|Taylor models combine the advantages of numerical
methods and algebraic approaches of efficiency, tightly controlled
and the ability to handle very complex problems with the advantages of
symbolic approaches, in particularly the ability to be rigorous and to
allow the treatment of functional dependencies instead of merely points.
The resulting differential algebraic calculus involving an algebra with
differentiation and integration is particularly amenable for the study
of ODEs and PDEs based on fixed point problems from functional analysis.
We describe the development of rigorous tools to determine enclosures
of flows of general nonlinear differential equations based on Picard
iterations. Particular emphasis is placed on the development of methods
that have favorable long term stability, which is achieved using
suitable preconditioning and other methods.
Applications of the methods are presented, including determinations of
rigorous enclosures of flows of ODEs in the theory of chaotic dynamical