and ${\mathbf{c}}$ is the new amplitude variable. The exponential decay allows a proper modelling of the dissipative aspect of the motion and can be related with a complex natural frequency (by analogy with linear complex modes), $$\lambda = - \zeta \pm i \omega$$ It was also assumed that this exponential decay is slowly varying.

Now, introducing the formalism of complex modes, defined by equation (\[eq:FormComplexModes\]), in system (\[eq:SlowMoyFree1\]) and noting $a_i(\eta)=c_i(\eta) e^{-\zeta \eta}$ for $i=x,v$, one obtains:

\[eq:CNM\] $$c_x^{\prime} = \zeta c_x - \dfrac{\epsilon_0}{2\omega}c_v\sin(\varphi_x-\varphi_v)$$ $$c_v^{\prime} = \zeta c_v  + \dfrac{\epsilon_1}{2\omega}c_x\sin(\varphi_x-\varphi_v) +\dfrac{1}{2\omega} c_r\sin(\varphi_v-\varphi_r)$$ $$c_x \varphi_x^{\prime} = \frac{\omega_0^2+\epsilon_0-\omega^2}{2\omega}c_x -\dfrac{\epsilon_0}{2\omega} c_v\cos(\varphi_x-\varphi_v)
      \label{eq:CNM3}$$ $$c_v \varphi_v^{\prime} = \frac{\epsilon_1-\omega^2}{2\omega}c_v -\dfrac{\epsilon_1}{2\omega}c_x\cos(\varphi_x-\varphi_v) +\dfrac{1}{2\omega}c_r\cos(\varphi_v-\varphi_r)
   \label{eq:CNM4}$$

It should be noticed that along with the assumptions made on the form of the displacements, equation (\[eq:FormComplexModes\]), the nonlinear term $a_r(\eta)$ is assumed to have the same exponential decay that the displacement. This assumption can be justified by looking at the expression of the averaged nonlinear term. In equation (\[eq:TermNLHarm\]), if we substitute for $\dot v$ the new complex form (\[eq:FormComplexModes\]), the exponential decay is reported outside the summation symbol and as a consequence, equation (\[eq:TermNLHarm\]) can be updated with the complex form defined by (\[eq:FormComplexModes\]).

If we finally combine equations (\[eq:CNM3\]) and 