is consequently more important.

In this section, some numerical results on the nonlinear modes on these two representative examples are presented. The nonlinear complex modes are derived by numerically solving the fixed points problem defined by system (\[eq:CNMr\]) along with the energy relationship given in equation (\[eq:Energy\]) which aims at finding the unknown eigenfrequency $\omega$ and at normalizing the modes. Since our motivation is to investigated the vibration control of weakly damped structures, the main oscillator is assumed to be conservative and the only source of dissipation is the absorber.

### Example 1. {#example-1. .unnumbered}

We first study the case of a weakly dissipative hysteretic cycle which build using the Bouc-Wen parameters $\beta=10^{-3}$ and $\gamma=-0.5$. The modal quantities are depicted in figures \[fig:ModesNL1\].

The modal amplitudes of the linear and nonlinear degrees of freedom, $c_x$ and $c_v$, natural frequency $\omega$, and modal damping $\zeta$ are plotted versus the global energy $\mathcal{H}$ of the system.

There are two main branches (in solid and dashed lines) and an unstable region appears in the solid line branch symbolized by crosses. We can discriminate several regions with distinct dynamical behaviours:

Low energies region: 

:   in the solid line branch, the motion appears to be 