On bifurcations of symmetric elliptic orbits

Abstract

We study bifurcations of symmetric elliptic fixed points in the case of \emph{p}:\emph{q} resonances with odd $q\geq 3$. We consider the case where the initial area-preserving map $\bar z =\lambda z + Q(z,z^*)$ possesses the central symmetry, i.e. is invariant under the change $z\to -z$, $z^*\to -z^*$. We construct normal forms for such maps in the case $\lambda = e^{i 2\pi \frac{p}{q}}$, where $p$ and $q$ are mutually prime integer numbers, $p\leq q$ and $q$ is odd, and study local bifurcations of the fixed point $z=0$ in various settings. We prove the appearance of garlands consisting of four $q$-periodic orbits, two orbits are elliptic and two orbits are saddle, and describe the corresponding bifurcation diagrams for one- and two-parameter families. We also consider the case where the initial map is reversible and find conditions when non-symmetric periodic orbits of the garlands are non-conservative (compose symmetric pairs of stable and unstable orbits as well as area-contracting and area-expanding saddles).