We experimentally investigate the nonlinear dynamics of a thermoacoustically self-excited aero-engine combustion system featuring a turbulent swirling liquid-fueled diffusion flame in a variable-length combustor. We focus on the steady-state dynamics via simultaneous measurements of the acoustic pressure in the combustor and the heat release rate (HRR) from the flame. When the combustor length is increased following the onset of thermoacoustic instability, we find that the pressure signal transitions from a period-1 limit cycle to chaos, whereas the HRR signal remains chaotic owing to the presence of an intrinsic hydrodynamic mode in the flame. When the hydrodynamic mode is filtered out of the data, we find that the pressure and HRR signals are in generalized synchronization. However, when the hydrodynamic mode is retained in the data, we find that the pressure and HRR signals are either weakly phase synchronized or desynchronized. This study has two main contributions: (i) it shows that a liquid-fueled diffusion-flame combustor can exhibit dynamics as complex as those of its gaseous-fueled premixed-flame counterparts and (ii) it highlights the need to be exceptionally careful when selecting a diagnostic signal from which to calculate nonlinear measures of self-excited thermoacoustic oscillations, because our experiments show that the pressure and HRR signals can be desynchronized by the presence of a hydrodynamic mode in the flame at a frequency different from that of the dominant thermoacoustic mode.