allows us to impose an additional condition. It is usual to choose that the velocity has a similar form to the linear case: $$\label{eq:CompDeplVit1}
    \dot{{\mathbf{z}}} =\frac{d{{\mathbf{z}}}(\tau,\eta)}{dt} = -\frac{d\tau}{dt}{{\mathbf{a}}}(\eta) \sin(\tau +\varphi(\eta))$$ The time scale $\tau$ can then be interpreted as the time scale of the periodic motion whereas, the time scale $\eta$ represents a perturbation time scale. Also, in equation (\[eq:CompDeplVit1\]), the term: $$\label{eq:Omega}
    \frac{d\tau}{dt}=\omega$$ represents an angular-frequency-like “variable” assumed to be constant in time. However, this angular frequency $\omega$ can be amplitude-dependent and *a priori* unknown as in the case of nonlinear modes (see section \[sect:NonlinearModes\]) or not as in the case of steady-state forced response where it corresponds to the excitation frequency (see section \[sect:ForcedResp\]).

Equation (\[eq:CompDeplVit1\]) now becomes: $$\label{eq:Vit}
    \dot{{\mathbf{z}}} = -\omega {\mathbf{a}}(\eta) \sin(\tau +{\mathbf{\varphi}}(\eta))$$

Differentiating (\[eq:solapprox\]) with respect to time and equating the result with (\[eq:Vit\]), we find: $$\label{eq:Slow1}
    {\mathbf{a}}^{\prime} \cos(\tau +{\mathbf{\varphi}}) - {\mathbf{a}} {\mathbf{\varphi}}^{\prime} \sin(\tau +{\mathbf{\varphi}}) = {\mathbf{0}}$$ where ${.}^{\prime}$ denotes derivatives with respect to the slow time scale $\eta$.

We have obtained the first equation governing the evolution of the slow flow variables ${\mathbf{a}}$ and ${\mathbf{\varphi}}$. The second one can be obtained by first differentiating (\[eq:Vit\]) with respect to $t$, which yields: $$\label{eq:Accel}
    \ddot{{\mathbf{z}}} = 