Physics 213 Previous Announcements
Fall, 2003

• There will be a homework discussion meeting this Friday from 3-4 pm in Physics 233.

• Changes in the course schedule:
  • All 10-minute presentations have been moved to after the Thanksgiving holiday. We will meet on Monday, Wednesday, and Friday at our usual time and place (December 1, 3, and 5) and have one extra class whose time and place needs to be set but will likely be Thursday afternoon, December 4. I include the updated dates and speakers below.

  • There will be regular lectures this Friday and Monday (November 21 and 24) but no class on Wednesday, November 26. Monday, November 24 will my last official lecture of the course, then you guys take over.

• Here are the references I gave earlier about chaos control with two extra. These articles go into much more depth and detail than what is necessary for Problem 1 of Assignment 5 so I do not recommend that you read them just for Problem 1.
  • Section 4.6 of Ott's book "Chaos in Dynamical Systems", Cambridge University Press, 1993. This is a brief version of the paper that started it all, "Controlling Chaos" by E. Ott, C. Grebogi, and J. A. Yorke, Physical Review Letters 64:1196-1199 (1990).
  • Resource Letter: CC-1: Controlling chaos by Dan Gauthier, American Journal of Physics 71(8):750-759 (2003).
  • "Using small perturbations to control chaos", Troy Shinbrot, Celso Greobgi, Edward Ott, and James A. Yorke, Nature 363:411-417 (1993).

Monday, November 17

1. Reading for the last week:
   • Strogatz Chapter 12, especially Section 12.4 on attractor reconstruction.

   • Ott, Section 1.6 on delay coordinates, Chapter 3 on fractal dimensions, and Chapter 6 on quasiperiodicity and phase locking (which I may just be able to discuss briefly this week).

   • The article "Low-Dimensional Chaos in a Hydrodynamic System" by A. Brandstater et al, Physical Review Letters 51(16):1442-1445 (1983) which confirms that a particular fluid turbulence can be understood in terms of the dynamics of some low-dimensional strange attractor. A key step is embedding various time series and computing one of the measure-based fractal dimensions.

2. In order to stay within the academic schedule, this Friday will, sadly, be the last official lecture of the course and we will then begin the 12-minute presentations (10 minutes to talk followed by 2 minutes of discussion, just like an official APS meeting). There will be four talks to a class, starting a week from tomorrow on Monday, November 24 and continuing on November 26 (Wed), December 1 (Mon), and December 3 (Wed) and possibly part of December 5 (Friday).

   I will be glad to meet with you during this coming week if you have any questions about your presentation or paper, especially if you would like some feedback on your presentation.

   If you will be using PowerPoint or OpenOffice (under Linux) to give your presentation, I would recommend that you email me your file at least 24 hours before you give your talk so that I can download the file to my laptop and make sure everything works. This will save the need for you to bring your own laptop to class and the time of switching laptops between presentations. If you have any other needs for your presentation, please let me know well in advance so I can try to help you.

3. The papers that go with your talks should be handed in to me by 5 pm on December 5. If I am away from my office when you come by, please slide your paper under my office door (Physics 047).

Wednesday, November 12

1. For those coming to lunch, I made a reservation at the Chinese restaurant NeoChina which is about 8 minutes from the campus by car. I'll give instructions how to get there at the end of class.

2. Reading:
   • All of Chapter 11 in Strogatz. You should understand the construction and concept of a Cantor set and how to calculate the box dimension of various simple sets.

   • Read all of Chapter 12 of Strogatz. Section 12.1, especially Examples 12.1.1 and 12.1.2 concerning the Baker's map, you will want to try to understand carefully.

   • For a bit more meat and detail, read Chapter 3 of Ott's book "Chaos in Dynamical Systems".

Friday, November 7

1. The last assignment, Assignment 5, is now available.

2. The midterm date has been shifted to Monday, November 17, to make sure that there is enough time to cover the lectures on fractals and time to hold a discussion class concerning homework problems and course material.

3. We will have a second class lunch (for those who didn't make the first lunch) next Wednesday, November 12, at NeoChina right after class. (NeoChina is about a ten-minute drive from the campus, we will car pool back and forth.) Please send me email by this Monday to confirm if you plan to attend so I can reserve a table of the right size.

4. There was a handout in class today, the paper "Low-Dimensional Chaos in a Hydrodynamic System" by A. Brandstater et al, Physical Review Letters 51(16):1442-1445 (1983).

5. A reference for those who might want to understand better the relation between the non-chaotic dynamics of a coin toss versus the random behavior of the final state: "How random is a coin toss" by Joseph Ford, Physics Today, April 1983, pages 40-47.

