Dynamic expansion points: an extension to Hadjidemetriou’s mapping method

Series expansions are widely used objects in perturbation theory in Celestial Mechanics and Physics in general. Their application nevertheless is limited due to the fact of convergence problems of the series on the one hand and constricted to regions in phase space, where small (expansion) parameters remain small on the other hand. In the mapping case, to overcome the latter problem, e.g., different expansion points are used to cover the whole phase space, resulting in a set of dynamical mappings for one dynamical system. In addition, the accuracy of such expansions depend not only on the order of truncation but also on the definition of the grid of the expansion points in phase space. A simple modification of the usual approach allows to increase the accuracy of the expanded mappings and to cover the whole phase space, where the series converge. Convergence problems due to the nonintegrability of the system can never be ruled out of the system, but the convergence of the series expansions in mapping models, which are convergent can be improved. The underlying idea is based on dynamic expansion points, which are the main subject of this article. As I will show it is possible to derive unique linear mappings, based on dynamically expanded generating functions, for the 3:1 resonance and the coupled standard map, which are valid in their whole phase spaces.