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SN~ (~6) lff 2It 3k_~ , 5 ",,x_J {b)

SN~ (~6) lff 2It 3k_~ , 5 ",,x_J {b)

I 2 118 5 6 7 9 10 11 13

I 2 118 5 6 7 9 10 11 13 I/, 16 21 Figure 9: Phase portraits corresponding to the regions marked in Figure 8. (o) SL S 3 (b) SL a Figure 10: Phase portraits for (9) at the global bifurcations. B

119 extent the dynamics is influenced by such symmetry breaking perturbations. In order to investigate this we have added to the r. h. s. of (9b) a term of the form er 3 cos ¢ and assumed that the phase evolution can be approximated by ¢ = t. ttere e is a small parameter. Of special interest is the dynamical behaviour near one of the global (SL~ or SL,~) bifurcations. For the unfolding (9) these occur in the forms shown in Figure 10. For the 3-d perfect (i. e. Sl-symmetric) system the phase portraits of Figure 10 are to be rotated about the x-axis. In the case of SL, this means that the two-dimenslonal unstable manifold of the right and the stable manifold of the intermediate equilibrium coincide. Because close to $'L, one period on the torus is very large we find either amplitude- modulated oscillations or relaxation towards the left equilibrium, depending on the initial conditions. On the other side of the SLs bifurcation only the relaxation to the equilibrium survives. Our numerical analysis indicates that this kind of behaviour also persists when the imperfection (e ~ 0) is switched on, provided e is not too large (e < 3). On the other hand, for larger E (e ,~ 10), the dynamics appears chaotic which in particular is suggested by a broad spectrum. This indicates transversal intersections of the stable and unstable manifolds and thus the appearance of horseshoes. These results are consistent with those described in [23] where the same kind of simulations is performed for Langford's P + C-normal form. While for sufficiently small e the dynamics near SL~ is adequately described by that of the perfect system (e = 0), the situation changes drastically near SLy,. Already for very small e (e ,,, 10 -s) we find phase locked periodic orbits on the torus with an odd period ratio that depends strongly on the distance from the SLa-locus in the unfolding space. Such a periodic orbit then undergoes a period doubling sequence when e is increased further. Presumably this kind of behaviour can be explainded by quadratic tangencies of the stable and unstable manifolds as discussed by Gavrilov and Shilnikov [8], Newhouse [21] and also recently by Gaspard and Wang [7]. Appendix We briefly describe the relevance of the singularity introduced and discussed in this paper for optically bistable systems. More details can be found in [4]. In the mean field limit the Maxwell-Bloch equations for a ring cavity are [19] = -pl [(1 + ie)a - A] - ~p 15 = a(1- d)- (1 + i$)p (A.1) 1 d = p2[~(a-~+'gp)-d]. Here a, p and d are proportional to slowly varying envelopes of the electric field, the macroscopic polarization and the atomic inversion, pl, p2 are ratios of relaxation con- stants, a is a field-matter coupling constant, A describes the amplitude of the incoming field and 8 and $ are proportional to, respectively, the atomic and cavity detuning. Since a and p are complex and d is real, (A.1) is a 5-dimensional system of differential equa- tions. The parameters Pl, P~, A, 0, 5, a are all real. Calculations with (A.1) greatly

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