(\[eq:Energy\]).

This formalism introduces the notion of complex nonlinear modes which correspond (as in the linear domain) to special solutions of the free vibrations problem where the system’s degrees of freedom oscillate with the same frequency but, in contrast with non-linear *normal* modes, with a phase difference between them. The dissipative terms introduce a phase difference between the coordinates and a *modal* damping $\zeta$ which, as the natural frequencies and deformed shapes depends on the system’s energy. It may also be note that, in contrast with classical nonlinear normal modes, these complex modes as defined here are not necessary normal to the iso-energetic curves. With this complex modal description, one can handle the dissipative effects of the nonlinearity directly without any assumption regarding their importance. This particular feature will be used in the following numerical applications to determine design rules on the dissipation rates. This is a major concern, in engineering applications, when dealing with very weakly damped (linear) structure; in these situations, the nonlinear oscillator has to ensure the resonance capture phenomenon as well as the dissipation of the vibratory energy.

Stability
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The stability of the nonlinear modes can be determined using the system (\[eq:CNMr\]), which is in 