scavo@cie.uoregon.edu (Tom Scavo) (01/26/91)
In article <9101242103.AA10740@lilac.berkeley.edu> Guan <guan@CHICONE.CS.MISSOURI.EDU> writes: > > I am planning to teach an undergraduate level course on chaotic >dynamical systems for junior or senior level science and engineer majors. >The goal is to introduce some basic mathematical ideas and phenomena >in chaotic dynamical systems for someone without a great deal of mathematical >background. > > My plan is to use primarily part I and part II of Bob Devaney's >book "An Introduction to Chaotic Dynamical Systems" to give a >reasonable mathematical background. Also let students observe >the phenomena by running their own experiments on computers using one >of the several dynamics programs available, e.g. "Dynamics" by Jim Yorke, >"MacMath" by Hubbard and West, or the one by Parker and Chua. I am >quite sure that similar courses must have been done somewhere else >before. I would like to get some general comments and opinion from >people who taught, or took such a course. I would also like to see if >I can get answer on the following questions : > > 1. Is there published report or discussion on a course like this >anywhere? See the January issue of the _College Math Journal_ for some additional insight. One thing I learned in this special issue devoted to chaos and fractals is that the NCTM has recently published a report _Discrete Mathematics across the Curriculum_ which apparently has some concrete suggestions on how to get dynamics into the classroom. Many other references will be found here. > 2. Is there other text book that might be appropriate, either as a >alternative, or as supplement to Devaney's book? Devaney's _Intro to Chaotic Dynamical Systems_ definitely requires a strong background in calculus. Although I haven't taught out of this book, I'll bet that students not having an analysis course will be blown away. A more elementary text would be Devaney's _Chaos, Fractals, and Dynamics_. (See the issue of CMJ mentioned above for a review.) You might also take a look at _Discrete Dynamical Systems_ by James T. Sandefur (Oxford University Press, 1990). Here the emphasis is more on difference equations and their applications rather than chaos and the qualitative theory of dynamical systems, and the level is more sophisticated than _CFD_ (i.e. it requires calculus). Both are recommended, however. > 3. What is the minimal prerequisite for a course like this? Some >linear algebra, or just calculus would do? Do we really need >advanced calculus, as indicated in Devaney? The basic concepts of dynamical systems theory can be under- stood by anyone with a strong background in algebra. In particular, functions and functional notation, absolute value, inequalities, and some familiarity with the graphs of elementary functions would be desirable. Complex arithmetic (complex arithmetic operations, modulus, polar coordinate, and square root) and a little trigonometry would be nice, but these topics might be introduced as the course develops, time per- mitting. Of course calculus will make some things easier, but it's amazing how much you can actually accomplish without it. > 4. Is there review, or systematic evaluation of these dynamics >programs published anywhere? If there isn't, my plan is to take the >opportunity of this course to perform such a review. If a review is >to be done, what programs should be included? I'm not aware of such a comprehensive review article (but see a software review in the special issue of CMJ). In ad- dition to the programs you mention above, I would also sug- gest Phaser (Springer-Verlag), Feedback (Peanut Software), and Fractint (public domain). But all of these are PC pro- grams; unfortunately, there just isn't much software available for the Mac. I'd also send away to Aerial Press (P.O. Box 1360, Santa Cruz, CA, 95061), Media Magic (P.O. Box 507, Nicasio, CA, 94946) and Lascaux Graphics (3220 Steuben Ave., Bronx, NY, 10467) for their excellent catalogues. You'll find lots of neat stuff here: books, software, videos, and more. By the way, I wouldn't recommend INSITE (which you implicitly refer to above) as an educational tool. It and a program called Dynamics Software (Dynamical Systems, P.O. Box 35241, Tucson, Arizona, 85740) are more oriented towards researchers. Probably the same could be said about Yorke's program, Dynamics. Personally, I would like to hear about Chaos in the Classroom (Dynamical Systems), Models (Lascaux Graphics), Chaos the Software (Autodesk), and other packages to be found in the catalogues mentioned above. > 5. To those who taught such a course before, is it possible for me to >get a copy of your course outline or syllabus, and some comments on how >things went in the class? Please send any text or TeX file would be >just fine. You can get a copy of the syllabus used at West Point by sending a letter to Col. D.C. Arney Mathematics Department U.S. Military Academy West Point, NY 10996 (I hope he isn't inundated with requests however; I found out about this in a Minicourse given by Jim Sandefur at the Math Meetings in San Francisco.) You can also request some lecture notes ($10) used at Oberlin College in a course (or module) entitled _Discrete Modeling Projects using Spreadsheets_ from M. Henle Mathematics Department Oberlin College Oberlin, OH 44074 I also have a syllabus, but it has yet to be tested. Tom Scavo scavo@cie.uoregon.edu
delliott@cec1.wustl.edu (Dave Elliott) (01/26/91)
In article <1991Jan25.172420.422@ariel.unm.edu> scavo@cie.uoregon.edu (Tom Scavo) writes: >In article <9101242103.AA10740@lilac.berkeley.edu> Guan <guan@CHICONE.CS.MISSOURI.EDU> writes: >> >> I am planning to teach an undergraduate level course on chaotic >>dynamical systems for junior or senior level science and engineer majors. >>The goal is to introduce some basic mathematical ideas and phenomena >>in chaotic dynamical systems for someone without a great deal of mathematical >>background. >> >> My plan is to use primarily part I and part II of Bob Devaney's >>book "An Introduction to Chaotic Dynamical Systems" to give a >>reasonable mathematical background. Also let students observe >>the phenomena by running their own experiments on computers using one >>of the several dynamics programs available, e.g. "Dynamics" by Jim Yorke, >>"MacMath" by Hubbard and West, or the one by Parker and Chua. I am >>quite sure that similar courses must have been done somewhere else >>before. I would like to get some general comments and opinion from >>people who taught, or took such a course. I would also like to see if >>I can get answer on the following questions : ...much good stuff omitted... >I'm not aware of such a comprehensive review article (but >see a software review in the special issue of CMJ). In ad- >dition to the programs you mention above, I would also sug- >gest Phaser (Springer-Verlag), Feedback (Peanut Software), >and Fractint (public domain). But all of these are PC pro- >grams; unfortunately, there just isn't much software available >for the Mac. ... [more omitted] There is a free program called Orbits, written by Stephen Eubank, Buff Miner, Jim Wiley, Toshi Tajima Institute for Fusion Studies Physics Department University of Texas Austin, Tx. 78712 and obtainable from Dr. Eubank, which works on most Macs; it permits the simultaneous plot of trajectories from several initial conditions and has in its distribution files examples of well-known dynamical systems and iterated maps.. pendulum, Duffing, Rayleigh, ... Henon, standard, Ushiki,... It probably is available somewhere by ftp, since it is redistributable [intact]. Another good program (I do not know how it is distributed) for Macintosh has been made available to Universities: DEGraph differential equation solver and grapher, by Henry C. Pinkham, Mathematics Department Columbia University New York, NY 10027. > >Tom Scavo >scavo@cie.uoregon.edu David L. Elliott Dept. of Systems Science and Mathematics Washington University, St. Louis, MO 63130 delliott@CEC1.WUSTL.EDU David L. Elliott Dept. of Systems Science and Mathematics Washington University, St. Louis, MO 63130 delliott@CEC2.WUSTL.EDU