Seminar Number and Title: Physics 10433, Utter Chaos
Instructor Bruce N. Miller
Office SWR 315 Email B.Miller@tcu.edu Phone x 7123 Office hours MW4:20-5:00
Purpose
To introduce students to new ideas of physics and to teach students the methods physicists use to reach conclusions about natural phenomena.
A common misconception is that science is ultimately infallible and all things are predictable once we get the science “right”. However, in the last couple of decades, we have had to confront the mounting evidence that nature is not ultimately predictable over long times. Examples of chaotic systems are the weather, the solar system, and the human heart. Starting with simple models, in this seminar we will investigate how chaos occurs in nature and how it can be controlled to our advantage. The course will include readings about seminal work and the scientists who contributed to the chaos “revolution”. We will also explore known computer models which capture the essential features of chaotic systems, and provide the opportunity to design your own model starting from scratch. No previous computer proficiency or advanced mathematics will be required.
Course Objectives
(1) In addition to fulfilling the general objectives of freshman seminars, students will be able to explain the elements of chaos theory that describe the randomness and unpredictability of natural phenomena. They will also understand that chaos is an inevitable consequence of the nonlinearity of physical law and cannot be eliminated by “improvements” in basic science.
(2) Students will explore the richness of the different forms which chaotic phenomena can manifest through exposure to the simple models which helped the field develop. They will be able to trace the development of chaos theory by describing these models and being able to identify the characteristic form of chaos that each exhibits. They will also be able to describe the properties of self-similar geometric structures, such as fractals and strange attractors, which occur in nature.
(3) Students will learn how to use available computer software to investigate the properties of specific systems. They will learn that changing a single parameter in a model may dramatically transform the behavior from chaotic to stable, or the reverse. Students will also have the opportunity to use computer modeling to design an “original” chaotic system.
(4) It should be kept in mind that while students discover how chaos revolutionized science, they will also be brought into contact with the classical concepts from which the revolution sprung. Therefore, they will be able to characterize the earlier classical models and assumptions about natural process which were transformed by the chaos revolution.
Syllabus
Each week this seminar course will address a specific aspect of chaotic behavior. We will roughly follow the historical development of the chaos revolution. However, some of the specific topics to be covered will be adjusted to reflect the student’s interest as the course progresses.
Assigned readings will be drawn from the four designated books listed below with code letters:
B1: Chaos: Making a New Science James Gleick, (Viking, 1987).
B2: The Essence of Chaos , Edward Lorenz
(University of Washington Press, 1993).
B3: Exploring Chaos edited by Nina Hall (WW Norton, 1993)
B4: Mathematics and the Unexpected, Ivar Ekeland (University of Chicago Press)
We will chiefly follow the historical development of the subject as presented in Gleick’s book (B1). This popular volume looks at chaos in terms of the individuals who carried out the seminal work in the field, so it is partly biographical in nature, and extremely accessible. Supplemental readings will also be included from the three other required books. Ed Lorenz (B2) gives an outstanding objective overview of chaotic dynamics for the lay reader and B2 will be used as a complement to B1. There are mathematical appendices for hardier spirits. Hall’s book (B3) is a collection of essays by a number of researchers in the field for the lay audience and will generally be reserved for specific topics of student interest. Ekeland’s book clearly discusses mathematical concepts which arise in the study of chaos and will chiefly serve as a handy reference.
In addition to reading and discussion, there will be a computer component to the seminar. A computer laboratory in the Economics trailer, located at 2900 W. Lowden at the corner of Lowden and Greene, will be reserved for one class period each week. First, the computer will be used to explore specific chaotic models of historical interest, i.e. the models featured in the texts. Each student will use a spreadsheet to directly examine and compare stable and unstable dynamical systems. In addition we will make use of some of the excellent program packages, such as the Chaos Explorer, that are available from Physics Academic Software. We also have some original computer programs developed in our department which demonstrate essential elements of chaotic behavior. Students will make use of the software to perform their own experiments. The number of computer periods scheduled will partially depend on the students interest and previous computer experience.
