Throughout this thesis, when referring to mathematical variables,
**boldface** will refer to vectors, and *italic* will refer to
scalars, matrices and functions. Scalars are written in *small*
letters and matrices in *CAPITALS*.

Some of the following definitions are taken, with minor modifications,
from Grebogi, Hammel, Yorke, and Sauer [11], hereinafter referred
to as GHYS.
The terms *trajectory*, *orbit*, and
*solution* are used interchangably throughout this thesis.

*Definition.* A *true trajectory* of *f*
satisfies for .
We are interested in the case where *a* and *b* are finite integers.
For a chaotic
map, *f* may be a simple equation, such as the logistic equation
, which always maps the interval [-1,1] onto itself.
For an ODE system like the *N*-body problem,
represents the true solution of integrating the phase-space co-ordinates
for one timestep.

*Definition.* is a
-*pseudo-trajectory*,
also called a *noisy orbit*, for *f* if
for , where is the noise amplitude.

*Definition.* For , the *1-step error*
made between step *i* and step *i*+1
of the pseudo-trajectory is the vector
.
Thus, a true trajectory is one whose 1-step errors are identically zero.

*Definition of shadowing.* The true trajectory
-*shadows* on if
for . The two
stages, of proving that a shadow exists, are *refinement* and *
containment*, defined below.

*Definition.* A *shadow step* is an interval, that can be larger
than the internal timestep of a numerical integrator, across which
a 1-step error is computed.

*Definition.* The pseudo-trajectory has a
*glitch* at point if for some relevant there
exists a true trajectory that -shadows
for , but no true trajectory that -shadows
it for , for .

The first group of chaotic systems for which it was proven that shadow orbits
exist was *hyperbolic* systems [2, 7].
In a 2-dimensional hyperbolic system,
there are two special directions called the *unstable* (or *
expanding*) and the *stable* (or *contracting*) directions, which
are generally not orthogonal. Small perturbations along the stable
direction decay exponentially in forward time, while small perturbations
in the unstable direction grow exponentially in forward time.
The two directions reverse roles in backwards time.
A ``trajectory'' for such a system can be imagined as a point moving in a 2D
plane, evolving through a third dimension, representing time.

For such a system it was shown that, if the angle between the
stable and unstable directions is uniformly bounded away from 0,
then a noisy trajectory can
be shadowed for all time. For non-hyperbolic systems, it appears that
shadows may exist only for finite time. The most important
question in this regard is, how *long* can a noisy orbit be shadowed?
If the time is at least as long as most typical simulations of
chaotic non-hyperbolic systems, then simulations have great validity;
if the shadowing time turns out to be too short, then a
less stringent error measure must be resorted to,
such as one listed previously.

*Definition.*
*Refinement* is an iterative process
that perturbs each point of a noisy orbit in an attempt to produce a
nearby orbit with less noise.
A refinement iteration is *successful*
if the trajectory before the iteration has noise
and the trajectory after the iteration has noise ,
and , for some reasonable
. Otherwise the iteration is *unsuccessful*.

*Definition.* *Containment* is a rigorous method to prove
the existence of a shadow orbit. See GHYS for details. Although
this is a good area for further work, containment is beyond the
scope of this thesis.

The refinement algorithm that concerns us in this thesis is the one first presented in two dimensions by GHYS, and generalized to handle arbitrary Hamiltonian systems by Quinlan and Tremaine [29], hereinafter referred to as QT.

QT make the distinction between *
dynamical noise* and *observational noise*. Observational noise
does not effect the future evolution of the system.
Laboratory measurements of a macroscopic system are usually of this type;
another example is computer output that prints fewer digits than are
represented internally. In contrast, dynamical noise does effect
the future evolution of the system. The noise introduced by numerical
solution of a system of ODE's is dynamical. QT studied
some existing noise-reduction algorithms in an attempt to refine noisy
trajectories of chaotic systems, but none worked as well as that
presented by GHYS. QT postulates that this may be because the other
noise reduction algorithms were designed to reduce observational
noise, whereas the GHYS procedure was designed to
reduce dynamical noise in a chaotic system.

Sun Dec 29 23:43:59 EST 1996

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