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

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

120 simplify if the

120 simplify if the additional assumption is made that the detuning parameters are opposite. Thus we have set 8 = -$ and so are left with five independent parameters. To determine the singularity introduced in Section 2 we have to impose four conditions, namely, two conditions for a cusp, one condition for the Itopf bifurcation and one further condition to make the Hopf bifurcation degenerate. Since we wish to determine numerical values for the parameters, we have to impose one further condition. We do this by requiring that the transversality condition for the Hopf bifurcation with respect to A (which is the appropriate "distinguished parameter") is broken as suggested by Armbruster [1]. By making extensive use of computer algebra (MAPLE), we were able to reduce these five conditions analytically to one single polynomial equation for Pl. The degree of this poly- nomial is very high (~ 250), but it could be factorized into polynomials of degree smaller than or equal to 27 by MAPLE. Then the zeroes of these polynomials were determined numerically. Only one of them turns out to be physically meaningful, thus we obtain a unique point (pl,p2,a,$,A) = (1.6, 1.0,212.2,4.0,87.7) in the parameter space where a degenerate Hopf bifurcation and a cusp coincide and where the transversality condition for the Hopf bifurcation is broken. After a center manifold reduction followed by a normal form transformation the coefficient e in (8) is found to attain the value c = -6.9. References [1] D. Armbruster, Z. Phys. B 53 (1983), 157-166. [2] D. Armbruster, G. Dangelmayr and W. Giittinger, Physica 16D (1985), 99-123. [3] G. Dangelmayr and D. Armbruster, Proc. London Math. Soc. 46 (1983), 517-546. [4] G. Dangelmayr and M. Wegelin, in: Proceedings of the NATO-ARW on continua- tion and bifurcations: Numerical techniques and applications, 18-22 September 1989, Leuven, to appear. [5] F. Dumortier, t{.. ttoussalre and J. Sotomayor: Generic three-parameter families of vector fields on the plane, unfolding a singularity. The cusp case of codimension three. Preprint. [6] J. Field and M. Burger: Oscillation in homogeneous chemical reactions, Wiley 1984. [7] P. Gaspard and X.-J. Wang, J. Stat. Phys. 48 (1987), 151-199. [8] N. K. Gavrilov and L. P. Shilnikov, Math. USSI~ Sbornik 17 (1972), 467-485 and 19 (1973), 139-156, [9] M. Golubitsky and W. F. Langford, J. Diff. Eq. 41 (1981), 375-415. [10] M. Golubitsky and D. Sehaeffer: Singularities and groups in bifurcation theory, Vol. I, Springer 1985. [11] J. Guekenheimer, in: D. A. Rand and L. S. Young (eds.): Dynamical systems and turbulence, Warwick 1980, Springer 1981.

121 [12] J. Guckenhelmer, SIAM J. Math. Anal. 15 (1984), 1-49. [13] J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bi- furcations of vector fields, Springer 1983. [14] J. P. Keener, SIAM J. Appl. Math. 41 (198I), 127-144. [15] W. F. Langford, SIAM J. Appl. Math. 37 (1979), 22-48. [16] W. F. Langford, in: G. J. Barenblat, G. Iooss, and D. D. Joseph (eds.): Nonlinear dynamics and turbulence, Pitman 1983. [17] W. F. La~gford, in: Differential Equations: Qualitative theory, Colloq. Math. Soc. Janos Bolyai 47 (1986). [18] L. A. Lugiato, V. Benza and L. M. Naxducci, in It. Haken (ed.): Evolution of order and chaos in physics, chemistry, and biology, Springer 1982. [19] L. A. Lugiato, L. M. Narducci and It. Lefever, in It. Graham and A. Wunderlin (eds.): Lasers and Synergetics. A colloquium on coherence and self-organization in nature, Springer 1987. [20] J. Marsden and M. McCracken (eds.): The Hopf bifurcation and its applications, Springer 1976. [21] S. E. Newhouse, Publ. Math. I. H. E. S. 50 (1979), 101-152. [22] T. Poston and I. Stewart: Catastrophe theory and its applications, Springer 1978. [23] P. Richetti, J. C. Roux, F. Argoul and A. Arneodo, J. Chem. Phys. 86 (I987), 3339- 3356. [24] E. Sch611: Nonequilibrium phase transitions in semiconductors, Springer 1987. [25] F. Takens, J. Diff. Eq. 14 (1973), 476-493.

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