MLAP 31500

Natural Sciences Elective

 

Order and Chaos in the Natural World

 

Spring Quarter 2014

 

CLASS NOTES

SECOND CLASS

April 8, 2014

 

I.   LORENZ, CHAPTER 1: GLIMPSES OF CHAOS

 

1.     What does Lorenz accomplish in the introductory chapters of his book?  Does what he accomplishes there differ from what Stewart accomplishes in his early chapters?

 

2.     Consider again what we should understand as the meanings of the terms “deterministic systems,” “random processes,” and “chaos?”  Do Stewart and Lorenz agree on the meanings of these terms?  What is the relationship, if any, between chaotic behavior and a random process?

 

3.     Stewart and Lorenz introduce the reader to chaotic behavior in the first chapters of their books.  They get to the point of introducing the subject by rather different routes.  What is the difference?

 

4.     The two authors illustrate their introductions to chaos by describing the behaviors of particular dynamical systems or models of dynamical systems.  What are those illustrative examples?

 

5.     In what respects are the systems that Stewart and Lorenz use in order to illustrate chaotic behavior similar?  In what respects are those systems different?  Why do the authors choose systems that are so similar or so different?  Which is the better system with which to illustrate chaotic behavior?  Why?

 

 

II.    DETERMINISTIC SYSTEMS, RANDOM PROCESSES, AND CHAOS ACCORDING TO IAN STEWART

 

A.    General Considerations

 

1.     What does Stewart accomplish in the introductory chapters of his book?

 

2.     What should we understand as the meanings of the terms “deterministic systems,” “random processes,” and “chaos?”  Is the technical usage of such terms significantly different from common usage?

 

3.     What might be examples of a deterministic system, a random process, and chaos?

 

4.     The book Chaos by James Gleick has the subtitle Making a New Science.  Is the study of order and chaos a science?  Or is the study of order and chaos a part of a particular scientific discipline such as astronomy, biology, mathematics, physics, or another discipline?

 

B.    Deterministic Systems in the Natural World

 

1.     What is the function of mathematics in the study of the natural world?

 

2.     What does the discovery of “laws of nature” contribute to an understanding of the natural world?

 

3.     In what ways are mathematical formulations of the laws of nature useful?  In particular, for what purposes do we reduce the laws of nature to differential equations?  (By the way, what is a differential equation?  As a student of liberal learning, try to answer this question without making use of technical aspects of mathematics.)

 

4.     What is the connection between the reduction of the laws of nature to differential equations and the conclusion that those laws describe deterministic systems?

 

C.    The Encounter of Voyager 1 and 2 with Hyperion.

 

1.     Give a precise description of the hypothetical experiment presented in the section “Voyage to Hyperion” in Does God Play Dice?

 

2.     What is the expected result of that experiment?  Why?

 

3.     What is the “observed” result?

 

4.     What is the apparent paradox revealed by a comparison of the expected and observed results?

 

5.     What does this discussion suggest about the relationship between the determinism of a physical system and the predictability of that system?

 

D.   Introducing Chaos

 

1.     Stewart illustrates his introductions to chaos by describing the behaviors of a particular dynamical system or model of a dynamical system.  What is that illustrative example?  Why does that model deserve to be called a dynamical system?

 

2.     In what respects is the system that Stewart uses in order to illustrate chaotic behavior similar to Hyperion?  In what respects are those systems different?  Which is the more realistic system with which to illustrate chaotic behavior?  Why?

 

3.     In what respects are systems governed by discrete mappings and systems governed by differential equations similar?  In what respects are they different?

 

 

 

 

LINKS:

 

Return to Course Page: mla315spring2013.html

 

Return to Peter Vandervoort's Home Page:  pov.html

 

Go to the home page of the Department of Astronomy and Astrophysics

of the University of Chicago:  http://astro.uchicago.edu/