Each week the students will be assigned a set of readings, a written report, and a short quiz.. The report should be one page long and should include a) the answer to assigned questions, b) an outline of the central ideas and conclusions found in the readings, and c) a discussion of what the student found most interesting or important in the readings. In addition students will be required to maintain an up to date laboratory workbook containing informal notes and results for the computer laboratory component of the course. The workbooks will be collected at the end of the semester and used as a component of the final grade.
Course Outline
Week 1. What is the butterfly effect? We will discuss Lorenz’ work on the weather and its generalization to sensitive dependence on initial conditions.
a) What is a dynamical system?
b) What is meant by the state of a dynamical system?
Reading B1 chapter 1, B2 Chapter 1, pp 1-15.
Week 2. What is a scientific revolution? We will discuss Kuhn’s theory as applied to the Copernican revolution. Then we will discuss whether Kuhn’s paradigm applies to the Smale horseshoe, as an idea that transformed dynamics.
a) What is stability?
b) What is periodic motion?
c) What is equilibrium?
Reading B1 chapter 2, B2 Chapter 1, pp 15-24.
Week 3. Life’s ups and downs, or how do the ideas of chaos impact population dynamics?
a) Is the Malthusian scenario correct?
b) What is a more realistic, but still simple, model that takes into account finite resources.?
c) Are realistic models chaotic?
d) What is a bifurcation?
e) Do nearby orbits always diverge?
Reading B1 chapter 3, B2 Chapter 2, pp 25-39.
Week 4. The geometry of nature; We will discuss self similarity and its many examples in nature, from coastlines to capillaries.
a) What is the cantor set?
b) What is a fractal?
c) What is fractal dimension?
Reading B1 chapter 4.
Week 5. Turbulence and fluid mechanics: We see chaos in a plume of smoke, a rotating fluid, or the atmosphere. What is its nature?
a) What is vorticity?
b) What is viscosity?
c) What is turbulence?
d) What is an attractor?
e) What makes an attractor “strange”?
Reading B1 chapter 5, B2 Chapter 2, pp 40-55.
Week 6. Universality: It’s origins in thermodynamics and its discovery in dynamics by Mitchell Feigenbaum. In the1960s and 70s it was found that phase transitions, for example from liquid to gas, have distinct universal properties. Using these ideas Feigenbaum examined the population model (the logistic equation) with a better “magnifying glass”. With just a hand calculator he discovered that all similar models have definite, underlying, quantifiable, features in common.
a) What are universal properties of chaos?
b) What is a point of accumulation?
Reading B1 chapter 6, B2 Chapter 2, pp 55-61.
Week 7. Chaos and convection: Experiments which show the transition from laminar flow to convective instability. This is what happens in the atmosphere each day as the earth’s surface warms, but the experiments were done on liquid helium near absolute zero.
a) What is laminar flow?
b) What is convection?
c) What is a Poincare section?
d) What is the chaotic sea?
Reading B1 chapter 7, B2 Chapter 2, pp 61-76.
Week 8. Images of Chaos: The boundary between definitive outcomes is not sharp or small, but fractal. The prime example is the Mandelbrot set. It has been the source of both scientific insight and computer art.
a) What is a basin of attraction?
b) What is a basin boundary?
c) What is the Mandelbrot set?
d) What is the Julia set?
Reading B1 chapter 8.
Week 9. Our Chaotic Weather.Following Lorenz book, we return to our weather system with greater insight. We examine “pre-revolutionary” models of the weather and the atmosphere in more detail, as well as pattern formation in some classic experiments. We find both chaos and structure in actual weather data, and we see why the older models failed to predict atmospheric conditions beyond a couple of days.
a) Compare and contrast the two main approaches to weather prediction.
b) What are the main ingredients required to make a global predictive model of the weather?
Reading B2 Chapter 3.
