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Lean premixed combustion chambers are susceptible to combustion instabilities arising from the coupling between the heat release rate perturbations and the acoustic disturbances. These instabilities are not desirable and knowledge of this complex mechanism is necessary in order to prevent or at least suppress them. A low order model is developed comprising a linear acoustic network and an improved analytical form for a flame describing function (FDF). The latter includes both the saturation of the amplitude of heat release rate perturbations and the change of phase lag relative to oncoming acoustic velocity fluctuations when the instability grows into a limit cycle. A stability map is constructed by moving the flame along the Rijke tube based on the eigenvalues resolved from the network. The acoustic model is then converted into the time domain and combined with the flame describing function to determine the …