**Numerical and Computational Challenges
in Science and Engineering Program**

**Wayne Hayes, The Fields Institute**

*LECTURE*S

### February 15, 2002, 10:00 am

*Shadowing Numerical Solutions of ODEs, with Applications*

Many dynamical systems studied today are "chaotic", which
implies that two solutions whose initial conditions differ by an arbitrarily
small amount will diverge exponentially away from each other. This is
worrisome to those who perform numerical studies of these systems, because
it means that numerical errors cause numerical solutions to diverge
exponentially away from the exact solution with the same initial conditions.
Surprisingly, for some (but not all) chaotic systems, it turns out that
there may exist an exact solution to the differential equation, called
a "shadow", that remains near the numerical one, although
it is not the exact solution that started with the same initial conditions
as the numerical one. If the study of the system does not depend upon
the precise choice of initial conditions, then the existence of a shadow
is strong evidence that the numerical solution faithfully exhibits the
properties of exact solutions to the problem being studied. In this
talk, I will outline how shadows of numerical solutions of ODEs can
be found, and discuss shadowing of the large gravitational N-body problem.

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