Periodic orbits for reversible, symplectic mappings (Q1117725)

A 2N-dimensional symplectic mapping which satisfies the twist condition is obtained from a Lagrangian generating function F(q,q'). The q's are assumed to be angle variables. Reversible maps of this form have \(2^{N+1}\) invariant symmetry sets which are Lagrangian manifolds. We show that such maps have at least \(2^ N\) symmetric periodic orbits for each frequency \(\omega =(m,n)\). Furthermore we show there is at least one orbit which minmizes the periodic action for each \(\omega\), and generically at least \(2^ N-1\) other ``minimax'' orbits. There is a ``dominant'' symmetry plane on which the minimizing orbit is never observed to occur. As a parameter varies, the symmetric orbits are observed to undergo symmetry breaking bifurcations, creating a pair of non-symmetric orbits. A pair of coupled standard maps provides a four- dimensional example. For this case the two-dimensional symmetry planes give a cross section of the resonances and a visualization of the Arnol'd web.