Resonance condition, geometrical

Fig. 3 Geometrical representation of the resonance condition. The phase difference after two reflections at the boundary layers has to be a multiple of In. Additionally, a phase shift of occurs at each interface...

In quasi-integrable systems we do not find only KAM tori, but also resonant motions, and among resonant motions we find the chaotic ones. If e is small and h satisfies a suitable geometric condition (convexity of h is sufficient) the Nekhoroshev theorem proves the exponential stability of the actions for all initial conditions, including the resonant ones. More precisely, there exist positive constants eo, a,b,Io,to such that if < so, for any (I(0),ip(0)) e B x T it is I(t) — /(0) < Io a for any time t satisfying the exponential estimate ... 

It is very important to stress that changes in the geometrical shape of the integration domain can induce bifurcations in the drift velocity field [31, 47, 50, 52]. Let us consider, for example, the drift velocity field computed for an elliptical domain with major axis o = 3A and minor axis b = a/1.1. As shown in Fig. 9.13(a), instead of the stable limit cycle of the resonance attractor in the circular domain of radius Rg, = 1.5A we have two pairs of fixed points where the drift velocity vanishes. In each pair, one fixed point is a saddle and the other one is a stable node. Depending on the initial conditions, the spiral wave approaches one of the two stable nodes. Trajectories of the spiral center obtained by numerical integration of the Oregonator model (9.1) are in perfect agreement with the predicted drift... 

Also the analysis of relative intensities (as a function of primary energy and geometric conditions) is important to assess the type of mechanism acting on the excitation. In polystyrene and for electronic excitations, that analysis showed that the behavior of relative intensities as a function of primary energy is typical of resonant interactions. This is compatible with the fact that triplet excitation (optically forbidden for spin reasons) intensity is favored relative to singlet excitation. 

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