MLA 31500

Natural Sciences Elective

Order and Chaos in the Natural World

Spring Quarter 2008

CLASS NOTES

SEVENTH CLASS

May 10, 2008

I. STEWART, CHAPTER 8: RECIPE FOR CHAOS (Review pp. 144-154 only; the chapter was assigned reading for April 12, 2008. See also pp. 147-148 in Lorenz, a part of the assignment for May 3, 2008.)

The logistic map is an iterated map of the form x_N = kx_N-1(1-x_N-1). In the 1970s, Robert May investigated the logistic map as an ecological model of a population of animals or other organisms. The idea is that x_N represents the population in year N, and the formula determines the population in one year in terms of the population in the previous year. The possible values of x_N lie between 0 and 1, where x_N=1 represents the maximum population that the ecosystem can sustain.

1. What properties of the formula for the logistic map make the map a plausible model for the changes of a population in an ecosystem?

2. In an application of the logistic map, how do we know when the representative point of the system in the phase space has reached an attractor? And what is the attractor?

II. STEWART, CHAPTER 10: FIG-TREES AND FEIGENVALUES

1. On what particular problem (or model) was Feigenbaum working when he made his famous discovery?

2. “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?”

3. What is the process of “renormalization,” and how is it related to scaling and self-similarity?

4. Likewise, “universality” is also portrayed as an important idea here. What does universality mean in this context?

5. Are the results of Feigenbaum’s work to be regarded as a discovery, a creation, or as something else?

6. How does Albert Libchaber’s experiment seem to have altered the perception of Feigenbaum’s work among physicists? Why should the experiment have made such a difference? In a similar way, how did the work of Swinney and Gollub alter the perception of Ruelle’s work?

7. Edward Lorenz had been working on thermal convection years earlier. Why didn’t the work of Lorenz stimulate interest in Feigenbaum’s work in the way that Libchaber’s experiment seems to have done.

8. The experiment of Harry Swinney and Jerry Gollub (described in Chapter 9 of Stewart) was a study of the onset of turbulence in Couette-Taylor flow. Libchaber’s experiment was a study of the onset of turbulence in thermal convection. In what respects are these experiments and their results comparable? In what respects do they differ? Does this comparison suggest that the mechanism for the onset of chaotic behavior, i.e., turbulence in this case, is universal?

II. RUELLE, CHAPTER 11: CHAOS: A NEW PARADIGM & CHAPTER 12: CHAOS: CONSEQUENCES

1. Ruelle describes Feigenbaum’s work on period doubling and renormalization without any reference to the logistic map. Why might he neglect to mention the logistic map? Is his account incomplete as a consequence?

2. 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?

3. How does one decide whether or not a given model can be used in order to represent phenomena to be investigated?

4. Consider Robert May’s investigations of the logistic map in ecology, Mitchell Feigenbaum’s discovery of scaling and universality, and Albert Libchaber’s experiment on thermal convection in liquid helium. How are these three lines of investigation connected? How do they complement each other? In what respects might these three efforts and their connections be something of a paradigm in the development of the “science of chaos?”

LINKS:

Return to Course Page: mla315spring2008.htm

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/