Synchronization is a universal concept in nonlinear science but has received little attention in thermoacoustics. In this numerical study, we take a dynamical systems approach to investigating the influence of harmonic acoustic forcing on three different types of self-excited thermoacoustic oscillations: periodic, quasi-periodic and chaotic. When the periodic system is forced, we find that: (i) at low forcing amplitudes, it responds at both the forcing frequency and the natural (self-excited) frequency, as well as at their linear combinations, indicating quasi-periodicity; (ii) above a critical forcing amplitude, the system locks in to the forcing; (iii) the bifurcations leading up to lock-in and the critical forcing amplitude required for lock-in depend on the proximity of the forcing frequency to the natural frequency; (iv) the response amplitude at lock-in may be larger or smaller than that of the unforced system and the system can exhibit hysteresis and the jump phenomenon owing to a cusp catastrophe; and (v) at forcing amplitudes above lock-in, the oscillations can become unstable and transition to chaos, or switch between different stable attractors depending on the forcing amplitude. When the quasi-periodic system is forced at a frequency equal to one of the two characteristic frequencies of the torus attractor, we find that lock-in occurs via a saddle-node bifurcation with frequency pulling. When the chaotic system is forced at a frequency close to the dominant frequency of its strange attractor, we find that it is possible to destroy chaos and establish stable periodic oscillations. These results show that the open-loop application of harmonic acoustic forcing can be an effective strategy for controlling periodic or aperiodic thermoacoustic oscillations. In some cases, we find that such forcing can reduce the response amplitude by up to 90 %, making it a viable way to weaken thermoacoustic oscillations.