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).

我们研究在奇数\(q\geqslant 3\)的p : q共振情况下对称椭圆不动点的分叉。我们考虑初始保面积映射\(\bar{z}=\lambda z+Q(z,z^{*})\)具有中心对称性，即在变量变化下不变的情况\ (z\to-z\) , \(z^{*}\to-z^{*}\)。我们在\(\lambda=e^{i2\pi\frac{p}{q}}\)情况下构造此类映射的范式，其中\(p\)和\(q\)是互质整数，\(p\leqslant q\)和\(q\)是奇数，并研究不动点\(z=0\)在各种设置下的局部分岔。我们证明了由四个\(q\)周期轨道组成的花环的出现，其中两个轨道是椭圆形的，两个轨道是鞍形的，并描述了相应的一参数族和二参数族的分岔图。我们还考虑初始映射可逆的情况，并找到花环的非对称周期轨道非保守的条件（包含稳定和不稳定轨道的对称对以及面积收缩和面积扩张的鞍座）。