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

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

## 168 We consider the

168 We consider the following non-autonomous reversible ODE in R" : = f(t, x, A) (I) where f: R I+"+/--, It" is regular enough, let us say C °o to avoid any problems, T-periodic in t and f(t, 0, 0) = 0, Vt. Moreover, reversibility for (1) means that there is an involution R G GL(n,R), ie. R -I =/~, such that f(-t,]~z,A) = -\$gf(t,z,A), A E R t representing the parameters. In perturbed bifurcation theory they are usually split into two groups : the "main" bifurcation parameters (in general only one) and the perturbation parameters, also called the unfolding parameters. The distinction is important only in the discussion of the precise structure of the bifurcation diagrams, and so we shall come back to it later on. We are looking for qT-periodic solutions of (1), q = 3,4... (subharmonics) , ignoring period doubling. Among the numerous examples of reversible systems there are : 1. Hamiltonian systems. (a) The classical example is the case of the Hamiltonians of the form kinetic plus potential energy. (b) Systems like the restricted 3-body problem with the involution (q~, q2,Pl,P2) s~, (ql,-q2,-Pl,P2) (in the canonical coordinates). 2. Systems of ODE's. (a) Mathieu equation ... (b) Some coupled oscillators. 3. Stationary solutions of 1-D PDE's (reversibility is in space). (a) Reaction-diffusion systems with Neumann or periodic boundary conditions. The bifurcation of subharmonics correspond to secondary branching with (1, q)- mode interaction.The other mode interactions will be the subject of a subse- quent paper. (b) Fourth order equations Kuratomo-Shivashinsky, Malomed-Tribel'skii ... ([14],[15]). (c) Falkner-Skan equation: "ff + uii + 3 (1 - 6 2) = 0. Michelson [171: "ff + 6 + ½u 2 - c 2 = 0 , fu = O, for an asymptotic solution of the parabolic Kuratomo-Shivasinsky equation. The integral condition is in particular satisfied by odd functions, wlfich are reversible in that situation. Let us note that the involutions for those two equations are of opposite signature. In the second section we shall describe the functional analysis set-up and the more important hypotheses for the problem, leading by a Lyapounov-Schmidt reduction to the Dq-equivariant bifurcation equations. In the third section we study different aspects of the bifurcation equations, in particular emphasizing the case when n = 3. Here it is essential to take into account the different actions of Dq on the source and the target. A simple verification shows that we still are in the range of application of Damon's theory [4],

169 and so we give in a fourth part the classification up to topological codimension 1 of the Dq-equivariant problems in R 3 with one distinguished parameter (for the involution rt with only one eigenvalue at -1). We are not going to discuss any stability questions. Our set-up resembles a mode interaction problem. In that situation a more careful analysis is needed, in particular to look at the Birkhoff normal forms. As far as the trivial solution of (1) is concerned we do not have to require it to be the origin. We are actually looking at the bifurcation near a solution (zo, ;~0) where xo is reversible, ie. Xo(-t) = Rxo(t). With a translation to the origin we can keep the reversibility of the problem. It is well known (Sevryuk [20]) that if dim(FixR) > n/2 these solutions appear in families : the trivial branch or manifold. Autonomous systems are not excluded from our analysis, but some care must be taken to fully exploit the results exposed here. If x0 is a stationary solution there is no a-priori period. The classical approach is to introduce it as a parameter and this has been done, along with the case of the equivariance of the equation with respect to any compact group, in Golubitsky-Krupa-Lim [10]. In this work we deal with another situation : when xo is nontrivial. In that case we have a natural period and the translated equation becomes nonautonomous. Nevertheless, because the problem was originally autonomous there is an SLorbit of solutions generated by x0 which must be factored out to improve the efficiency of the analysis. This has been done for general systems in Vanderbauwhede [24]. His approach can be extended to reversible systems and we end up with an equation of type (1), perhaps in a reduced space. Acknowledgements I would like to thank the members of the Mathematics Institute of the University of Warwick for their hospitality, the SERC for a research grant and the referee for valuable suggestions. 2 Bifurcation Equations We carry out the procedure followed by Vanderbauwhede [23] where a detailed account of most of the well-known results can be found. We define (1) in a function space setting. Let us consider the following Banach spaces (equipped with their respective supnorm) : Xq = {x e Cl(it, it") l z is qT-periodic}, Yq = {Y e C°(tt, R") I Y is qT-periodic}, For computational convenience we shall need sometimes complex valued functions, but the extension is dear. (1) is equivalent to the equation F(x, A) = -~ + f(., z, A) = 0 (2) for F : Xq × 1~ t --* Yq. We have supposed that F(0, 0) = 0. F is Dq-equivariant, with two different actions on the source and the image, defined by : 6, the rotation generator, acts on Xq,Yq as a phase shift: ~z(t) = x(t + T), a, the reflexion generator, acts on Xq by az(t) = Rz(-t), on Yq by ay(t) = -Ry(-t).

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