nearly localized in the linear oscillator which modal curve is a straight line: the nonlinear system’s behaviour is linearizable. Note that, in this region, the natural frequency of the linear oscillator is constant. The dashed line branch shows, on the other hand, a prominent motion in the absorber and a decreasing natural frequency due to the softening characteristic of the nonlinearity.

High energies region: 

:   there, the situation is quite reversed, the solid line branch shows a strong localization in the nonlinear oscillator with a decreasing natural frequency; and the dashed line branch displays a prominent motion in the linear system.

Intermediate energy region: 

:   between the previous asymptotic states, the system experiences a bifurcation phenomenon of its modes. When the global energy increases, the motion of the linear oscillator jumps from a high to a low level; on the other hand, the motion of the nonlinear oscillator jumps from low to high level. Note that this phenomenon appears when the nonlinear natural frequency joins the linear one, the system of two coupled oscillators enters an internal resonance [@Nayfeh-Mook].

### Example 2. {#example-2. .unnumbered}

The second example involves an hysteretic nonlinearity with a rather high level of dissipation (see figures 