# AiS Challenge Team Interuim

Team Number: 76 School Name: Santa Fe High School

Area of Science: Chaos Theory

Project Title: A Computational Approach to the Chaos Theory

Problem Definition: Chaos math is a growing part of all sciences, a broad and interesting field encompassing many different areas. Gone is the day of vague uses for the once enigmatic chaos theory; today, these have been replaced by complex math models and simulations used for everything from traffic control to population growth.

We will be simulating water flow on different surfaces using chaos math. By reproducing these currents on a molecular level using a lattice format, we can show anything from a small, unobstructed stream to a plane covered in various obstacles. This program could be used for a variety of things: flood, river, stream, and dam simulation, etc. We chose this project because not only is chaos math a fast-growing yet largely unexplored field that we are interested in, we also chose it because of the fact that we live in New Mexico and this program could also help with the modeling of the erosion on all of our mountainous and hilly areas.

Problem Solution:

The basic format of our program is as follows. An array of identical cells to which rules can be applied, a cellular automaton, is set up. These cells interact with each other to model all sorts of things: in our case, fluid movement. Cellular automaton consist of three basic parts:

• State: a variable for each cell (In our case, some cells will have a molecule variable placed on them to represent a real water molecule. So each cell will be a 0 or a 1 depending on the presence of a molecule)
• Neighborhood: the set of cells that a given cell interacts with. Usually the adjacent cells.
• Program: the set of rules that govern each cells movement and
interaction with other cells. (In our case, we will set up rules such as which way the molecules move if a collision occurs with another molecule.)

In order to govern this cellular automaton, we will be using Lorenz equations, a math model based upon the physics field of fluid dynamics. These equations are used to model the action of the chaotic behavior of a gaseous system. They are as follows:
dx/dt = delta * (y x) dy/dt = r * x y x * z dz/dt = x * y b *z

In these equations, delta represents the ratio of the fluid viscosity of a substance to its thermal conductivity. Thankfully, the exact ratio need not be known to create an accurate model. R represents the difference in temperature between the top and bottom of the gaseous system. Lorenzs original fraction for this was 8/3. X represents the rate of rotation of the cylinder or container. B is the width to height ratio of the container being used to hold the gas. Y represents the difference in temperature at opposite ends of the container, and Z represents the deviation of the system from a linear portrayal of temperature. Computers are immensely useful for equations such as this, because of the sheer amount of calculations needed to produce a useful simulation. We will have a graphic representation of the water movement.

Progress to Date:

Although we have not written any of the program yet, we have done extensive research into the equations and code needed for this program, and understand exactly what we will be writing.

Expected results:

When we are finished, our program will be able to provide a graphic representation of water flow on a given surface. Eventually, the program will grow in complexity until the user will be able to place different obstacles in the current, change temperature, and alter the program in other ways. Once we finish a working cellular automaton of about 1000 x 1000, we will expand on that and make it even bigger. Our program could ultimately form a base for other students wishing to experiment with lattice gases and similar modeling programs. And in the future world, an improved and more advanced form of our program could prove useful in any fields involving gas/water currents, rivers, etc.

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