Wednesday, November 5

1. Quiz 4 was returned with solutions.

2. Everyone should have a project identified by the end of today!

3. Some Java examples to play with of the Hopfield model of associative memory.
   • Java applet by Olivier Michel.
   • Java-Based Observation of the Hopfield Network.

Monday, November 3

1. A nice Java example of chaos control by T. Vincent. Here an unstable fixed point of the 2d Henon map is stabilized by the Ott-Grebogi-Yorke method discussed in class. Try increasing the noise in modest increments and watch how control is ultimately lost. Also get a sense of how the chaotic orbit must meander ergodically until it gets close enough to the fixed point to become stabilized. This related link shows a stabilized periodic orbit of the Lorenz equations.

Friday, October 31

1. Please try to attend the next CNCS seminar this coming Tuesday (November 4, 2:15 pm, Physics 113) by Dr. Emilio Frazzoli. A pdf file of his abstract is available here.

Tuesday, October 28

1. There will be a 15-minute quiz this Friday, October 31, to cover some of the basic concepts discussed recently in class. Some of the topics and skills that the quiz may cover are:
   • You should know the concept of a trapping region and how to demonstrate that some surface traps orbits.
   • You should know the definition of nullclines and how to obtain insight from known nullclines.
   • You should know the implications of the Poincare-Bendixson theorem for dynamics in a two-dimensional phase space.
   • You should know how to compute the average dissipation of an orbit of a flow and map.
   • You should understand some implications of dissipation, e.g., that a flow can't have planar periodic orbits if the trace of the Jacobian is a nonzero constant everywhere.
   • You should understand the concepts of an invariant set, of a basin of attraction, of an attractor, and of stable and unstable manifolds of a fixed point or of a periodic orbit.

2. There will be a "late midterm" test on Friday, November 14, that will last the entire class period. About one third of this test will be similar to the previous quizzes and will address concepts discussed since the previous quiz (mainly fractals). The remainder will test your knowledge at an elementary level of the basic concepts that we have discussed over the semester. This test will count twenty percent of your total course grade.

3. Here is a link to a description of the course project.

4. Some references regarding chaos control:
   • Resource Letter: CC-1: Controlling chaos by Dan Gauthier, American Journal of Physics 71(8):750-759 (2003).
   • "Using small perturbations to control chaos", Troy Shinbrot, Celso Greobgi, Edward Ott, and James A. Yorke, Nature 363:411-417 (1993).
   • Section 4.6 of Ott's book "Chaos in Dynamical Systems", Cambridge University Press, 1993.

5. Some interesting links related to chaotic mixing of a fluid:
   1. Jerry Gollub lab page on fluid mixing, some outstanding visualizations of mixing in a fluid.
   2. Mpeg movie of a blinking vortex flow, shows rather nicely how fluid is wound up in an intricate structure.
   3. Chaotic mixing associated with an atmospheric polar vortex.

Friday, October 24

1. The next set of lecture notes 10-22-03.txt is available, please read them in parallel with the lectures.

2. The video I showed briefly in class today was "Chaos and Fractals in Simple Physical Systems as Revealed by the Computer", Frank Varosi and James Yorke. You can sign this video out from the Vesics Library, call number 003.3 C461 1991. If this kind of exploratory mathematics appeals to you, you may enjoy playing with the software that comes with the book "Dynamics: Numerical Explorations: Accompanying Computer Program Dynamics 2 (Applied Mathematics Sciences, Vol 101)" by Helena Nusse and James Yorke.

3. Please start reading the Hopfield 1982 paper on a attractor model of associative memory. There are several Java applets that you can play with to get some intuition about how the Hopfield model works:
   • Java applet by Olivier Michel.
   • Java-Based Observation of the Hopfield Network.

Friday, October 17

1. The second 213 class on Monday will be held from 5-6 pm in our usual room, Physics 158.

2. In preparation for Monday's lectures, please review some of your basic multivariate calculus, especially the meaning of the gradient grad(f) of some function f that defines a surface f=0, the physical meaning of the divergence div(V) of some vector field V, and the divergence theorem that relates a surface integral of a vector field to a volume integral of the divergence of that vector field.

3. There were two handouts today, the graded Quiz 3 and answers for the quiz questions.

4. If you would like to join me for lunch next Wednesday (Oct 22) right after class, please send me email. If there are more than about 6 people, we will have a second lunch for those who can't make the first one.

5. As a supplement to Assignment 4, please visit within the next week the North Carolina Botanical Garden, which is an easy 15-minute drive down 15-501 from the Duke campus. Your specific goal is to find the part of the garden where the insect-eating pitcher plants live and study the vector field defined by the hairs within the throat of these plants. Don't forget to look at the inside bottom of the pitchers to see the trapped insects.

