theoretical concept of non-linear modes is found to provide interesting dynamical descriptions as well as a valuable design tool. The derivation of non-linear modes is first performed analytically. To do so, we first apply the averaging method described in section \[sect:AsymptoticAnalysis\] to the initial equations of motion (\[eq:mvtR\]) (without forcing and damping); then, the study of the fixed points of the resulting system leads to the definition of an eigenvalue problem. In a second time, numerical results are presented for several types of Bouc-Wen hysteretic restoring forces and to possibility of localization phenomenon and possible energy pumping are demonstrated.

Free vibrations analysis – Non-linear modes {#ssect:FreeVib_NLM}
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The free vibrations problem of the two degrees of freedom system is defined by:

\[eq:mvt\_free\] $$\ddot{x}(t) + \omega_0^2 x(t) + \epsilon_0\left(x(t)- v(t)\right)  = 0$$ $$\ddot v(t)  + r(\dot v(t),t)+ \epsilon_1 \left(v(t)-x(t)\right)  = 0$$

Using the averaging method described in section \[sect:AsymptoticAnalysis\], the two displacements variables take the form of (\[eq:solapprox\]), [*ie*]{}:

$$x=a_x(\eta) \cos(\tau +\varphi_x(\eta))$$

and $$v=a_v(\eta) \cos(\tau +\varphi_v(\eta))$$ \[eq:NLM-solapprox\]

where $\tau=\omega t$, and $\omega$ is the angular frequency of the oscillations.

The term $G(a,\varphi,\tau)$ in equations (\[eq:SlowStand\]) can be simply derived and introduced in equations (\[eq:SlowMoy\]), which leads to:

\[eq:SlowMoyFree1\] $$2\omega a_x^{\prime} = 