Rigorous integration of maps and long-term stability

In spite of its importance, it has been very difficult to estimate long-term stability of particles in repetitive systems in a fully rigorous way. One of the main causes of the difficulty is the inaccuracy of the maps of the system; while to any fixed order they can be computed easily using Differential algebraic (DA) techniques, it has so far not been possible to determine bounds for the remainders. Another difficulty is that most methods to rigorously formulate the problem lead to the need for global optimization of highly complex multi-dimensional objective functions. The remainder-enhanced differential algebraic (RDA) method, an extension of the DA method that simultaneously provides rigorous bounds for the remainders, can solve both problems. The Taylor maps are evaluated rigorously with interval remainders, using the verified integral method within the framework of RDA. And rigorous RDA global optimization allows to efficiently get bounds on long-term stability.