CSCI 4446/5446 Course materials:
The zoom link for lectures is on the course Canvas page (not here
because this page is publicly accessible and I don't want to get us
The zoom link for my office hours
General information and administrivia
version of CSCI 4446 is available through
the Complexity Explorer
MOOC platform housed by the
Santa Fe Institute. We'll be supplementing the course
with some of these materials during CSCI 4446/5446 this spring. These
MOOC materials may be useful to you in other ways as well, especially
if you have to miss a lecture. Please go to that website, register
for the course (which is called
"Nonlinear Dynamics: Mathematical and Computational Approaches"),
and look around a bit (including through the ``supplementary
Problem Set 1: logistic map. You can use the logistic map app on
the Complexity Explorer MOOC to check that your solutions are correct
(look in the "supplementary materials" tab). Also, you may wish to
take some time this week to review section 1 of the ODE notes listed
below ("Liz's written notes and videos") if your knowledge of
differential equations is at all rusty.
- Problem Set 2: bifurcation diagrams
and Feigenbaum's constant. Again, you can use the Complexity Explorer
logistic map app mentioned above to check your solutions.
- Problem Set 3: fractals. For some
examples of fractals in the wild,
Project Guidelines: you can find tech reports that compile
projects from some previous semesters (2010, 2011, 2012, and
Search for the title "Projects in Chaotic Dynamics..."
Problem Set 4: Runge-Kutta and the driven pendulum equations.
Problem Set 5: Poincare sections.
Final Project Details
Problem Set 6: variational equation. See the notes listed below
("Liz's written notes and videos").
The following materials may be useful to you as you do this problem
- A detailed list
the assigned reading for PS7-9 ("Reading assignments for
Instructions on getting the data for this problem set
an example of how to embed a data set .
TISEAN time-series analysis toolkit is available in our cloud IDE
at https://coding.cs.colorado.edu via JupyterHub. This is the
recommended method for accessing these binaries, as they are already
pre-compiled and ready for use. Login with your CU Credentials and
select the "4446/5446 Chaotic Dynamics" tile to get started. (If you
do not have a stable internet connection, the offline option is to use
the CU CS Foundations Virtual Machine which also has the pre-compiled
- If you want to install TISEAN on your own machine,
helpful hints regarding installing and using it and this link to
the pytisean wrapper
on github may be useful. 'brew tap brewsci/science' followed by
'brew install tisean' has worked in the past on Macs if you're a brew
user, but leaves the commands prefaced by a '-' ... i.e.,
Jay Kominek's mpeg movie of what happens as you change tau in
embeddings of data from the Lorenz system.
- There are lots of other references and resources for this
problem set in the "interesting links" section below.
Problem Set 8: Lyapunov exponents.
The following materials may be useful to you as you do this problem
Problem Set 9: fractal dimension.
here for a detailed list of the assigned reading for this topic
here for a scan of some of that reading (pp166-191 of Parker
Problem Set 10: playing with bike wheels, writing Lagrangians, and
starting to explore the two-body problem for a binary star. This
material is covered in the first few sections of the classical
mechanics notes listed below ("Liz's written notes and videos").
here for a picture defining true
here for an interactive simulator that you can use to explore
here for a wonderful lecture on dynamical toys like tops and
Problem Set 11: integrating the two-body equations. See section 4
of the classical mechanics notes listed below. Here's
link that Kristine Washburn found about a variant of this problem.
You may also wish to check out the n-body section of Colonna's webpage
(listed below). Here is
Hits Wall Street" article that's on the reading assignment.
Problem Set 12: integrating the three-body equations for a
binary-field star collision. See section 4.2 of the classical
mechanics notes listed below. The "visualization of dynamical
systems" page in the "interesting links" list below has source code
for a lovely visualization of this problem.
Liz's videos and written materials:
Some useful and/or interesting links: (caveat emptor!)
- A great article from Quanta magazine entitled
Hidden Heroines of Chaos" about the people who carried out
Lorenz's computer simulations.
- xkcd's takes on chaos (and
- A nice
about fractals (21 min)
- An amazing
animated bifurcation diagram
Riding around on the Lorenz
transcript of Lorenz's 1972 speech to
the AAAS entitled "Predictability: Does the flap of a butterfly's
wings in Brazil set off a tornado in Texas?"
- Pendulum stuff:
- Henri Poincare didn't only play a formative role in the
foundation of the field of nonlinear dynamics. Among other things, he
came up with the theory of relativity and wrote down e=mc^2 before
Einstein did. Read a bit about
- The Fyre tool for producing
artwork based on histograms of iterated chaotic functions ,
written by an alumnus of this course.
- Michael Skirpan's
fractal tree generator (= the mother of all solutions to PS3).
site license for Matlab now covers student computers!
visualization of dynamical systems page from the Nonlinear
Dynamics and Time Series Analysis Group at the Max Planck Institute
for the Physics of Complex Systems.
- Video recordings of the lectures from Steve
Strogatz's introductory course on nonlinear dynamics and chaos
- Complexity, the flip side of
dynamics of a flock of starlings. Here's
the Vimeo version of that
video if you prefer that channel.
- Movies of metronomes synchronizing (modern-day equivalent of
Huyghens' pendulum clocks): an array of five
array of 32 (!)
PhET project, an interactive simulator that you can use to explore
all sorts of interesting systems. Click on "Play with sims" and go to
"Physics" for the n-body simulator (called "My Solar System").
- Analog computers for nonlinear dynamical systems: the
mechanism and the
orrery (built by Liz's advisor)
- "Guide to
Takens' Theorem" paper (heavy going, mathematically, but very
- Rigid body
dynamics in zero gravity on the international space station.
- Jim Roberge's fabulous
lectures about control theory (RIP, wonderful mentor).
- A gorgeous youtube video that zooms in on the
- Another gorgeous video of an
evolving 3D fractal surface.
- A 'chalkmation' youtube video - complete with music - about the
set (warning: a bit of foul language at the end).
TISEAN time-series analysis toolkit. The TISEAN site has binaries
for UNIX & windows. 'brew install tisean' works on Macs if you're a
- Chaos in the path of a Roomba
- Chaotic music & dance stuff:
- NASA's movie of
- Remember that wonderful
"powers of ten" video from high-school physics?
- SIAM's dynamics
tutorials, many of which were contributed by grad students in courses
like this one.
- Wolfram's Mathworld site.
FAQ for sci.nonlinear. A fabulous resource.
- The Santa Fe Institute,
which has a couple of
great educational programs for graduate students (the Complex
Systems Summer School) and undergraduates (called "Research
Experiences for Undergraduates").
- Some Java
demos developed by Michael Cross, who teaches the CSCI4446-equivalent
course at Caltech.
- The Chaos
- Helwig Hauser's visualization
of dynamical systems page. The pages above that are interesting,
- Jean-Francois Colonna's
"virtual space-time travel" page, which includes lots of stuff
about the Lorenz system, pendula, the n-body problem, etc. Very nice
- Some sources of interesting time series data:
- Would you like your own double pendulum?