This paper concerns the influence of the phase of the heat release response on thermoacoustic systems. We focus on one pair of degenerate azimuthal acoustic modes, with frequency coo. The same results apply for an axial acoustic mode. We show how the value phi(0) and the slope -tau of the flame phase at the frequency coo affects the boundary of stability, the frequency and amplitude of oscillation, and the phase phi(qp) between heat release rate and acoustic pressure. This effect depends on phi(0) and on the nondimensional number tau omega(0), which can be quickly calculated. We find for example that systems with large values of tau omega(0) are more prone to oscillate, i.e. they are more likely to have larger growth rates, and that at very large values of two the value phi(0) of the flame phase at coo does not play a role in determining the system's stability. Moreover for a fixed flame gain, a flame whose phase changes rapidly with frequency is more likely to excite an acoustic mode. We propose ranges for typical values of nondimensional acoustic damping rates, frequency shifts and growth rates based on a literature review. We study the system in the nonlinear regime by applying the method of averaging and of multiple scales. We show how to account in the time domain for a varying frequency of oscillation as a function of amplitude, and validate these results with extensive numerical simulations for the parameters in the proposed ranges. We show that the frequency of oscillation omega(B) and the flame phase den at the limit cycle match the respective values on the boundary of stability. We find good agreement between the model and thermoacoustic experiments, both in terms of the ratio omega(B)/omega(0) and of the phase phi(qp), and provide an interpretation of the transition between different thermoacoustic states of an experiment. We discuss the effect of neglecting the component of heat release rate not in phase with the pressure p as assumed in previous studies. We show that this component should not be neglected when making a prediction of the system's stability and amplitudes, but we present some evidence that it may be neglected when identifying a system that is unstable and is already oscillating (C) 2017 The Combustion Institute. Published by Elsevier Inc. All rights reserved.