given by the following differential equation : $$\dot r (t)= A \dot v(t) - \nu \left( \beta |\dot v(t)| |r(t)|^{n-1} r(t) - \gamma \dot v(t) |r(t)|^n \right)
    \label{eq:BoucWen}$$ where $A$, $\nu$, $\beta$, $\gamma$ and $n$ are the loop parameters of the Bouc-Wen model. A proper choice of these parameter allows to describe a wide range of hysteresis loops, with softening or hardening behaviour, different levels of nonlinearity, with various intermediate states possibilities (smooth or bilinear). In what follows, the parameter $n$ is set to $1$.

Asymptotic analysis {#sect:AsymptoticAnalysis}
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In this section, we will use an averaging method to derive the periodic solutions of a nonlinear system such as system (\[eq:mvtR\]). Let’s consider the following general dynamical problem: $$\label{eq:mvt}
    \ddot{{\mathbf{z}}}(t) + g\left(\dot{{\mathbf{z}}},{\mathbf{z}},t\right)={\mathbf{0}}$$ in which ${\mathbf{z}}$ is a displacement vector and the term $g$ includes along with linear (stiffness) terms, any nonlinear term and excitations.

We seek a solution of (\[eq:mvt\]) in the following form: $$\label{eq:solapprox}
   {\mathbf{z}}(t)={\mathbf{z}}(\tau,\eta)={\mathbf{a}}(\eta) \cos(\tau +{\mathbf{\varphi}}(\eta))$$ where the amplitude ${\mathbf{a}}$ and phase ${\mathbf{\varphi}}$ are slowly varying quantities (time scale $\eta$) with respect to the fast time scale $\tau$. This transformation from the displacement dependent variables (one variable respectively) to the amplitudes and phases dependent variables (two variables respectively) 