for several values of load amplitude $P$.

Numerical results
-----------------

The following results were obtained by numerically solving the nonlinear system (\[eq:SlowMoyApp\]) using Newton-like solver combine with an arc-length continuation method [@Nayfeh-Applied]. We used the first example Bouc-Wen hysteretic cycle ($\gamma = 10^{-3}$, figure \[fig:Cycle2\]).

The nonlinear frequency response are compared with the frequency responses of the linear system alone; the backbone curves have also been represented in order to see how the prediction of the nonlinear modes are in accordance with the forced response.

In the first example (figure \[fig:RF\_LowEnergy\]), the forcing level is quite small and we can see in both the linear and nonlinear oscillator have a linearizable behaviour.

As the level of the excitation is increased, the system’s behaviour differs from the linear case and several interesting phenomena appear. An example of nonlinear response is plotted in figure \[fig:RF\_HighEnergy\].

In this case, the vibratory energy is strongly localized in the nonlinear oscillator (absorber) in the vicinity of the resonance peak. The absorber appears to be efficient. One can also notices that the nonlinear response remains quite close to the backbone curves which attests the quality of the prediction of the nonlinear modes.

Conclusions
===========

The results 