Topological degree theory and local analysis of area preserving maps

We consider methods based on the topological degree theory to compute periodic orbits of area preserving maps. Numerical approximations to the Kronecker integral give the number of fixed points of the map provided that the integration step is "small enough." Since in any neighborhood of a fixed point the map gets four different combinations of its algebraic signs we use points on a lattice to detect the candidate fixed points by selecting boxes whose corners show all combinations of signs. This method and the Kronecker integral can be applied to bounded continuous maps such as the beam-beam map. On the other hand, they cannot be applied to maps defined on the torus, such as the standard map which has discontinuity curves propagating by iteration. Although the use of the characteristic bisection method is, in some cases, unable to detect all fixed points up to a given order, their distribution gives us a clear picture of the dynamics of the map. ©2003 American Institute of Physics.