  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.
