Overview
This website is intended to supplement an
article (PDF, 1.8MB) that was published in the August, 2004
issue of the American Journal of Physics. The article presents a computational approach to teaching
conservative chaos in an upper-level, undergraduate Classical Mechanics course.
Specifically, I have used this approach to teach students in the second
semester of a two-semester, Sophomore-Junior level Classical Mechanics sequence
about chaos in conservative systems. The basic background that the students need
to have is as follows:
- Some experience with Hamiltonian mechanics for integrable systems.
- Multi-variable calculus (partial derivatives, etc.).
- Some linear algebra (eigenvalues and eigenvectors of matrices,
matrix-vector multiplication).
- Some knowledge of differential equations would be helpful, but is not
strictly necessary because we examine maps rather than continuous systems.
The basic idea is for the instructor to use Mathematica to create several figures that can help
illustrate different ideas that are important in the study of chaotic,
conservative systems. The instructor then makes the Mathematica code used
to create the figures available to the students. The students are then
asked to use that Mathematica code to solve a number of small-scale
computational problems (problems that typically involve making only
minor changes to the instructor's code). After the student develop
some experience with these computational tools they are asked to
apply them to the study of a new system (which will involve more substantial
modification of the instructor's code).
All of the examples in the article and in the Mathematica notebook (below)
are for the Standard Map (also called the Chirikov-Taylor map). This
two-dimensional area-preserving map is ideal for presenting a number of
the features of conservative chaos in the simplest way possible.
The Mathematica Code
Below are links to the Mathematica notebook that was used to create all of the
figures in the article. Along with the code is a brief description of
what the code actually does. Hopefully this information, combined with the
descriptions in the article, will allow anyone to use these computational
tools on their own. Please feel free to use, modify, and distribute my Mathematica
code as you see fit (although an attribution would be nice).
The Mathematica notebook was created using Mathematica 4.2. It works
correctly in Mathematica 4.1, but I have not tried it on any earlier versions.
You can read the file (without executing it) using the free
MathReader software
from Wolfram Research.
Animations
The links below are to the animations mentioned in the article. Each animation
is in QuickTime format, so you will need QuickTime (Mac) or Windows Media Player with
QuickTime (Windows) to view them. Please read the article for a more detailed description of the
animations.
- Liouville Flow (24 kB): This
animation illustrates the flow of trajectories in phase space and the
fact that the area occupied by a group of trajectories is invariant
under the mapping (Liouville's Theorem).
- Stable and Unstable Directions (16 kB): This
animation illustrates the flow of trajectories along the unstable manifold
of an unstable fixed point (using the forward map) and along the stable
manifold of the same fixed point (using the inverse map).
- Homoclinic Tangle (44 kB): Illustrates the
first several steps in the construction of the homoclinic tangle around
the unstable fixed point at (0.5,0) for the Standard Map with K=1.5.
These animations can also be downloaded as a zip archive (24 kB).
Try this if your browser has trouble with the animations.
Lecture Slides
This section includes PDFs of the lecture slides that I created for my lectures
on conservative chaos. The sequence of topics basically follows Chapter 11
of Hand and Finch.
WARNING: These lectures almost certainly include several errors. I tried to
fix errors as I discovered them while teaching the course, but I make no
guarantees that I did not miss some things (or even that I fixed all of the
ones that I did find -- sometimes I forget!).
To get the lecture slides, just download and unpack this
zip archive (500 kB).
The unpacked archive should contain the following files:
- ConservativeChaos.pdf (an overview of chaos in conservative systems)
- PoincareSections.pdf (a discussion of Poincare Sections and 2-D maps)
- GoldenTorus.pdf (breakup of KAM tori, continued fractions, etc.)
- PoincareBirkhoff.pdf (overview of Poincare-Birkhoff Theorem)
- TangentMap.pdf (the tangent map and determining stability of fixed points)
- Tangles.pdf (stable and unstable manifolds and homoclinic tangles)
- Lyapunov.pdf (exponential divergence and the Lyapunov exponent)
When I teach this material I present the lectures in the order listed above, although
other orderings are certainly possible.
Sample Assignments
Below are links to two assignments that I gave when I taught the course. The
first is a traditional homework assignment, although it includes some computational
problems. The
second is a large-scale student project that was assigned to all students. Students
worked in pairs and each pair studied a different chaotic map. As you will see if
you read it, the project is very open-ended. The benefit of this is that
it led to some interesting exploration on the part of some students (see Sample Student Work
below).
- Hamiltonian Chaos Homework: PDF (48 kB)
- Research Project: PDF (56 kB)
Sample Student Work
This section contains
work done by some of my students on the project mentioned in
the previous section.
- WTComp3.pdf (8 MB): the report submitted by one of my students (Wes Taylor)
- WesCP3.mov (1 MB): a movie created by a student (Wes Taylor) illustrating the
evolution of a chaotic map as the nonlinearity parameter is increased. The
formation of nonlinear resonances and the rise of chaos can be clearly seen.
- Fractal.mov (2.7 MB): a movie created by two students (Chad Grennor and Matt Wilson) illustrating the
fractal nature of the phase space for a chaotic map. The movie zooms in on a
region of phase space near the edge of a nonlinear resonance and shows that
nonlinear resonances are seen on many size scales.
My thanks to Wes Taylor, Chad Grennor, and Matt Wilson for allowing their work to appear here.
The material above can also be downloaded as a single zip archive
(9.2 MB). Try this if your browser won't display the animations properly. You
will need QuickTime (or Windows Media Player with QuickTime) to view the animations.
Recommended Texts
The following books are recommended as resources for teaching conservative chaos.
- Analytical Mechanics by Louis Hand and
Janet Finch (Cambridge University Press, 1998): This text is too advanced
for the course that I teach, but the coverage of conservative chaos in
Chapter 11 is excellent. I based my lectures primarily on that material.
- Chaos and Nonlinear Dynamics by Robert Hilborn (Oxford University
Press, 2000): This book is an excellent resource for all things chaos.
- Chaos and Integrability in Nonlinear Dynamics by Michael Tabor
(John Wiley and Sons, 1989): Chapter 4 of this text is another good
reference for conservative chaos.
- The Transition to Chaos In Conservative Classical Systems: Quantum
Manifestations by Linda Reichl (Springer-Verlag, 1992): This book
(written by my dissertation advisor) is a comprehensive reference work
on chaos in conservative systems and its manifestations in the quantum
mechanics. Chapter 3 has some excellent material on area-preserving maps,
particularly the Standard Map.
Todd K. Timberlake (ttimberlake@berry.edu)