A detailed description of the model is addressed in section \[sect:ModelDescription\]. In section \[sect:AsymptoticAnalysis\], an asymptotic method for the study of periodic responses is used. It consists of a two variables expansion combined with an averaging procedure The nonlinear modes and their stability, are first studied in section \[sect:NonlinearModes\] and numerical results are discussed to highlight to energy pumping phenomenon. A parametric study emphasize the importance of the dissipation rate of the non-linear absorber and some design criteria are found. Results on transient response show a correct prediction from the nonlinear modes. In section \[sect:ForcedResp\], the behaviour of the system in forced response (harmonic excitation) is then addressed and it is shown that, in accordance with the modal prediction, the system experiences some localization phenomenon along with jump phenomena.

Model description {#sect:ModelDescription}
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A system of two oscillators linearly connected is considered. The main oscillator, with mass $M$, represents an approximation to some continuous elastic system; the small one, with mass $m$, is the absorber and is strongly nonlinear. The coupling is assumed to be weak. Also, the system remains technologically realistic since the mass ratio about 2% for the numerical applications.

The motion of this system is governed by 