[en] In recent years there have been several applications of the nonlinear numerical continuation approach to aeroelastic systems with freeplay. While some of these have been successful, the general application of the method to such systems remains problematic. Numerical continuation can fail in the presence of complex bifurcations, numerous nearby periodic solution branches and other factors. In this paper, a three-part procedure for applying numerical continuation to aeroelastic systems with freeplay is proposed, designed to ensure that the complete periodic behavior is identified, even for systems with complex bifurcation diagrams. First, the equivalent linearization approach is used to determine approximations to the periodic solution branches of the nonlinear system. Then, a shooting-based technique is applied separately to each linearized approximation in order to pinpoint the nearest exact periodic solution. This process results in a cloud of periodic solutions, representing all the branches and sub-branches. Finally, a branch-following shooting procedure is applied to this cloud of points in order to obtain a complete description of every branch of periodic solutions. The procedure is demonstrated on a simple 3-DOF mathematical aeroelastic system with freeplay; it is shown that an extremely complex bifurcation is fully captured. The system's bifurcation diagram features multiple branch crossings, folds and loops. Its complete calculation allows the justification of several interesting LCO phenomena, such as aperiodic LCOs.