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All of the results of the previous sections are valid for the familiar domain of the real numbers. However, when we work in any fixed-precision system, we have two difficulties:

- Not all numbers are even representable in the system, and
- Arithmetic doesn't have the properties we are used to.

To return to the analogy of the introduction, we consider the domain
of machine representable numbers not as a smooth circle but as a
slotted ring, with the number of slots on the ring corresponding to
the number of machine-representable numbers in the interval ,
where all numbers in ** u** are ``lumped together''.
In this analogy, ** u** corresponds to the approximate width of
the slots. Now our bead can occupy one of the slots on the ring,
and not just any arbitrary position, and the floating-point Gauss
map takes the bead from one slot, winding around the ring as
many times as are indicated by the integer part, and finally putting
the bead into another slot. We see now that the maximum winding
number of the floating-point Gauss map is finite, and the slot next
to the origin is the one with this winding number.

A more evident difficulty is that all the representable points are
* rational*, and we know that the exact Gauss map takes these
initial points to zero eventually. So if we define a floating-point
Gauss map as

- Are there any orbits which
*don't*go to 0? - Is the behaviour of the floating-point Gauss map similar to the exact Gauss map? In particular, is chaotic?
- Can we define an appropriate Lyapunov exponent for this map?
- Is numerical work with useful at all for study of
**G**?

Not surprisingly, some orbits under do terminate at 0, though
often not when we expect them to. However, on some machines, some orbits
never hit **0**, being periodic. For example on the HP28S the initial
point gives ,
, and , with period
**3**. Note that under the exact Gauss map the second iterate ()
of this initial point is zero. Since the set of machine-representable
numbers is finite, * all* orbits are ultimately periodic (perhaps
with period **1**, as at **x=0**). Note that the behaviour of
depends strongly on the floating-point implementation. For example,
with the Apple SANE numerics implementation, the starting point gives an orbit with either a long transient regime or a long period,
with no regularity detected in the first **65,000** elements of the orbit.
A Maple implementation of Floyd's algorithm for
finding when the cycle starts.

Since all orbits are ultimately periodic, and there are only a finite
number of such orbits, the floating-point Gauss map (and indeed * any*
machine simulation of * any* dynamical system) is ** not** chaotic
in the usual sense. Arbitrarily small perturbations in the initial
conditions are not even allowed, so the sensitivity of the map to such
perturbations is moot. The alternate definition of S. I. C. requires arbitrarily
close initial conditions, which cannot happen in a finite set.
Similarly, the usual definition of the Lyapunov exponent for
the exact Gauss map seems not to be relevant here: the presence of the
derivative in the definition of the Lyapunov exponent measures the
effect of arbitrarily small perturbations. However, if we define
an approximate Lyapunov exponent for the first **N** iterations of the
floating point Gauss map as

But what if the calculated orbit has no counterpart in the exact system?
If roundoff errors introduced into the calculation produce an orbit that
is unlike any in the exact system, this approximate Lyapunov exponent would
be spurious. We will give a proof in the following section, which uses
the techniques of floating-point error analysis, that shows orbits under the
floating-point Gauss map are ``machine close'' to corresponding orbits
under the exact Gauss map. A general theorem of this nature has been
proved for hyperbolic invariant sets, by Bowen (see [10]).
Here a direct proof is more appropriate and
informative. This means that the approximate Lyapunov exponent defined
above will accurately reflect the Lyapunov exponent of * some*
orbit of the exact Gauss map, provided **N** is large enough that
transient effects have been minimized, and not so large that
accumulated roundoff error in the sum degrades the result.
This is usually not a large problem.

We contrast this behaviour with what happens when continuous maps are
made discrete by (e.g.) finite difference schemes.
Yamaguti & Ushiki [29]
and Ushiki [28] have shown that finite difference formulae
applied to non-chaotic continuous systems may produce chaotic
numerical solutions if the stepsize is not too small, assuming
that the calculations are carried out * exactly*. In contrast
we have noted here that a chaotic discrete map becomes nonchaotic
when implemented in fixed-precision arithmetic.
A further difficulty is that all of the orbits of are ultimately
periodic, and periodic orbits of **G** have Lyapunov exponents that are
different from the almost-everywhere value (which is usually the exponent
of physical interest).

It is not immediately clear that these Lyapunov exponents calculated
from will tell us anything useful about the exact map **G**.

On closer examination, however, we see that if the period of an orbit is long, then the orbit behaves for a long time as if it were aperiodic, reflecting the effect of `nearby' initial points that are aperiodic.

Hence we may expect that the computed Lyapunov exponent of a long
period orbit will be close to .
This is what happens in practice, since many initial points seem to give
long period orbits.
For example, if we compute the first 100,000 elements of the orbit
of under on the HP28S, we get a computed
. This is within % of the expected value of
the Lyapunov exponent of the exact Gauss map.
Notice, though, that the Lyapunov exponent of the orbit of the exact
map **G** starting at is not even * defined*---we * rely*
on the roundoff error to give us our results, which is somewhat unusual.
We will expand more on this in a later section.

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