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Appendix

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.



Robert Corless
Wed Feb 4 14:46:04 EST 1998