MLA 31500

Natural Sciences Elective

Order and Chaos in the Natural World

Spring Quarter 2008

CLASS NOTES

SIXTH CLASS

May 3, 2008

I. STEWART, CHAPTER 9: SENSITIVE CHAOS

1. According to Stewart, what was the conclusion of Ruelle and Takens regarding the Landau or Hopf-Landau scenario for the onset of turbulence?

2. What positive claim did Ruelle and Takens make regarding the onset of turbulence.

3. One can think about several questions regarding the experiment on Couette-Taylor flow that was performed by Harry Swinney and Jerry Gollub. What was the physical arrangement of the experiment? What was the experimental procedure? What were the results, and how did those results differ from expectations?

4. In what respects did the experiments of Harry Swinney and Jerry Gollub support the claims of Ruelle and Takens.

5. An important test of the picture of Ruelle and Takens is to exhibit a strange attractor in the results of a computer experiment on some model of a dynamical system. Ruelle, Takens and Packard proposed a scheme for using data for this purpose. Suppose that your data were the output of a computer solving the Lorenz equations. How would you organize and plot the data in order to exhibit the Lorenz attractor.

6. Likewise, how do you organize and collect the data from real measurements of a dripping faucet in order to show that there is a strange attractor in the dynamics.

7. What seem to be the most important issues in this chapter for Stewart?

II. LORENZ, CHAPTER 4: ENCOUNTERS WITH CHAOS

1. The first few sections of this chapter address subjects that we have encountered previously in Stewart and, to some extent, in Ruelle. Is there anything new or distinctive in Lorenz’s account, or are the first few sections redundant for us?

2. Lorenz makes a distinction between “chaos,” which he also calls “full chaos” and “limited chaos.” What is that distinction? (The glossary in the book is helpful on such points.)

3. Lorenz takes up the question as to whether or not Poincaré considered the phenomenon of full chaos in the modern sense in which Lorenz describes the concept. What conclusion does Lorenz draw?

4. In accounts of the history of the subject, we find Smale introducing the term “differentiable dynamical systems,” Ruelle and Takens the term “strange attractor,” Li and Yorke the term “chaos,” and Lorenz the “butterfly effect.” Lorenz remarks that, for some years, he resisted the term “chaos” in favor of “irregularity.” And he describes his work as a search for “non-periodic” behavior in models of weather and turbulent convection. What does this history tell us about the scientific tastes and perspectives of pioneers in the field and about the role of nomenclature in promoting the development of a field?

5. In his computer investigation of a model of terrestrial weather, what was Lorenz looking for? What did he find? Was what he found inconsistent with what he was looking for?

6. Lorenz makes a link between non-periodic or aperiodic behavior and unpredictable behavior. How do we explain that link? (Hint: Why can’t periodic behavior be chaotic?)

7. What are the attributes of the Lorenz attractor that make it a strange attractor?

III. RUELLE, CHAPTERS 7, 8, 9, & 10

1. In Chapter 7, Ruelle describes the motion of a pencil in the neighborhood of the state of unstable equilibrium in which it balances on its point. He asserts that such motion exhibits exponential growth and that it is an example of sensitivity to initial conditions. Can both assertions be correct? Would Stewart and Lorenz agree? (See, e.g., page 119 in Lorenz.) Ruelle’s analysis would also apply to the motion of an idealized pendulum near its state of unstable equilibrium. Sensitivity to initial conditions is a sine qua non for chaotic behavior. Is the motion of an idealize pendulum chaotic?

2. On the other hand, Ruelle describes the idealized motion of a ball on a billiard table with bumpers as an example of motion that is sensitive to initial conditions. Would Stewart and Lorenz agree?

3. According to Ruelle, sensitivity to initial conditions was a concept whose relevance to natural phenomena was well understood at the beginning of the twentieth century by Hadamard, Duhem, and Poincaré. Why was the concept not appreciated more widely among physicists until the 1960s?

4. On pages 55 and 56 in Chapter 9, Ruelle describes his doubts in 1968 concerning the Landau-Hopf scenario for the onset of turbulence. What were those doubts? Does he really explain the emergence of the idea of strange attractors in a way that we can understand?

5. In his introduction of strange attractors, how does Ruelle represent the behavior of a dynamical system (e.g., a turbulent fluid)? What aspects of the behavior of such a system and the occurrence of chaos were Ruelle and Takens seeking to represent in a strange attractor? How does this relate to Stephen Smale’s horseshoe map?

6. In Chapters 7-10, Ruelle discusses chaotic behavior for the first time in the book. Yet, he does not introduce the term chaos until Chapter 11. Why not?

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/