Sensitive dependence on initial conditions (S.I.C.)

This is a more powerful definition than you find in most books. It
is published in R.M. Corless,
``Continued Fractions and Chaos'', Proceedings of the Organic
Mathematics Workshop.
Suppose
our problem is

ie. , subject to

We ignore the boundary conditions on . If there exist
constants **M > 0** and (**M** not too large, not too small) such that for every
there exists a
with such that the
solution of

has for some then we say that the problem

is S.I.C

**Remark:** This definition says that *some* perturbations of
the initial condition lead
to finite separation
after an exponentially short length of time -- that is, initial
errors *may* grow exponentially,
for a short while.

The weasel words ``**M** not too large, not too small'' are
there for practicality. Notice
the limit is not involved. This makes it a
different definition than the
others in the literature.

Wed Feb 4 14:46:04 EST 1998