Natural Sciences Elective
Order and Chaos in the Natural World
Spring Quarter 2013
May 7, 2013
I. GLEICK, pp. 59-80: LIFE’S UPS AND DOWNS
(SEE ALSO STEWART, pp. 144-154.)
1. How does the use of models in fields such as ecology, economics, and psychology differ from the use of models in fields such as astronomy and physics? What about biology and chemistry?
2. How does one decide whether or not a given model can be used in order to represent phenomena to be investigated?
3. Models can be based on solutions of differential equations or on solutions of difference equations. What is the distinction? Is the distinction significant?
4. What are the essential ideas underlying functional iteration (the construction of an iterated map)? In this connection, what is the meaning of the formula xnext = F(x)?
5. Gleick characterizes a process of functional iteration in terms of a feedback loop (pg. 61). In ecological models of the kind that he is considering, what are examples of processes that are involved in feedback? In this connection, what is unsatisfactory about the model of a population based on the map xnext = rx?
6. On pages 65 -69, Gleick interrupts the account of the logistic map with a section about James Yorke and Stephen Smale. Is this a digression, or does the passage have a point?
7. Think through the investigation of the logistic map along the lines described by Gleick. What would be the steps required in order to construct the diagram on page 64? What would be the steps required in order to construct the large diagram on page 71?
8. What are the important properties of the logistic map represented in the figures on pages 64, 71, 74, and 75? Which of those properties are representative of chaotic behavior? Which are representative of ordered behavior?
II. GLEICK, pp. 157-187: UNIVERSALITY
STEWART, CHAPTER 10: FIG-TREES AND FEIGENVALUES
1. These chapters center on a famous contribution to nonlinear dynamics and the theory of chaotic behavior made by Mitchell Feigenbaum while he worked at the Los Alamos National Laboratory. As Gleick describes that history, it appears that Peter Carruthers brought Feigenbaum to Los Alamos to work on problems in a particular field of research. What was that field? And did Feigenbamn work on such problems?
2. On what particular problem (or model) was Feigenbaum working when he made his famous contribution?
3. “Scaling” is portrayed as an important idea in Feigenbaum’s work. What is scaling, and how does it manifest itself in the problem on which he was working? What is the connection between “scaling” and “self-similarity?”
4. What is the process of “renormalization,” and how is it related to scaling and self-similarity?
5. Likewise, “universality” is also portrayed as an important idea here. What does universality mean in this context?
6. Did Feigenbaum’s work involve applications of what Thomas Kuhn might call “normal science?” If so, what was the normal science that was applied?
7. Feigenbaum’s scientific and professional background had both conventional and unconventional aspects. What conventional aspects of his background must have contributed to the work? What unconventional aspects?
8. Are the results of Feigenbaum’s work to be regarded as a discovery, a creation, or as something else?
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