The search of high-order periodic orbits has been typically restricted to problems with symmetries that help to reduce the dimension of the search space. Well-known examples include reversible maps with symmetry lines. The present work proposes a new method to compute high-order periodic orbits in twist maps without the use of symmetries. The method is a combination of the parameterization method in Fourier space and a Newton–Gauss multiple shooting scheme. The parameterization method has been successfully used in the past to compute quasi-periodic invariant circles. However, this is the first time that this method is used in the context of periodic orbits. Numerical examples are presented showing the accuracy and efficiency of the proposed method. The method is also applied to verify the renormalization prediction of the residues’ convergence at criticality (extensively studied in reversible maps) in the relatively unexplored case of maps without symmetries.

高阶周期轨道的搜索通常仅限于对称性问题，这些问题有助于减小搜索空间的尺寸。众所周知的示例包括带有对称线的可逆贴图。本工作提出了一种新的方法，可以在不使用对称性的情况下计算扭曲图中的高阶周期轨道。该方法是傅里叶空间中的参数化方法与牛顿-高斯多重射击方案的结合。过去已经成功地使用参数化方法来计算准周期不变圆。但是，这是第一次在周期性轨道的背景下使用此方法。数值算例表明了该方法的准确性和有效性。