We use recurrence analysis to investigate the forced synchronization of a self-excited thermoacoustic system. The system consists of a swirl-stabilized turbulent premixed flame in an open-ended duct. We apply periodic acoustic forcing to this system at different amplitudes and frequencies around its natural self-excited frequency, and examine its response via unsteady pressure measurements. On increasing the forcing amplitude, we observe two bifurcations: from a periodic limit cycle (unforced) to quasiperiodicity (weak forcing) and then to lock-in (strong forcing). To analyse these bifurcations, we use cross-recurrence plots (CRPs) of the unsteady pressure and acoustic forcing. We find that the different time scales characterizing the quasiperiodicity and the transition to lock-in appear as distinct structures in the CRPs. We then examine those structures using cross recurrence quantification analysis (CRQA) and find that their recurrence quantities change even before the system transitions to lock-in. This shows that CRPs and CRQA can be used as alternative nonlinear tools to study forced synchronization in thermoacoustic systems, complementing classical linear tools such as spectral analysis.