ELSEVIER Physica D 73 (1994) 388-398 Rotationally-ordered periodic orbits for multiharmonic area-preserving twist maps J.A. Ketoja a, R.S. MacKay b,l a Research Institute for Theoretical Physics, P.O.Box 9, FIN-O0014 University of Helsinki, HelsinkL Finland b Nonlinear Systems Laboratory, Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK Abstract The Poincar~-Birkhoff theorem guarantees existence of at least two rotationally-ordered periodic orbits of each rational rotation number for each area-preserving twist map. For many maps, however, there are more than two. We prove this for maps near a non-degenerate multi-well anti-integrable limit, and deduce an intricate bifurcation diagram for rotationaUy-ordered periodic orbits in so-called "multiharmonic" families. Our results are motivated and supported by numerical investigations of the reversible 2-harmonic family. We believe the results will be helpful for understanding the breakup boundary for invariant circles in multiharmonic families, which numerically exhibits a Cantor set of cusps. 1. Introduction Area-preserving twist maps are the simplest non-trivial class of Hamiltonian dynamical sys- tem. They also arise from variational problems relevant to incommensurate solid-state struc- tures. Area-preserving twist maps have the form (x',y') = F(x,y) (1) for some function F from R2 to itself such that F(x + 1,y) = F(x,y) + (1,0), detDF = + 1, Ox'/Oy > O. (2) (3) (4) The associated class of variational problems is to find the sequences x = (Xn)n~z which make the formal sum oo W(x) = ~ h(xn,xn+l) (5) n~ --oo stationary with respect to all variations of com- pact support. The correspondence is given by y' dx' - y dx = dh (x, x'). (6) Rotationally-orderedperiodic orbits play a fun- damental role for area-preserving twist maps and the associated variational problems. These are periodic orbits whose points come in the same horizontal order as for a rigid rotation by some rational rotation p/q, i.e. Xn+q = xn + p, (7) (8) I Nuffield Foundation Science Research Fellow 92-93. Yn +q ----Yn, 0167-2789/94/$07.00 (~ 1994 Elsevier Science B.V. All fights reserved SSD10167-2789 (94)00061-T