Thermoacoustic instabilities within liquid rocket engines (LRE) result from a coupling between the dynamics of the injection, the combustion process and the acoustics of the propulsive system. This is a long-standing issue, which affects engine performance and may lead to its destruction. Quantifying the oscillation amplitude remains a critical and challenging issue, as it is a complex balance between driving and damping processes. This is true in the small-amplitude linear phase of the process, associated with the triggering of the self-sustained oscillation, and in the non-linear one, leading to the so-called limit cycle related to various saturation and energy transfer processes. Several combustion dynamics models exist accounting for the unsteady rate of heat release under various acoustic solicitations [1]. On the contrary, few studies offer a precise analysis of the mechanisms underlying the numerous acoustic damping processes which can be involved. The NPCC setup, a cold-flow test rig mimicking the geometry of a LRE [2], was designed in this purpose. It features a chamber and a dome linked by three injectors. Using a perforated wheel, the device can be efficiently forced at its eigenfrequencies. The competition between forcing and damping phenomena lead to a limit-cycle for three different eigenmodes: 1T, 1T1L (which is coupled with the 1T mode of the dome) and 1T2L chamber modes. The chamber 1T mode triggers strong non-linear harmonic response, while the other two exhibit weak non-linearities. Previous studies have shown that Large Eddy Simulation (LES) was able to retrieve this behaviour for all three eigenmodes [3]. Lowering the forcing amplitude makes it possible to stay in the linear acoustic regime. Disturbance energy budgets [4] are used in the LES code to characterize the damping phenomena during chamber-only (1T2L) and coupled dome-chamber (1T1L) forcing. Using the fluxes from the disturbance energy balance, we retrieve the correct global damping of the system and extract the local damping contributions (jets, chamber, injectors, dome). Then, a non-linear term representing the energy transfer to the harmonics observed in the experiment is derived from non-linear acoustics theory and tested on a purely numerical 1D test case, retrieving the correct limit-cycle amplitude. Finally, this non-linear approach is tested on the NPCC test rig combined with the linear model from [3] to represent the response at the beginning of the excitation. The limit-cycle of the 1T mode is correctly retrieved by this model.