Applied Mathematics is a way to use math to solve problems in the real world, like in economics, engineering, and medicine. 

This experiment was designed to examine chaos under controlled conditions. This experiment looks at chaos and how it appears in iterative processes that differ by a constant and are computed using different levels of precision. The experiment tests how varying levels of precision affect outcomes with processes with different constants.

Does rounding a number change whether a math sequence spirals into chaos? Iterative processes repeat the same equation over and over. A tiny change in precision can shift the outcome from a steady pattern to total unpredictability.

You use the equation x(n+1) = rx(n)(1-x(n)) in a spreadsheet. You test different values of the constant r from 2.70 to 3.70. For each value you run 26 rounds at precision levels of 2, 3, 4, and 15 decimal places.

Some constants produce sequences that settle on one number. Others make the sequence alternate between two values. Still others create chaos with no pattern at all. You discover that for certain constants, just changing the rounding level switches the outcome from alternating to steady.

Project Title: A Controlled Look at Chaos: The Varying Effects of Precision on
Iterative Processes

Objectives/Goals

This experiment was designed to examine chaos under controlled conditions. This
experiment looks at
chaos and how it appears in iterative processes that differ by a constant and
are computed using different
levels of precision. The experiment tests how varying levels of precision affect
outcomes with processes
with different constants.

Methods/Materials

The experimenter compared outcomes for different values of the constant r and
different levels of
precision for the iterative process x(n+1) = rx(n)(1-x(n)). Each process was
repeated for 26 iterations,
starting with x(0) = 0.5. Microsoft Excel was used to calculate the iterations.
The experiment tested
different values of r, from 2.70 to 3.70 in increments of .02. The experiment
also tested the effects of
precision on sequences, using precision levels of 2, 3, 4, and 15 digits past
the decimal point for each
value of r.

Results

The experimenter found that the constant used had a great effect on the type of
sequence that resulted:
convergent (repeating a single number), bifurcating (alternating between two
numbers), and chaotic
(never settling down or tending toward any number). The experimenter also found
that, for some
constants, changing the level of rounding had such a strong effect that the same
sequence would either
bifurcate (alternate between two numbers) or converge (repeat a single number),
depending on the
precision. For example, with constant 2.92, sequences with lower levels of
precision bifurcate while
higher ones converge. The rounding, however, had a much smaller effect than
expected for chaotic
sequences that have no apparent attractors.

Conclusions/Discussion

Overall, the constants are an important part of the iterative process. Precision
has a distinct effect,
although it does not affect certain chaotic sequences as much as expected.

Summary Statement

This experiment reveals how chaos appears in iterative processes with differing
levels of precision.

Help Received

Parents and teachers proofread report; I learned about chaos theory by reading
books myself.

 

Precision and Chaos in Iterative Math

The Varying Effects of Precision on Iterative Processes

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Math is all about numbers, patterns, and solving problems. Computer science is about using math and technology to create computer programs and solve real-world problems. Together, they help us understand how things work, how to solve problems, and how to use technology to make our lives better!

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Precision and Chaos in Iterative Math

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