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\title{Verified Integration and Generation of Poincare Maps Using
Taylor Models}
\author{Martin Berz}

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\markright{APIC'95, El Paso,
Extended Abstracts,
A Supplement to the international journal of {\rm Reliable
Computing}\ \ \ \ \ \ \ \ \ \ \  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ }

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\auffil{The author is with 					
Department of Physics and Astronomy,
Michigan State University,
East Lansing, MI 48824,
e-mail BERZ\@NSCL.NSCL.MSU.EDU,
517-333-6313 (Phone),
517-353-5967 (Fax).}


The field of nonlinear dynamics is simultaneously one of the oldest and one of
the newest areas of research in mathematical physics; originating in the study
of planetary motion and the stability of the solar system, it has experienced a
renaissance in many scientific disciplines. One of the many currently important
problems is the study of long term stability of motion in large particle
accelerators, which formally can be characterized by moderately nonlinear
differential equations also characteristic of planetary motion. 

In case nonlinear dynamics is governed by differential equations, it is 
often described by the so-called Poincare map which expresses the functional 
dependence of final coordinates on initial coordinates for a discrete step 
in the independent variable. Using nonlinear normal form methods 
\cite{monthnf}, we have recently shown \cite{neklaf} how a Poincare map can 
be used to extract guaranteed bounds for stability times of dynamical 
systems. In particular, the methods were applied to the study of dynamics 
in particle accelerators for several practical cases \cite{disshoff94}.

The substantial complexity associated with the practical use of the method,
reflected in execution times of hours to days, requires careful control of blow
up as well as efficient strategies to perform verified six-dimensional
optimization. For both of these problems it proved necessary to depart from
plain interval arithmetic and to represent functional dependencies by Taylor
polynomials with interval remainder bounds. These so-called Taylor models
\cite{rdaic} represent a hybrid of high-order multivariate automatic
differentiation tools \cite{adalgo} to obtain the Taylor polynomials and
interval methods to bound the Taylor remainder. 

The rigor associated with the method requires a rigorous knowledge of the 
Poincare map. Furthermore, any error bounds for it have to be tightly 
controlled since they inversely proportionally translate into the stability 
times that can be guaranteed with the normal form methods. To determine the 
value of the Poincare map at individual initial conditions requires the use 
of verified numerical integrators. We will show how Taylor model methods can 
be combined with Banach fixed point and differential algebraic techniques to 
obtain verified solutions of differential equations. The error bounds are 
obtained directly from within the method and do not rely on conventional 
estimates requiring the separate bounding of higher derivatives. 

Since the Poincare map does not only represent the functional dependence of 
coordinates on time but also on initial conditions, it is necessary to 
extend the integration methods in such a way as to describe this dependence 
in a functional form. This requirement in particular is conveniently 
achieved with the use of Taylor models. 

Applications of the method to complex practical problems in four and six 
dimensions from the field of accelerator physics will be given. 

\begin{thebibliography}{99}

\bibitem{monthnf}
M. Berz,
``High-Order Computation and Normal Form Analysis of
               Repetitive Systems'', in: M. Month (Ed),
               {\bf Physics of Particle Accelerators},
AIP 249, American Institute of Physics, 1991, p. 456 ff.

\bibitem{neklaf}
M. Berz and G. Hoffst{\"a}tter,
``Exact estimates of the long term stability
               of weakly nonlinear systems
               applied to the design of large storage rings'',
{\it Interval Computations} (in press).

\bibitem{disshoff94},
G. H. Hoffst\"atter,
{\bf Rigorous estimates of survival times in storage rings
               and efficient computation of fringe--field transfer
maps}, Ph.D. Thesis,
Michigan State University, Michigan, {USA}, 1994.

\bibitem{rdaic}
M. Berz and G. Hoffst{\"a}tter,
``Computation and Application of Taylor Polynomials 
               with Remainder Bounds'',
submitted to {\it Interval Computations}, 1994.


\bibitem{adalgo}M. Berz,
``Forward Algorithms for High Orders and Many Variables'',
In: {\bf Automatic Differentiation of
               Algorithms: Theory, Implementation and Application},
SIAM, Philadelphia, 1991.

\end{thebibliography}
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