# Wayne's Thesis Research

In chronological order, my main thesis papers are my
Master's Thesis (January 1995),
Ph.D. Qualifying Depth Paper (June 1996),
Research Proposal (February 1998),
Thesis Proposal (January 1999), and my
Ph.D. Thesis.

I'm looking at the **Reliability of Computer Simulation of the Large
Gravitational N-body Problem**. These are the kinds of simulations of
galaxies and globular clusters that astronomers like to do. Currently,
I'm looking from a *shadowing of chaotic systems* point of view.

N-body systems have been known to be chaotic for several decades. This
means that they display "sensitive dependence on initial conditions",
so that the phase space distance between two nearby solutions increases
exponentially with time. Since numerical solutions of these systems
introduce small errors like roundoff and truncation, a numerical
solution is guaranteed to diverge exponentially from the true solution
with the same initial conditions. This is the case even if quantities
such as the energy and momenta of the system are conserved to high
accuracy, because there are an infinite number of solutions all having
the same energy, but vastly different phase space trajectories. Thus,
the validity of a numerical simulation of an N-body system requires
careful consideration.

A counter-intuitive result of chaos theory is that it is sometimes
possible that a true solution exists that stays nearby to the numerical
solution, although the two never actually meet. If such a true
solution exists, it is called a "shadow" of the numerical solution.
Thus "shadowing" is a reasonably stringent method of global error
analysis where the measures of error are the nearness of the shadow to
the numerical solution, and how long the shadow stays close before
diverging from the numerical solution. If a shadow exists, it is a
very powerful statement that your numerical solution is faithfully
following the dynamics of the system.

**Currently**, I'm catching up on lots of background reading on
chaotic dynamical systems in general.
I'm also looking into more general work on the
reliability of N-body simulation, and also trying to learn more about
chaotic systems theory in general, which I am currently a bit rough in.

### Committee

My supervisor is
Ken Jackson.
In addition, my committee includes
Wayne Enright,
Tom Fairgrieve,
Rudi Mathon,
Ted
Shepherd of the Department of
Physics,
and Scott Tremaine of
the Canadian Institute of Theoretical
Astrophysics.

## Techreports

**A Fast Shadowing Algorithm for High Dimensional ODE
Systems**, techreport for a talk I gave at the Dynamical Numerical
Analysis conference at Georgia Tech, Dec 14-16, 1995. It is essentially
a highly condensed version of chapters 1-3 of my Master's thesis, with
a few more optimizations discussed.
Here's the abstract, and the gzip'd postscript of
the full text.

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