Physics 513 Syllabus
Fall Semester, 2003

Professor Henry Greenside

hsg@phy.duke.edu     (919) 660-2548     Physics 047

Welcome      Prerequisites      Time and Place      Office Hours      Computer Accounts      Grading      References

Welcome

Welcome to Physics 513, one of the core courses of Duke University's Center for Nonlinear and Complex Systems. This course is an introduction to and survey of nonlinear dynamics, an exciting and interdisciplinary area of research concerning how natural and man-made systems evolve in time. Besides being interesting in its own right, nonlinear dynamics has proved useful to many disciplines including biology, chemistry, computer science, engineering, environmental science, geology, mathematics, medicine, meteorology, physiology, and physics.

A goal of the course will be to study the successes, limitations, and implications of a modern discovery, that simple deterministic nonlinear evolution equations can generate complex behaviors that quantitatively agree with experimental observations of many physical systems. It will take a while to give you the background to appreciate the scope of this discovery and so the course will introduce and discuss many of the following topics:

  1. Concepts related to a geometric and global way of thinking about nonlinear evolution equations. These concepts include: phase space, dissipative versus conservative systems, attractors, basins of attraction, elementary bifurcation theory, linear stability theory, Poincare sections and maps, strange attractors, Lyapunov exponents, transition scenarios (Feigenbaum, Ruelle-Takens, and intermittency), universality, synchronization, fractals, fractal dimensions, and analysis of time series by embedding.
  2. Comparisons of theory with high-precision laboratory experiments, e.g., the sequences of transitions that lead to chaos in a convecting fluid of moderate lateral extent.
  3. Applications of nonlinear dynamics to different disciplines, e.g., ecology, engineering, neurobiology, and fluid dynamics.
  4. Strategies and algorithms for simulating, analyzing, and controlling nonlinear systems, including the integration of differential equations, and the calculation of power spectra, Lyapunov exponents, and fractal dimensions. We will also discuss fundamental limitations imposed on numerical simulation by nonlinear dynamics, e.g., the difficulties of accurate long-term forecasting in the presence of chaos.
Students enrolled in 513 should attend the biweekly seminars of the Center for Nonlinear and Complex Systems. These seminars are generally held on Tuesdays in Physics 113 from 2:15-3:15 pm. Students in this course should also consider enrolling in the course NCS 201/Physics 201.This course meets once a week on Thursdays and is taught by a team of 3-4 CNCS faculty, who in turn discuss their research and its relation to nonlinear dynamics.

Prerequisites

The course is intended for advanced undergraduates and graduate students with diverse backgrounds. Students will need to have a working knowledge of multivariate calculus and of linear algebra at a sophomore undergraduate level. They should feel comfortable with the concepts of limits, complex numbers (you should especially be familiar with the identity exp(ix) = cos(x) + i sin(x)), integrals, derivatives, the gradient and divergence operators, multivariate Taylor series, Fourier series, matrices, determinants, eigenvalues and eigenvectors, big-oh notation, and ordinary differential equations.

Students should have had at least one introductory undergraduate physics course so that concepts such as vectors, forces, momentum, energy, energy conservation, flux, Newton's equations of motion, temperature, and heat are familiar.

Some experience with a programming language like Fortran, Java, or C++ is also needed although not at a sophisticated level, e.g., the computer statement "i = i + 1" should make sense to you. Some homework assignments and lecture demonstrations will use previously written programs for Mathematica, a powerful computer program that provides an interactive symbolic, numerical, and graphical environment for mathematics and data analysis.

Although many of these mathematical, physical, and computational concepts will be reviewed in class as they are used, the review will usually be terse. Students not comfortable with these concepts should talk to me before enrolling.

Time and Place

The class will meet Monday, Wednesday, and Friday from 11:50 am-12:40 pm in Physics 158. There will be occasional supplementary meetings, e.g., to discuss homework assignments, to see and discuss a video, or to tour the lab of some CNCS researcher.

At least an hour before each class, please get in the habit of viewing the Announcements section of the 513 home page. This section will be updated frequently to mention possible last minute changes in the class schedule, availability of homework assignments and solutions, seminars and colloquia of interest to the class, and the time and place of supplementary classes.

Office Hours

I do not have fixed office hours. However, I will make my best effort to meet with you if you have any questions at all about the course (or more generally about nonlinear dynamics or about physics or about Duke of if you would just like to chat). If you are in the Physics building, please free to drop by my office Physics 047 and say hi.

To meet with me at some specific time, please email me at the address hsg@phy.duke.edu or call me at my Physics office at 660-2548.

Feel free to send me e-mail at any time. I am often logged on in the evenings and on the weekends and will be glad to discuss the course or homework with you.

Computer Accounts

Students taking the course will need access to electronic mail, to an Internet browser capable of running Java applets, and to the computer mathematics program Mathematica. (The latter is available to students for free at this Duke OIT link.) Lectures, homework assignments, electronic documents, data sets, and graphical files will be available through the Miscellaneous Files link from the 513 home page.