Wednesday, October 14

1. Two extra classes: Don't forget that we have a supplementary homework and course discussion this Thursday afternoon from 4-5 pm in Physics room 233.

   We will also have the second makeup class this Monday from 5-6 pm, tentatively in Physics 113 as before.

2. Small extension for Assignment 4: You can hand Assignment 4 in this Monday, October 20, since I will likely not finish grading Assignment 3 until that date.

3. Recommended reading: The new reading addresses the concepts of phase space, dissipation, attractors, and basins of attraction and will occupy us for about the next 3-4 lectures. We will then discuss the Hopfield paper and then spend a week discussing fractal sets and fractal dimensions.
   1. Start by reading my new set of lecture notes 10-15-03.txt. I will cover in class most but not all of the information in these notes and expect you to read and understand what I don't cover.

   2. Berge, Pomeau, Vidal: pages 101-136. This is the next best place to continue reading from a conceptual point of view and you should start here. Note: I have asked that the Vesics Engineering-Math-Physics library add this book to the 213 reserve so that you can find it more easily.

   3. Strogatz: Sections 2.1, 2.5-2.8; 6.1 and 6.2; 7.0, 7.1, 7.3; 9.2 and 9.3, especially pages 311-314 and 324-325. The discussion is not as unified or general as I would prefer for the level of this course, but Strogatz as usual has an excellent discussion with many examples worked out.

Monday, October 6

1. Please try to attend tomorrow's CNCS seminar by Bruce West, titled Complexity Produces Strange Kinetics and Dynamics. The seminar is at 2:15 pm in Room  113 of the Physics Building.

2. The third quiz of the course will be given this Friday morning promptly at the beginning of class. The questions will focus on material since the last quiz, e.g., the concept of a quasiperiodic time series with incommensurate frequencies, how a power spectrum can be used to distinguish periodic, quasiperiodic, and chaotic time dynamics, details of the Gollub-Benson paper (make sure you read this carefully), and some details of how power spectra are calculated.

3. There will not be class a week from today, Monday October 13, because of Duke's Fall Break.

4. We will have a supplementary homework discussion class on Thursday afternoon, October 16, from 4-5 pm. Assume for now we will meet in Room 233 of Physics.

5. We will have our second makeup class on Monday, October 20, from 5-6 pm. Assume for now that we will meet in our room 158.

6. A fun article to read, There's chaos on the buses about a researcher who suggests that arrival time of shuttle buses should be chaotic. Might be a fun topic for someone in the class to present for their end-of-the-semester talk and paper.

Tuesday, September 30

1. Assignment 4 is now available.

2. The next class will be on Monday, October 6.

Monday, September 29

1. A new file of lecture notes, 9-29-03.txt, is now available that discusses the background needed to appreciate the Gollub-Benson paper and the mathematics and assumptions of using the power spectrum for analyzing time series from dynamical systems.

   You should read these notes carefully for the next several lectures. You should also read carefully the Gollub and Benson paper and also the xeroxed chapter on the Fourier Transform from the Berge et al book "Order Within Chaos".

2. A reminder: there is an extra lecture today from 5-6 pm in Physics Room 113, and there are no classes this Wednesday and Friday since I will be out of town at a meeting.

3. I will post the next homework assignment by Tuesday morning, 9/30.

Thursday, September 25

1. Today's homework discussion class will meet in Physics 233 instead of in our usual room (which is unfortunately busy).

2. The extra lecture on Monday from 5-6 pm will also meet in a different room, namely Physics 113.

Wednesday, September 24

1. Assignment 3 is now due this Monday at the beginning of class, instead of this Friday.

2. There were six handouts in class today:
   1. Graded Quiz 2.
   2. Answer sheet for Quiz 2.
   3. Experimental paper by Gollub and Benson, please start reading this.
   4. Chapter III of the Berge et al book on Fourier Analysis of dynamical systems. Please start reading this also.
   5. Paper by John Hopfield on an attractor model of associative memory, which we will discuss later this semester.
   6. The classic 1963 paper by Lorenz. This is not required reading but I strongly enourage you to read and enjoy this seminal paper.

Monday, September 22

1. The homework discussion class will meet this Thursday, Sept 25, at 4 pm. I am still trying to confirm that our usual room is free but assume that it is Room 158 for now.

2. We will have an extra class this coming Monday, September 29, at 5 pm in Physics 158. Since I will be away next week at a scientific meeting (starting on Tuesday), there will be no class next week on Wednesday Oct 1 or on Friday Oct 3. We will have one more make up class after I return on October 6.

