Where & When: JBH359 10am-11am June 28 2007
Advisor: Dr. Schubert
Committee: Drs Botting and Gomez

Title: A SECURE CLIENT/SERVER JAVA APPLICATION
MATWIN:JA JAVA TOOL FOR EXPERIMENTING AND
COMPUTING IN DYNAMICAL SYSTEMS

ABSTRACT
The dynamic of any situation refers to how the situation changes over the 
course of time. A dynamical system is a physical setting together with 
rules for how the setting changes or evolves from one moment of time to the
next. It is the tool with which scientists make mathematical models to 
represent the real systems. One basic goal of the mathematical theory of 
dynamical systems is to determine or characterize the long-term behavior of
the system.

A basic observation of dynamical systems, largely due to the advent of 
computers and computer graphics, is how extraordinarily complicated the 
motions of a system can be, even when the underlying rules are extremely 
simple. An example for this is the NewtonUs description of the motion of 
bodies under gravity. The forces are extremely simple: bodies attract to 
each other by a force proportional to the product of their masses and 
inversely proportional to the square of the distances separating them. Yet
the motions caused by these forces are extremely complex, resulting, for 
instance, in the braided rings of Saturn.

One of the main approaches to dynamical systems is Differential equations.
Given any initial state, a differential equation, or a system of 
didderential quations are used to describe the forces, or directions in 
which a system is being pushed. Some of these differntial equations are 
easy to solve using integration methods to get exact solutions, while in 
some other cases, it is hard or even impossible to find exact solutions to
other system of differential equations. In such these cases where exact 
solutions are hard to find, approximate solutions can be calculated using 
numerical solution methods such as Euler and Runge Kutta. These numerical 
methods require a huge amount of calculations that need to be repeated over
and over for huge number of times, which is a waste of time plus the fact 
that these repeatd calculations can lead to huge mistakes in the final 
values calculated.

In the 1940s an entirely new tool of analysis also came on the scene-the 
digital computer. In addition to its obvious brute-force capabilities of 
grinding out nummerical solutions of differential equations, it presents a
flexible, interactive tool for the purpose of discoveries. This interplay 
between computations and analysis has proved to be of great importance, and
represents one of the major methods of ucovering dynamic properties. 
Indeed, the area of computer science has rapidly progressed to a state in 
which it can now make fundamental contributions to our knowledge, rather 
than acting oly as the servant of other methods of analysis.

Here, I prsent MATWIN, a toolkit for Windows computers, consisting of four
graphics programs for solving some systems of differential equations. These
graphics programs include functions to plot grapghs, tangents, derivatives,
calculate the roots, maxima and minima, and definite intgrals. It also 
includes a number of programs for drawing the soltion to a system of 
differential equations in a plane along with slope field lines. The initial
condition is determined by a mouse click at the desired point. With 3Dview
program, it is possible to have a 3D view for a system of three 
differential equations. The view can be rotated, translated or moved using
mouse clicks or via inserting the desired angle values in the program.
MatWin supports mathematical computation and experimentation in dynamical 
systems. This software was developed for a college level courses that teach
dynamical systems or differential equations. Also, it can be of great help
to high school calculus students too.