(\[eq:CNM4\]) and introduce the phase difference variables $\varphi_{ij}=\varphi_i - \varphi_j$, with $i,j=x,v \mbox{ or } r$, we obtain the following system:

\[eq:CNMr\] $$c_x^{\prime} = \zeta c_x - \dfrac{\epsilon_0}{2\omega}c_v \sin \varphi_{xv}$$ $$c_v^{\prime} = \zeta c_v  + \dfrac{\epsilon_1}{2\omega}c_x \sin \varphi_{xv} +\dfrac{1}{2\omega} c_r \sin \varphi_{vr}$$ $$c_x c_v \varphi_{xv}^{\prime} = \frac{\omega_0^2+\epsilon_0-\epsilon_1}{2\omega}c_x c_v -\dfrac{1}{2\omega}\left(\epsilon_0 c_v^2 -\epsilon_1 c_x^2 \right)\cos \varphi_{xv}-\dfrac{1}{2\omega} c_r \cos \varphi_{vr}$$

This unforced system is in (averaged) standard form and its fixed points are the main approximation of the nonlinear modes.

Along with this, we have to introduce an additional relation between the coordinates in order to normalize the modes. To do so, we define $\mathcal{H}$ the global energy of the system by: $$\label{eq:Energy}
    \mathcal{H} = \mathcal{T} + \mathcal{U} -\mathcal{W}_d$$ where $\mathcal{T}$, $\mathcal{U}$ are respectively the global kinetic and potential energies of the system; $\mathcal{W}_d$ is the energy dissipated by the hysteretic force during one cycle of the motion.

Also note that, if the non-linear modes can be approximated by the fixed points of equations (\[eq:CNMr\]), these will depend on the value of the angular frequency $\omega$ which is an unknown of the free dynamical problem. This angular frequency $\omega$, which represents the eigenfrequency of the modes, is determined using the additionnal energy relation 