Computational Approach to Teaching Conservative Chaos
This website is intended to supplement an article (conschaos) 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,
- 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
You can download a zip archive with the Mathematica notebook that was used to create all of the figures in the article here: StandardMap.nb. 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.
You can download a zip archive containing the animations mentioned in the article here: Animations. Each animation is in QuickTime format, so you will need QuickTime (Mac) or Windows Media Player with QuickTime (Windows) to view them. Brief descriptions of each animation are given below. Please read the article for a more detailed description of the animations.
- Liouville Flow: 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: 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: 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.
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: Slides. 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.
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).
Sample Student Work
This section contains work done by some of my students on the project mentioned in the previous section. The materials can be downloaded as a single zip archive: Student. A brief description of each file in the archive is given below.
- WTComp3.pdf: the report submitted by one of my students (Wes Taylor)
- WesCP3.mov: 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: 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 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 quantum mechanics. Chapter 3 has some excellent material on area-preserving maps,
particularly the Standard Map.