A wavelet analysis on the output signal of a non-linear system in the neighbourhood of a Hopf bifurcation (i.e., a limit-cycle oscillation) has been performed to point out the linear and non-linear signatures of the system. Indeed, this kind of non-linear behaviour is characterized not only by a simple harmonic oscillation in developed steady state condition, but also by an initial transitory phase with a complex time evolution of the spectral signal content. Both these issues could be described in an analytical way via a singular perturbation analysis but they could be also directly analyzed by a signal-processing tool via wavelet analysis (Continuous Wavelet Transform, CWT): this is obtained by using the wavelet capability in describing e$ciently the time evolution of the spectrum of the signal (i.e., a non-linear &&signature'' of a Hopf bifurcation). Furthermore, the analytical results given by the singular perturbation analysis for the same class of non-linear problems, have allowed one to interpret correctly the results obtained by the wavelet analysis. In this paper this theoretical link between the wavelet analysis and the output signal of a dynamical system (recently achieved for linear mechanical damped systems) has been extended to non-linear systems in the neighbourhood of a Hopf bifurcation. This kind of analysis is of natural interest in aeroelastic applications where an e$cient (and noise insensitive) analysis of the output signal recorded during the #ight tests performed in the neighbourhood of the critical condition (#utter) may be desirable together with the capability to record the non-linear system signature. The time-scale decompositions (CWT) have given the opportunity to analyze pre-and post-critical transient behaviour of a non-linear aeroelastic system and, thus, upon considering the analytical tool given by the singular perturbation analysis, to identify its linear and non-linear features. Applications include either two non-linear coupled oscillators or a typical section in a supersonic #ow.