-\nabla U(q(t)) $$ in the presence of symmetries of a compact Lie group $\Gamma.$ We look for non-stationary periodic solutions of this system in a neighborhood of a $\Gamma$-orbit of critical points of the $\Gamma$-invariant potential $U.$ Our results generalize that of [13, 14]. As a topological tool, we use an infinite-dimensional generalization of the equivariant Conley index due to Izydorek, see [9].

Article information

Adv. Differential Equations, Volume 25, Number 1/2 (2020), 1-30.

First available in Project Euclid: 6 February 2020

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Primary: 34C25: Periodic solutions 37G40: Symmetries, equivariant bifurcation theory


Kowalczyk, Marta; Pérez-Chavela, Ernesto; Rybicki, Sławomir. Symmetric Lyapunov center theorem for orbit with nontrivial isotropy group. Adv. Differential Equations 25 (2020), no. 1/2, 1--30.

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