standard form, and studying the eigenvalues of the Jacobian matrix $D_{{\mathbf{z}}}F({\mathbf{z_0}},M)$ at the equilibrium points. $$\begin{gathered}
    \label{eq:Jacob}
    D_{{\mathbf{z}}}F({\mathbf{z_0}},M)=\\
    \begin{bmatrix}
        \zeta & -\dfrac{\epsilon_0}{2\omega}\sin \varphi_{xv} & - \dfrac{\epsilon_0}{2\omega}c_v\cos \varphi_{xv} \\[0.5cm]
        \dfrac{\epsilon_1}{2\omega}\sin \varphi_{xv} & \zeta + \dfrac{1}{2\omega} \dfrac{\partial (c_r\sin \varphi_{vr})}{\partial c_v} & \begin{array}{l} \dfrac{\epsilon_1}{2\omega}c_x\cos \varphi_{xv}\\\hspace{1.2cm}  + \dfrac{1}{2\omega} \dfrac{\partial (c_r\sin\varphi_{vr})}{\partial \varphi_{xv}}\end{array}\\[0.5cm]
            \dfrac{\omega_0^2+\epsilon_0-\epsilon_1}{2\omega} c_v +\dfrac{\epsilon_1}{\omega}c_x\cos \varphi_{xv} &  \begin{array}{l} \dfrac{\omega_0^2+\epsilon_0-\epsilon_1}{2\omega}c_v -\dfrac{\epsilon_0}{\omega}c_v\cos \varphi_{xv}\\\hspace{1.2cm} - \dfrac{1}{2\omega} \dfrac{\partial c_r \cos \varphi_{vr}}{\partial c_v}\end{array} & \begin{array}{l} \dfrac{1}{2\omega}\left(\epsilon_0 c_v^2 -\epsilon_1 c_x^2 \right)\sin \varphi_{xv}\\ \hspace{1cm}-\dfrac{1}{2\omega} \dfrac{\partial(c_r \cos \varphi_{vr})}{\partial \varphi_{xv}}\end{array}
    \end{bmatrix}\end{gathered}$$

Numerical results, localization and energy pumping {#ssect:NLM_Num}
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The hysteretic Bouc-Wen model is quite versatile and a large variety of hysteretic loops can be modelled. In this study, we focused on the class of softening hysteretic nonlinearity (*i.e.* $\gamma=-0.5\leqslant0$ for example in equation (\[eq:BoucWen\])) and we particularly investigated the influence of the (nonlinear) dissipation due to hysteresis. As an example, let’s consider the two hysteretic loops depicted in figures \[fig:Cycle1\] and \[fig:Cycle2\] and their equivalent stiffness and damping (first harmonic). Both of these examples feature a strongly nonlinear behaviour. Their equivalent stiffness are quite similar as opposed to their equivalent damping. Actually as the loop area (which is directly related to the energy dissipation) of the second example is larger, its equivalent damping 