Wednesday, September 17

1. The second quiz will be given this Friday, September 19, at the beginning of class and last about 15 minutes. It will cover some basic and elementary details of what we have discussed in class so far, e.g., how to put a set of odes in first-order form, the definition of the eigenvalue and eigenvector of a matrix, how to calculate a Jacobian matrix, how to carry out a multivariate Tayor expansion to lowest linear order, etc.

2. Assignment 3 is now available. Please get in touch with me if you have any questions about this assignment. I will also be setting up a supplementary class meeting next week to give the class a chance to discuss the assignment and work on it together.

3. Main reading for next two lectures are the current lecture notes 9-12-03.txt. There is further recommended reading in these notes.

4. Recent Science article about applications of nonlinear dynamics and chaos to a variety of scientific problems. Did you know that some microwaves now use a chaotic power pattern to defrost foods more quickly?

Friday, September 12

1. I put new lecture notes 9-12-03.txt at this link.

2. I also also put a new Mathematica notebook plotshiftmap.nb for you to download and play with. I will discuss this notebook briefly in today's lecture.

3. I have not finished grading the first assignment but should do so this weekend so I will return these to you on Monday. The next assignment will also be made available this weekend, probably on Sunday night.

4. The next quiz will be on Friday, Sept 19, and will begin promptly at the beginning of class. It will cover material since the last quiz, focusing mainly on the linear stability of fixed points of maps and flows.

Wednesday, September 3

1. The homework discussion class will meet today at 5 pm today in Physics 158. Bring any questions that you might have about the homework or the course.

2. The first quiz will be given this Friday promptly at the beginning of class. It will be closed book and last about 10 minutes. Questions could cover any material discussed in class, in my lecture notes, and especially on the homework. However, I will not ask you questions about reading in Strogatz and Gleick that I did not discuss in class or in my notes.

3. The latest set of my lecture notes is available here: 9-2-03.txt.

Friday, August 29

1. There will be a class this Monday, September 1, at 11:50 am in Room 158, despite the Labor Day holiday.

2. There will also be a supplementary 213 class this Monday evening, September 1, in room Physics 113 (room just to right as you enter Math/Physics from the main entrance) to watch and then discuss a 1-hour video of a 1990 NOVA TV show on the "Strange World of Chaos". The show introduces several key scientists who have contributed experimentally and theoretically to nonlinear dynamics and shows examples of their research. The TV show will be followed by an informal discussion of 15-20 minutes. You are welcome to bring any friends that you think might also enjoy this video. If you are not able to attend and still wish to see this video, you can sign it out from the Vesics Math-Physics-Engineering library.

3. The second homework assignment is now available and is due one week from today.

4. There are two Mathematica notebooks to download and play with in the Miscellaneous Files link, plotlogistics.nb and bifurcationdiagram.nb. I will show you how to use these notebooks in this Monday's lecture.

5. Please check that the information on the Participants link is accurate, also whether there is information that I should add or delete.

Wednesday, August 27

1. I made somewhat more specific the required reading through this Monday, see the announcement below for Monday, August 25.

2. There will be regular classes this Monday, even though it is Labor Day. :(

3. I have put two lecture note files on the Lectures link of the 213 home page. You should read through both of these files, which complement my next two lectures.

4. Please get outside at night and look at Mars , which is at its brightest in nearly 60,000 years. This is not entirely irrelevant for a course on nonlinear dynamics. Jack Wisdom of MIT has shown that the obliquity of Mars, how much its spin axis is tilted away from being normal to the plane of the planets, varies chaotically in time, which means that there would be severe variations of weather and solar winds that would make it difficult for life to survive. Calculations suggest that Earth's large moon has damped the possible chaotic variations of Earth's obliquity, which may explain why life developed here but not on Mars.

Monday, August 25

1. The first homework assignment is available and should be completed before this Wednesday's class on August 27.

2. Reading for this week:
   1. Read the following sections of Strogatz:
      • All of Chapter 1.
      • Sections 2.0. 2.1, 2.2, 2.5, 2.6
      • Sections 6.1, 6.2.
      • Sections 10.1 through 10.3.
      Except for the sections in Chapter 10, a qualitative quick reading is fine at this point, we will come back and fill in many details later on.

   2. Read the first three chapters of Gleick's book. Chapter 3, pages 59-80, complements especially well our discussion of the dynamics of the logistic map.

3. I will be polling the class to find an evening the following week during which we can watch and then discuss a NOVA video about chaos.

4. For those taking the complementary course NCS 201, the first lecture will be this Thursday afternoon and will be given by Professor Lawrie Virgin.

5. The first seminar of the Center for Nonlinear and Complex Systems will take place on Tuesday, September 9, at 4 pm. The speaker, Professor Ed Ott of the University of Maryland, is one of the leading pioneers in nonlinear dynamics research and so you should try to attend his seminar.