On Bifurcations of Symmetric Elliptic Orbits

Abstract

We study bifurcations of symmetric elliptic fixed points in the case of p:q resonances with odd $q\geqslant 3$. 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 of variables $z\to-z$, $z^{*}\to-z^{*}$. We construct normal forms for such maps in the case $\lambda=e^{i2\pi\frac{p}{q}}$, where $p$ and $q$ are mutually prime integer numbers, $p\leqslant q$ 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 saddles, 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 where nonsymmetric periodic orbits of the garlands are nonconservative (contain symmetric pairs of stable and unstable orbits as well as area-contracting and area-expanding saddles).