Which residual mode captures the energy of the dominating mode in second order Hamiltonian systems?

Structural instability of nonlinear plates modelling suspension bridges: Mathematical answers to some long-standing questions

“…The results in [9] tell us that the energy transfer occurs when the ratio between the torsional and longitudinal frequencies is small (close to 1). And in Remarks 5.1 and 6.8 we saw that the energy of the k-th longitudinal mode for k = 8, 9, 10 transfers earlier to the second torsional mode rather than to the first.…”

“…Theorem 4.5, which is proved in Section 8, raises the attention on the further technical assumption (39). We are confident that it might be weakened (see Remark 8.4 at the end of Section 8) and, perhaps, completely removed (see [9]). The reason is that, although they are not explicitly known, the stability regions for the Hill equations are very precise and, for the model problem (34), there is an additional energy parameter which could vary the stability regions.…”

“…minimal (close to 1) precisely for the tenth longitudinal eigenfunction: why small ratios yield strong instability is explained for a simplified model through Mathieu equations in [9]. However, in the case of a bridge, the sustaining cable yields a serious constraint.…”

“…We have so determined the constants in the model problem (9) and, defining u as in (12), we come to the adimensional problem (14) where…”

“…For a particular second order Hamiltonian system of the kind of ( 28), it is shown in [9] that there exist two increasing and divergent sequences {A n l } ∞ n=0 (l = 1, 2) such that:…”