Fig 1.
Interdependence between amplitude, frequency, and phase in the absence of internal resonance.
Left: Oscillation amplitude vs frequency detuning. Full and dotted lines respectively correspond to stable and unstable oscillations in the open-loop configuration. The light-grey curve stands for the backbone approximation. Right: Frequency detuning vs phase shift. Full and dotted lines correspond to the same stability properties as in the left panel. The units of amplitude and frequency are arbitrary, and the phase shift varies in the interval (0, π). The values of the parameters used to obtain the curves are given in Table 1.
Fig 2.
Internal resonance with a linear higher harmonic mode (I).
In this case, the coupling force acting on the main mode oscillator is proportional to the higher mode amplitude and to the square of the main mode amplitude. Rows A and B correspond to opposite signs of this coupling force. Full and dotted (respectively, dark- and light-green) lines correspond to stable and unstable oscillations in the open-loop (respectively, closed-loop) configuration. Arrows in row A mark curve segments referred to in the text. The light-grey curve is the backbone approximation. The corresponding parameters are given in Table 1.
Table 1.
Parameters used in Eq (6) to obtain the results plotted in Figs 1 to 4.
Fig 3.
Internal resonance with a linear higher harmonic mode (II).
As in Fig 2, when the coupling force acting on the main mode oscillator is proportional to its own amplitude and to the square of the higher harmonic amplitude.
Fig 4.
Internal resonance with a linear higher harmonic mode (III).
As in Fig 3, with the addition of a coupling force acting on the higher harmonic oscillator. This additional force is proportional to the higher harmonic amplitude and to the square of the main mode amplitude.
Fig 5.
Dependence of internal resonance on the amplitude of the driving force.
In all panels, larger force amplitudes are plotted with increasingly darked shades of green (see parameters in Table 2), and differences in stability are disregarded. (A) Effect of increasing the force on the resonance peak in the absence of internal resonance. (B,C,D) Effect on the internal resonance gap for the cases considered in Figs 2A, 3A and 3B, respectively.
Fig 6.
Internal resonance with a nonlinear higher harmonic mode: effect of the driving force.
All panels show the zone of the internal resonance gap, with curves for larger force amplitudes plotted with increasingly darker shades of green (see parameters in Table 3). Differences in stability are disregarded. Rows A to D respectively correspond to the coupling forces considered in Figs 2A and 2B and 3A and 3B.
Table 2.
Parameters used in Eq (6) to obtain the results plotted in Fig 5.
Table 3.
Parameters used in Eq (6) to obtain the results plotted in Fig 6.