Week 10. The Dynamical Systems Collective: In their attempt to understand nonlinearity, four graduate students at U. Cal. Santa Cruz blended the work of Russian mathematicians and an American electrical engineer to reformulate how we look at dynamical systems in simple terms. They applied their eclectic vision to the human immune system and a dripping faucet.
a) What is a Lyapunov exponent?
b) Can information be measured?
c) Can information be created or destroyed?
d) What is entropy?
e) What is time series analysis?
f) What is the embedding dimension?
Reading B1 chapter 9, B2 chapter 4, pp 111-136.
Week 11. Inner Rhythms: an examination of chaos in medicine. The methods of chaotic dynamics are applied to physiology and psychology. Schizophrenia, and the fibrillating heart are seen as chaotic modes of dynamical systems.
a) What is a circadian rhythm?
b) What is the BZT reaction?
c) What is self-organization?
Reading B1 chapter 10, B2 chapter 4, pp 136-157.
Week 12. Review and Synthesis: We outline the central features of deterministic chaotic systems and see that they fall into two distinct classes: Conservative and Dissipative. We then discuss the influence of noise, with such examples as stochastic resonance (e.g. we hear weak sounds better when there is a little background noise).
Reading B1 chapter 11, notes, B2 chapter 4, pp 157-160
Weeks 13 and 14. We will focus on recent applications chosen by the students.
Reading B2 chapter 5 and selections from B3.
Grading Process (percentages)
1. Weekly written reports will account for 30% of the final grade.
2. Weekly quizzes over the assigned reading will account for 20% of the final grade.
3. Classroom participation, i.e. evidence that the student is attentive and intellectually engaged during the class period, will account for 20% of the final grade.
4. Mastery of the computer as a tool for investigating chaotic dynamical systems will account for 20% of the final grade. This will not require programming or model construction. The latter will count for extra credit.
5. The individual presentation will count for 10% of the final grade.
Attendance Policy
Attendance in class at the scheduled time is mandatory. If a quiz is missed, a grade of zero will be recorded. No makeups will be given unless the absence is officially excused according to TCU policy. In the case of an excused absence, a missed quiz must be made up within two weeks following the absence. In addition, for every unexcused absence beyond the second, the final class average will be decreased by 5%. Please do not come to the class if you are ill. Please do go to the infirmary and obtain an official excuse with your prescriptions.
Students with Disabilities
Please inform me during the first two weeks of class of any difficulties, disabilities and/or learning needs you may have for which accommodations need to be made. Every effort will be made to accommodate your needs.
Academic Misconduct
Academic misconduct is taken seriously in this class. It is defined in the Bill of Student Rights and Responsibilities.
Representative Bibliography
Listed below are some outstanding and fascinating sources of enrichment. Most are non-technical in approach, but may have some technical elements. The strong exception is reference 3 which is a great place to begin a study of the mathematics underlying chaos theory.
1. Abraham and Shaw, Dynamics; the Geometry of Nature, (Addison-Wesley, 1982).
2. Casti. Searching for Certainty (Morrow, 1990).
3. Devaney, An Introduction to Dynamical Systems, (Benjamin/Cummings, 1986).
4. Ekeland Mathematics and the Unexpected (University of Chicago Press, 1988).
5. Ford, “How Random is a Coin Toss ?,” Physics Today 36, no. 4 (1983) pp 40-47.
6. Mandelbrot, The Fractal Geometry of Nature (Freeman, 1982).
7. May, Stability and Complexity in Model Ecosystems (Princeton, 1973).
8. Poincare, Science et Methode (Flammarion, 1912) English Translation Science and Method (Science Press, 1913).
9. Peitgen and Richter The Beauty of Fractals (Springer-Verlag, 1986).
Peterson The Mathematical Tourist (Freeman, 1988).
10. Ruelle, Chance and Chaos (Princeton, 1991).
11. Schroeder Fractals, Chaos, Power Laws : Minutes from an Infinite Paradise (Freeman, 1991).
12. Smale “How I got Started in Dynamical Systems” in The Mathematics of Time by Smale,
pp. 147-151 (Springer-Verlag, 1980).
13. Stewart Does God Play Dice (Basil Blackwood, 1989).
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