-\epsilon_0a_v\sin(\varphi_x-\varphi_v)$$ $$2\omega a_v^{\prime} = \epsilon_1a_x\sin(\varphi_x-\varphi_v) -\dfrac{1}{m} a_r\sin(\varphi_r-\varphi_v)$$ $$2\omega a_x \varphi_x^{\prime} = (\omega_0^2+\epsilon_0-\omega^2)a_x -\epsilon_0 a_v\cos(\varphi_x-\varphi_v)$$ $$2\omega a_v \varphi_v^{\prime} = (\epsilon_1-\omega^2)a_v -\epsilon_1a_x\cos(\varphi_x-\varphi_v) +\dfrac{1}{m} a_r\cos(\varphi_r-\varphi_v)$$

As the system (\[eq:SlowMoyFree1\]) is in standard first-order form, we can study its fixed points by making all time derivatives zero (left hand side). Then one obtains a system of nonlinear equations (with amplitudes $a_j$ and phases $\varphi_j$ along with the angular frequency $\omega$ as unknowns) which can be interpreted as a non-linear eigenvalue problem to find an approximation of the non-linear modes.

However, from the first equation of system (\[eq:SlowMoyFree1\]), one can notice that the fixed points necessary verify, $$\varphi_x-\varphi_v = 0 \mod(\pi)$$ Carrying this result in the second equation, one finds that $$a_r\sin(\varphi_v-\varphi_r)=0$$ which is absurd because first $a_r$ cannot be zero and second because the hysteretic and dissipative characteristics of the nonlinearity impose a non-zero phase difference between the restoring force $r$ and the displacement $v$.

In order to avoid this, the general form of the solutions (given by equations (\[eq:solapprox\]) or (\[eq:NLM-solapprox\])) needs to be adapted. The formalism was inspired by the complex mode definition for linear systems and is given by: $$\label{eq:FormComplexModes}
    {\mathbf{z}}(\tau,\eta)={\mathbf{c}}(\eta) e^{-\zeta \eta} \cos(\tau +{\mathbf{\varphi}}(\eta))$$ where $\zeta$ represents the modal damping 