# 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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