A concurrent computational and experimental study of self-excited combustion dynamics in a model configuration of a lean direct injection (LDI) gas turbine combustor is described. Both acoustically open and closed configurations are considered in the computations and analyzed using dynamic mode decomposition (DMD) to identify the frequency couplings. In the acoustically open case, simulations are carried out with unperturbed and perturbed inlet mass flow rates at the dominant frequencies observed in the acoustically closed geometry. In the unperturbed case, a vortex breakdown bubble (VBB) is shown to be the dominant flow structure, while the perturbed simulations indicate the presence of swirling hydrodynamic modes and the possibility of coupling between hydrodynamics and acoustics as a mechanism for sustaining thermo-acoustic instabilities. Detailed analysis of the acoustically closed combustor simulation reveals the presence of another important hydrodynamic mode, the precessing vortex core (PVC). The possibility of nonlinear coupling between the acoustics and PVC modes is also revealed. The thermo-acoustic coupling is further analyzed by the Rayleigh index frequency spectrum.