Abstract
Space exploration has often benefitted from the qualitative analyses of non integrable
problems enabled by numerical continuation procedures. Yet, standard
approaches based on Newton’s method typically end with discrete representations
of family branches that may be subject to misinterpretation and overlook important
dynamical features. In this research, we introduce novel continuation procedures
based on the differential algebra of Taylor polynomials. Our algorithms
aim at generating dense family branches as an atlas of polynomial charts that are
locally valid for a range of system and continuation parameters. Examples of
particular solutions will be shown within the framework of the Circular Restricted
Three-Body Problem, along with fold and period-doubling bifurcations that are efficiently
detected using automatic domain splitting and map inversion techniques.