Differential algebras with remainder and rigorous proofs of long-term stability

Abstract

It is shown how in addition to determining Taylor maps of general optical systems, it is possible to obtain rigorous interval bounds for the remainder term of the n-th order Taylor expansion. To this end, the three elementary operations of addition, multiplication, and differentiation in the Differential Algebraic approach are augmented by suitable interval operations in such a way that a remainder bound of the sum, product, and derivative is obtained from the Taylor polynomial and remainder bound of the operands. The method can be used to obtain bounds for the accuracy with which a Taylor map represents the true map of the particle optical system. In a more general sense, it is also useful for a variety of other numerical problems, including rigorous global optimization of highly complex functions. Combined with methods to obtain pseudo-invariants of repetitive motion and extensions of the Lyapunov- and Nekhoroshev stability theory, the latter can be used to guarantee stability for storage rings and other weakly nonlinear systems.