In this paper we propose a framework for studying fundamental limitations in the control of thermoacoustic instabilities described by nonlinear models. More generally, the framework applies to general interconnections of lightly damped linear oscillators and nonlinear static nonlinearities of saturation type driven by broad-band Gaussian disturbances. The key concept is to replace the static nonlinearities with their corresponding random input describing functions. This allows the application of classical fundamental limitations theory to the combustion models. In this paper we interpret the limitations results in the particular case of a model of a combustion process controlled by on-off actuators. We also formulate conservation principles to understand the dynamics associated with this combustion model.