Abstract: The describing function is a powerful tool for characterizing nonlinear dynamical systems in the frequencydomain. In some cases, it is the only available description ofa nonlinear operator characterizing a certain subcomponent of the system. This paper presents a methodology to provide a state-space realization ofone given describing function, in order to allow the study of the system in the time domain as well. The realization is based on Hammerstein models and Fourier–Bessel series. It can be embedded in time domain simulations of complex configurations with many nonlinear elements interacting, accurately describing the nonlinear saturation of the system. The technique is applied to an example application in the field ofcombustion instability, featuring self-excited thermoacoustic oscillations. We benchmark the performance of the tool comparing the results with a frequency domain analysis of the same system, obtaining good agreement between the two formulations.