166  M. W. Hirsch, C. C. Pugh and M. Shub. Invariant Manifolds, (Springer Lect. Notes Math., 583, 1977).  M. Krupa. 'Bifurcations of Relative Equilibria', to appear in Siara J. of Math. Anal.  S. S. Smale. 'Differentiable Dynamical Systems', Bull. Araer. Math. Soc., 73(1967), 747-817.  M. G. Golubitsky, M. Krupa and A. Vanderbauwhede. 'Secondary bifurcations in symmetric systems', Lecture Notes in Mathematics 118, (eds. C. M. Dafermos, G. Ladas, G. Papauicolaou), Marcel Dekker Inc., (1989). Center for Applied mathematics Sage Hall Cornell Ithaca, NY U.S.A. Department of Pure Mathematics The University of Sydney Sydney, N.S.W. 2006 Australia
On the Bifurcations of Subharmonics in Reversible Systems J.E.Furter Mathematics Institute University of Warwick Coventry CV4 7AL Abstract Following Vanderbauwhede's approach , the study of the local bifurcation of subharmonics in reversible systems leads to reduced equations equivarlant under the dihedral groups. Depending on the dimension of the space, or on the type of the involution, the bifurcation equations can change significantly. We investigate some unusual properties of those equations. In particular we classify up to topological codimension 1 the degenerate bifurcations when the dimension of the space is odd and the signature of the involution is +1. 1 Introduction We propose to extend the investigation of the local bifurcations of subharmonlc solutions of reversible systems (whose flow is reversed by an involution of the phase space). These systems have been most widely studied for their KAM-type theory. Bearing a lot of similarities with the Hamiltonian systems, that theory has been looked at by a string of authors from Moser  to Scheurle  and more recently in the book of Sevryuk  and the extension to the non-analytic case of Pluschke . Another direction of study has been the structure of the periodic solutions, often co- ming in families. From the initial works of Birkhoff , for the restricted 3-body problem, and Hale , systems with "property E', the theory unfolded to the global considera- tions of Wolkowisky , in the plane, or to more general systems. When the involution reverses half the coordinates, Devaney  gave an account of the generic properties of families of periodic solutions and a Lyapounov theorem. Some global results were given by Kirchg~sner-Scheurle . An extension of Devaney's results to involutions reversing less than half of the coordinates is given in Sevryuk . More recently Vanderbanwhede  studied the generic bifurcation of families of sub- harmonics in the case of a non-trivial crossing of a root of unity by a multiplier, providing an example of bifurcation with the symmetry of the dihedral groups Dq. As a consequence of the nonresonance condition no other multiplier can be at 1. This is never satisfied if the dimension of the space is odd or for some type of involution, where some multipliers are locked at 1. We would like to extend the analysis to these cases. We should point out that symmetry is another situation where multipliers are forced at unity (Golubitsky- Krupa-Lim ).