We experimentally study the nonlinear dynamics of a self-excited thermoacoustic system subjected to acoustic forcing. Our aim is to relate these dynamics to the behavior of universal model oscillators subjected to external forcing. The self-excited system under study consists of a swirl-stabilized turbulent premixed flame (equivalence ratio of 0.8 and thermal power of 13.6 kW) enclosed in a quartz tube with an open-ended exit. We coustically force this system at different amplitudes and frequencies, and measure its response with pressure transducers and OH* chemiluminescence from the flame. By analyzing the data with the power spectral density and the Poincare´ map, we find a range of nonlinear dynamics, including (i) a shifting of the self-excited frequency towards or away from the forcing frequency as the forcing amplitude increases; (ii) an accompanying transition from periodicity to two-frequency quasiperiodicity; and (iii) an eventual suppression of the self-excited amplitude, indicating synchronization of the self-excited mode with the forced mode. By further analyzing the data with the Hilbert transform, we find evidence of phase trapping, a partially synchronous state characterized by frequency locking without phase locking. All of these dynamics can be found in universal model oscillators subjected to external forcing. This suggests that such oscillators can be used to accurately represent thermoacoustically self-excited combusting systems subjected to similar forcing. It also suggests that the analytical solutions to such oscillators can be used to guide the reduction and analysis of experimental or numerical data obtained from real thermoacoustic systems, and to identify effective methods for open-loop control of their dynamics.