Reliable Computing 8: 21–41, 2003.
2003 Kluwer Academic Publishers. Printed in the Netherlands.
Controlling the Wrapping Effect in the Solution
of ODEs for Asteroids
JENS HOEFKENS, MARTIN BERZ
Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory,
Michigan State University, East Lansing, MI 48824–1321, USA, e-mail: email@example.com,
Department of Physics, University of Illinois at Urbana-Champaign, Urbana, IL 61801–3080, USA,
(Received: 13 November 2001; accepted: 2 April 2002)
Abstract. During the last decade, substantial progress has been made in ﬁghting the wrapping effect
in self-validated integrations of linear systems. However, it is still the main problem limiting the
applicability of such methods to the long-term integration of non-linear systems. Here we show how
high-order self-validated methods can successfully overcome this obstacle.
We study and compare the validated integration of a Kepler problem with conventional and high-
order methods represented by AWA and Taylor models, respectively. We show that this simple model
problem exhibits signiﬁcant wrapping that is particularly difﬁcult to control for conventional ﬁrst-
order methods. It will become clear that utilizing high-order methods with shrink wrapping allows
the system to be analyzed in a fully validated context over large integration times. By comparing
high-order Taylor model integrations with Taylor model methods subjected to an artiﬁcial wrapping
effect, we show that utilizinghigh-order methods to propagate initial conditions is indeed the foremost
reason for the successful suppression of the wrapping effect.
To further demonstrate that high-order Taylor model methods can be used for the integration
of complicated non-linear systems, we summarize results obtained from a fully veriﬁed and self-
validated orbit integration of the near earth asteroid 1997 XF11. Since this asteroid will have several
close encounters with Earth, its analysis is an important application of reliable computations.
Since interval methods have been developed in the 1960s, an important area of inter-
est has been their use in the integration of ordinary differential equations (ODEs).
Starting with early work by Moore , , Lohner’s development of AWA ,
 and more recent work by Nedialkov  have provided the ﬁeld with great
boosts in terms of accuracy and performance. By in essence propagating ﬁrst-order
dependence on initial conditions, the methods provide effective performance for
linear problems without unduly suffering from the wrapping effect. However, it
has been recognized that propagating only linear dependence on initial conditions,
i.e., enclosing the image of the initial conditions by parallelepipeds, is often not
sufﬁcient to avoid the wrapping effect in the integration of non-linear systems.