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Periodic orbits in $k$-symmetric dynamical systems. (English) Zbl 1194.37149
Summary: A map $L$ is called $k$-symmetric if its $k$th iterate ${L}^{k}$ possesses more symmetry than $L$, for some value of $k$. In $k$-symmetric systems, there exists a notion of $k$-symmetric orbits. This paper deels with $k$-symmetric periodic orbits. We derive a relation between orbits that are $k$-symmetric with respect to reversing $k$-symmetries and symmetric orbits of ${L}^{k}$. With this relation we set out an efficient method for finding systematically all periodic orbits that are $k$-symmetric with respect to reversing $k$-symmetries. This $k$-symmetric fixed set iteration (FSI) method generalizes a celebrated method due to DeVogelaere that applies to symmetric periodic orbits in reversible dynamical systems.We use the FSI method to study $k$-symmetric periodic orbits of a map of the plane ${ℝ}^{2}$ possessing a crystallographic reversing $k$-symmetry group. The explicit findings illustrate a typically $k$-symmetric phenomenon, consisting of a nontrivial relation between the symmetry properties of periodic orbits and their periods.
##### MSC:
 37N05 Dynamical systems in classical and celestial mechanics 70K99 Nonlinear dynamics (general mechanics)