Grading

Your final grade for the course will be based on your class participation, homework assignments, a midterm exam, and a final project. These will be weighted approximately as follows:

Activity Percent of Total Grade
Class participation 5%
Quizzes 10%
Midterm exam 20%
Final project (oral presentation and paper) 25%
Homework assignments 40%

There will not be a final examination.

Class participation.

I expect all members of the class to read and to think about the assigned material before lecture and to come prepared to ask questions and to discuss the material in class. If you don't understand something during lecture or from the assigned reading, please don't be shy, ask questions! If something catches your interest and you want to learn more, ask questions. Talking with me outside of lecture is also one way to participate in class. I want to see evidence that you are actively trying to learn and think about the course material.

Quizzes

There will be a 15-minute closed-book quiz roughly every two weeks. The quizzes will consist of a few short questions that you should be able to answer easily if you are keeping up the material, so the quizzes should give some useful feedback to you and to me as to whether the lectures are being successful.

Midterm exam.

This will be an overnight 3-hour open-book exam that will be given about half-way through the semester.

Final project of a presentation and paper.

Instead of a final exam, each member of the class will write a 10-page double-spaced paper (including figures and references) on some topic related to nonlinear dynamics that he or she is especially interested in. This paper can be based on reading some journal article, on carrying out some simple experiment, or by simulating or analyzing some mathematical model. Each student will give a 15-minute oral presentation to the class about their paper.

Homework assignments

There will be a homework assignment roughly every one to two weeks, for a total of about seven assignments. These are the heart of the course in that it is through these assignments that you will build up and apply your understanding of the various concepts and techniques. Please keep the following in mind as you work on and write up your assignments:
  1. Late homework is not accepted unless you give me a reasonable excuse at least three days before the homework is due.

  2. Your main two goals in writing up your homework are to be clear (so that I can understand what you have written) and to demonstrate insight. Writing clearly means using readable handwriting . You should avoid tiny script and avoid trying to cram many sentences and equations onto a single page. Leave plenty of space between symbols and between successive lines of equations. Leave plenty of space between the ending of one homework problem and the beginning of the next. Spread your answers out over many pages if necessary. (Paper is cheap compared to the time needed for you to complete the assignments and for me to grade your assignments.) If I can not read and understand your assignments easily, you will get little or no credit.

    Demonstrating insight means using complete sentences that explain what you are doing and why. Cryptic brief answers like "yes", "no", "24", or "f(x)" will not be given credit. Instead, explain what you are doing and why, e.g., as if to a friend who is not familiar with this course. Your homework must show that you understand how you got your answer and that you appreciate the significance of your answer. A well-written complete answer is one that you will be able to understand yourself a month after you have written the answer, even if you don't remember the original question.

  3. You are allowed to collaborate on the homework assignments (this is realistic, scientists collaborate all the time in research) but as much as possible you should attempt the assignments on your own since you will learn the most that way. Whether or not you collaborate, you must write up your homework on your own, in your own words, and with your own understanding. You must also acknowledge explicitly at the beginning of your homework anyone who gave you substantial help, e.g., classmates, myself, or other people. (Again, scientists usually acknowledge in their published articles colleagues that helped to carry out the research.) Failure to write your homeworks in your own words and failure to acknowledge help when given can lead to severe academic penalties so please play by the rules.

  4. The assignments will require typically a mixture of analytical, numerical, and graphical approaches. The mathematical derivations or analyses for the analytical problems should be written out by hand on paper. Please use ink, not pencil. Numerical and graphical answers involve output that are best printed out on a laserprinter, then stapled to your handwritten sheets. A hand-sketch of a graphical plot with essential features described is also acceptable.

  5. Please pay attention to details as you write your assignments. All symbols should be given names the first time you introduce them, e.g., say "the momentum p" or "the flux F" instead of just using the symbols p and F. Physical units should be given for any answer that is a physical quantity, e.g., say "the angular momentum was A=0.02 J-sec" or "the angle was µ=0.32 radians." Numerical answers should have the minimum number of significant digits that is consistent with the given data. For example, if you have a product or ratio of numbers of which the least accurate number has two significant digits, the final answer should have only two significant digits. Graphs should have their axes clearly labeled by the corresponding variables and by the variables' physical units. Each graph should have a title that explains the graph's purpose. A good way to learn how to write effectively is to imitate the style of published articles, e.g., those published in Physical Review Letters .

  6. If you use using Mathematica in a homework assignment, please do not give me the output of your entire session. Instead, just give me enough output to convince me that you have answered the question correctly. You should also include any Mathematica code that you write so that I can try to understand how you obtained your answers.

References

The course will be taught from my own notes and will not follow any textbook particularly closely. One required reference is the book Nonlinear Dynamics and Chaos by Steven Strogatz (Addison-Wesley, Reading, MA, 1994). This book is particularly well written and accessible and should be the first place to look for discussions and examples that complement my lectures. However, this book is a bit too elementary for the intended level of this course and does not cover some of the topics in enough detail.

The following are some other books that I encourage you to take advantage of during the semester and after. The first three are on reserve in the Vesic Library .

Nonlinear dynamics remains an active and exciting research frontier for many disciplines. You should get in the habit of browsing weekly through key scientific journals to get a sense of ongoing research. The following journals are often quoted for research related to nonlinear dynamics:
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