An Efficient Method for Locating and Computing Periodic Orbits of Nonlinear Mappings

The accurate computation of periodic orbits of nonlinear mappings and the precise knowledge of their properties are very important for studying the behavior of many dynamical systems of physical interest. In this paper, we present an efficient numerical method for locating and computing to any desired accuracy periodic orbits (stable, unstable, and complex) of any period. The method described here is based on the topological degree of the mapping and is particularly useful, since the only computable information required is the algebraic signs of the components of the mapping. This method always converges rapidly to a periodic orbit independently of the initial guess and is particularly useful when the mapping has many periodic orbits, stable and unstable, close to each other, all of which are desired for the application. We illustrate this method first on a two-dimensional quadratic mapping, used in the study of beam dynamics in particle accelerators, to compute rapidly and accurately its periodic orbits of periods p = 1, 5, 16, 144, 1296, 10368 and then obtain periodic orbits of its four-dimensional complex version for periods which also reach up to the thousands.