f (1) 2 1 2 = ⋅=
• 2 f (1) 2 2 1 4 = ⋅⋅=
• 3 f (1) 2 2 2 1 8 = ⋅⋅⋅=
1. If you continue this pattern, what do you expect would happen to the
numbers as the number of iterations grows? Check your result by conducting
at least 10 iterations.
2. Repeat the process with an initial value of −1. What happens as the number
of iterations grows?
f = ⋅+=
• 2 1 (4) (3) 3 1 2.5 2 f f = = ⋅+=
• 3 1 (4) (2.5) 2.5 1 2.25 2 f f = = ⋅ +=
3. What happens to the value of the function as the number of iterations
increases?
4. Choose an initial value that is less than zero. What happens to the value of
the function as the number of iterations increases?
5. Come up with a new linear function that has a slope that falls in the range
−< < 1 0 m . Choose two different initial values. For this new linear function,
what happens to the function’s values after many iterations? Are the
function’s values getting close to a particular number in each case?
6. Use the function gx x () 2 =− + with initial values of 4, 2, and 1. What happens
after many iterations with all three initial values? How do the results of all
three iterations relate to each other?
( ) 1.5 2 1.5 2 4 3 =− − + =
• ( ) ( ) 2
g g 1.5 3 2 3 2 4 2 = =− − + =
• ( ) ( ) 3 g g 1.5 2 2 2 2 4 4 = =− − + =
• ( ) ( ) 4
g g 1.5 4 2 4 2 4 0 = =− − + =
• ( ) ( ) 5 g g 1.5 0 2 0 2 4 0
7. What is the loop that forms after many iterations when the initial value of g
is 1.7?
8. What is the loop that forms after many iterations when the initial value of g
is 1.72?
9. Choose an initial value that is between zero and 4 and is not a whole
number. Iterate it using the function, f, ten times. If necessary, you can
round your results to the nearest ten-thousandth.
10. Choose a second initial value that is 0.01 greater than the initial value from
question 9. Iterate it using the function, f, ten times. If necessary, you can
round your results to the nearest ten-thousandth.
11. Is there a relationship between the ten values from question 9 and the ten
values in question 10?
12. Use the Internet to conduct research on real-world applications of chaos
theory. Some examples of search terms to use are chaos theory, the
butterfly effect, and fractal. Write 2–3 paragraphs on how chaos theory is
used in today’s world in various fields.
Chaos theory is a mathematical concept that studies complex systems that are highly sensitive to initial conditions, leading to unpredictable outcomes. One famous principle of chaos theory is the "butterfly effect," which suggests that small changes in initial conditions can have large effects on the long-term behavior of a system.
In today's world, chaos theory is used in various fields such as meteorology, economics, biology, and computer science. In meteorology, chaos theory is used to predict weather patterns and understand the dynamics of atmospheric systems. Small changes in temperature or pressure can lead to drastic changes in weather conditions, making accurate predictions challenging.
In economics, chaos theory is used to study the behavior of financial markets and understand the dynamics of stock prices and investment portfolios. The concept of chaos theory has also been applied in biology to study the behavior of ecosystems and population dynamics, where small changes in one species can have ripple effects throughout the entire system.
In computer science, chaos theory is used in developing chaotic systems for encryption and data security. Chaotic systems are often used to generate random numbers for secure communication and cryptography.
Overall, chaos theory has had a significant impact on various fields, providing insights into the complex and unpredictable nature of systems in the natural world and helping researchers and scientists understand and model these intricate relationships.