126 T.L. Hill et al. Nu D 3˛1up1um1u1 C2˛3up2um2u1 2˛3up1um1u2 C3˛5up2um2u2 Cı r 1 3 ˛4 u3 p2 Cu3 m2 3˛4 up1u2 m2 Cup1u2 m2 ! C (12.15) ıfr 1g 0 @ 3˛2 2up1um1u2 Cu2 p1um2 Cu2 m1up2 C˛3 um1u2 p2 Cup1u2 m2 C3˛4up2um2u2 3˛4 2u1up2um2 Cum1u2 p2 Cup1u2 m2 C˛3 u2 p1um2 Cu2 m1up2 C3˛2up1um1u1 1 AC Cıfr 3g 0 @ 3˛2 u2 m1up2 Cu2 p1um2 ˛2 u3 p1 Cu3 m1 1 A ; where ı represents the Dirac-delta function. 12.3 The Backbone Curves of the Example System In order to find the backbone curves, we must solve the time-dependent resonant equations of motion, Eq. (12.12), which first requires that the time-dependence is removed from these equations. In [8] it is shown that the ith element of the vector of resonant nonlinear terms, Nu, may be written Nui DNCui eCj!rit CN ui e j!rit ; (12.16) therefore, substituting Eqs. (12.5) and (12.16), the resonant equation of motion, Eq. (12.12), for the ith mode may be written !2 ni ! 2 ri Ui 2 e j i CNCui eCj!rit C !2 ni ! 2 ri Ui 2 eCj i CN ui e j!rit D0; (12.17) where the contents of the square brackets form a complex conjugates pair. Therefore, the contents of these brackets may each be equated to zero, i.e. !2 ni ! 2 ri Ui C2NCui eCj i D0; (12.18) where it can be seen that Eq. (12.18) is independent of time. Now, substituting Eq. (12.5) into Eq. (12.15) allows the complex components NCui to be identified. These may then be substituted into Eq. (12.18) to give 4 !2 n1 ! 2 r1 U1 C3˛1U3 1 C2˛3U1U2 2 Cı1=3˛4U3 2eCj d1;3 Cı33˛2U2 1U2eCj d3;1 Cı1 3˛2U2 1U2 2Ce j2 d1;1 C˛3U1U2 2eCj d1;1 C3˛4U3 2 eCj d1;1 D0; (12.19a) 4 !2 n2 r 2!2 r1 U2 C2˛3U2 1U2 C3˛5U3 2 Cı1=33˛4U1U2 2e j d1;3 Cı3˛2U3 1e j d3;1 Cı1 3˛2U3 1 C˛3U2 1U2e j d1;1 C3˛4U1U2 2 2CeCj2 d1;1 e j d1;1 D0; (12.19b) where the phase difference, di;j, is defined as di;j Di 1 j 2, and the Dirac-delta function is denotedık Dıfr kg. Note that !r2 Dr!r1 has been used. Equation (12.19) demonstrate that some terms are a function of the phase difference between the two modes (where the phase difference is dependent on r). As will be shown in the following sections, such terms enforce a specific phaserelationship for resonant responses described by the backbone curve, known as phase-locking. It therefore follows that backbone curves that are described by expressions which are not a function of the phase difference do not have a specific phase-relationship, and the responses they describe may therefore exhibit any phase value between the modes. Furthermore, it can be seen in Eq. (12.19) that the terms which exhibit a phase-dependence are also those that are dependent on r i.e. are conditionally resonant terms, suggesting a relationship between conditional resonance and phase-locking.
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