Directions: In this portfolio, you will use repeated function composition to explore elementary ideas that are used in the mathematical field of chaos theory. Items under the Questions headings will be submitted to your teacher as part of your portfolio assessment. For all questions, make sure to be complete in your responses. This can include details such as the function being iterated, the initial values used, and the number of iterations. The phrase many iterations is used in some of the questions. Interpret that to mean using enough iterations so that you can come to a conclusion. If necessary, round decimals to the nearest ten- thousandth. Introduction In this unit, you learned how to use function operations. One of the most important operations is function composition. Just as two functions, f and g, can be composed with each other, a function, f, can be composed with itself. Everytime that a function is composed with itself, it is called an iteration. Iterations can be noted using a superscript. You can rewrite , and as ( ) ( ) f f x 2 ( ) f x, ( ) ( ) f f f x  as 3 ( ) f x so on. For this work, it is recommended that you use technology such as a graphing calculator. Example 1 Start with the basic function . If you have an initial value of 1, then you end up with the following iterations. ( ) 2 f x x= • (1) 2 1 2 f= ⋅ = • 2 (1) 2 2 1 4 f= ⋅ ⋅ = • 3 (1) 2 2 2 1 8 f= ⋅ ⋅ ⋅ = Questions 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? Example 2 © 2015 Connections Education LLC. All rights reserved. For this example, use the function 1 ( ) 1 2 f x x = + and an initial value of 4. Note that with each successive iteration, you can use the previous output as your new input to the function. • 1 (4) 4 1 3 2 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 = = ⋅ + = Questions 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 ( ) 2 g x x = − + 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? Example 3 Nonlinear functions can lead to some interesting results. Using the function ( ) 2 2 4 g x x = − − + and the initial value of 1.5 leads to the following result after many iterations. • ( ) 1.5 2 1.5 2 4 3 g = − − + = • ( ) ( )2 1.5 3 2 3 2 4 2 g g = = − − + = • ( ) ( )3 1.5 2 2 2 2 4 4 g g = = − − + = • ( ) ( )4 1.5 4 2 4 2 4 0 g g = = − − + = • ( ) ( )5 1.5 0 2 0 2 4 0 g g = = − − + = At this point, further iterations of the function will repeatedly obtain zero. You can conclude that repeated iterations of the function with an initial value of 1.5 will lead to a value of zero. For most values of this function, repeated iterations will lead to looping values. The next set of questions will address this concept. © 2015 Connections Education LLC. All rights reserved. 2 Questions 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? Example 4 Repeated iterations will not always lead to a single number or a looping pattern. You can use the function 2 ( ) 4 f x x x= − to demonstrate this. Questions 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? Chaos Theory Chaos theory is the branch of mathematics that studies how small changes in inputs to functions can lead to vast changes in overall results. There is the famous idea that if a butterfly flaps its wings on one side of the world, it can lead to a hurricane on the other side of the world. This idea is part of chaos theory. Questions 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.

Question

Directions: In this portfolio, you will use repeated function composition to explore elementary ideas that are used in the mathematical field of chaos theory. Items under the Questions headings will be submitted to your teacher as part of your portfolio assessment. For all questions, make sure to be complete in your responses. This can include details such as the function being iterated, the initial values used, and the number of iterations. The phrase many iterations is used in some of the questions. Interpret that to mean using enough iterations so that you can come to a conclusion. If necessary, round decimals to the nearest ten- thousandth. Introduction In this unit, you learned how to use function operations. One of the most important operations is function composition. Just as two functions, f and g, can be composed with each other, a function, f, can be composed with itself. Everytime that a function is composed with itself, it is called an iteration. Iterations can be noted using a superscript. You can rewrite , and as ( ) ( ) f f x 2 ( ) f x, ( ) ( ) f f f x  as 3 ( ) f x so on. For this work, it is recommended that you use technology such as a graphing calculator. Example 1 Start with the basic function . If you have an initial value of 1, then you end up with the following iterations. ( ) 2 f x x= • (1) 2 1 2 f= ⋅ = • 2 (1) 2 2 1 4 f= ⋅ ⋅ = • 3 (1) 2 2 2 1 8 f= ⋅ ⋅ ⋅ = Questions 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? Example 2 © 2015 Connections Education LLC. All rights reserved. For this example, use the function 1 ( ) 1 2 f x x = + and an initial value of 4. Note that with each successive iteration, you can use the previous output as your new input to the function. • 1 (4) 4 1 3 2 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 = = ⋅ + = Questions 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 ( ) 2 g x x = − + 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? Example 3 Nonlinear functions can lead to some interesting results. Using the function ( ) 2 2 4 g x x = − − + and the initial value of 1.5 leads to the following result after many iterations. • ( ) 1.5 2 1.5 2 4 3 g = − − + = • ( ) ( )2 1.5 3 2 3 2 4 2 g g = = − − + = • ( ) ( )3 1.5 2 2 2 2 4 4 g g = = − − + = • ( ) ( )4 1.5 4 2 4 2 4 0 g g = = − − + = • ( ) ( )5 1.5 0 2 0 2 4 0 g g = = − − + = At this point, further iterations of the function will repeatedly obtain zero. You can conclude that repeated iterations of the function with an initial value of 1.5 will lead to a value of zero. For most values of this function, repeated iterations will lead to looping values. The next set of questions will address this concept. © 2015 Connections Education LLC. All rights reserved. 2 Questions 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? Example 4 Repeated iterations will not always lead to a single number or a looping pattern. You can use the function 2 ( ) 4 f x x x= − to demonstrate this. Questions 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? Chaos Theory Chaos theory is the branch of mathematics that studies how small changes in inputs to functions can lead to vast changes in overall results. There is the famous idea that if a butterfly flaps its wings on one side of the world, it can lead to a hurricane on the other side of the world. This idea is part of chaos theory. Questions 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.

Answer 1

Chaos theory is used in various fields to understand and predict complex systems. One application is in weather forecasting, where small changes in initial weather conditions can lead to significant differences in long-term weather patterns. This is known as the butterfly effect, where a small change in one part of the system can have a large impact on the overall outcome.

Another field where chaos theory is applied is in economics and finance. Stock markets and financial systems are inherently complex and can exhibit chaotic behavior. Understanding these chaotic dynamics can help economists and traders make better predictions and manage risks in the financial markets.

Chaos theory is also used in biology, particularly in the study of population dynamics and ecosystems. Small changes in environmental conditions or individual behavior can have cascading effects on the entire ecosystem. By studying chaos theory, biologists can better understand how these systems function and how to effectively manage them.

In physics, chaos theory is used to model and simulate complex systems, such as fluid dynamics and particle interactions. These systems are difficult to understand using traditional deterministic models, but chaos theory provides a framework for understanding the underlying dynamics.

Overall, chaos theory has found applications in a wide range of fields, from physics and biology to economics and weather forecasting. Its ability to capture the complex and unpredictable nature of these systems has revolutionized our understanding of the world around us.