Abstract A numerical method aimed at locating stable periodic orbits in strongly chaotic systems is presented. The method is based on the selection of trajectory segments which are characterized by a relatively low positive local Lyapunov exponent. Once such a selection is made, convergence to stable (or weakly unstable) periodic orbits is obtained by a Newton method. The algorithm is rather general and can be used for systems with more than two degrees of freedom. The proposed approach is tested on the quartic oscillator model and on the potential of the hydrogen atom in a strong magnetic field. In the latter case new stable periodic orbits are found in the region of strong chaotic motion. The possible quantum localization on these orbits is discussed briefly.