CSCI 4446/5446 Course materials:

In general, TA help hours will be held on mondays from noon-1pm in ECOT 831 starting on 27 January.

• General information and administrivia
• A version of CSCI 4446/5446 is available through the Complexity Explorer MOOC platform housed by the Santa Fe Institute. We'll be supplementing the on-campus course with some of these materials 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 materials" page.
• 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, click here or here.
• Final Project Guidelines: you can find tech reports that compile projects from some previous semesters here. Search for the title "Projects in Chaotic Dynamics..."
• Problem Set 4: Runge-Kutta and the driven pendulum equations. You can download a pdf of the Parker & Chua book in the optional reading list from the CU Libraries, but you'll need to be on a campus network (or the VPN).
• Problem Set 5: adaptive Runge-Kutta and the Lorenz and Rossler systems. The following materials may be useful to you as you do this problem set:
  • Some news posts from sci.nonlinear about numerical chaos .
  • Section 2.6 of the ODE Notes listed below (particularly for the last problem on this set)
  • A SIAM News piece about numerical dynamics in the solar system that was produced during a final project in this course
  • Some pictures showing how changing arithmetic precision affects the numerics.
• Final Project Details
• Problem Set 6: Poincare sections. You can download a pdf of the Parker & Chua book in the optional reading list from the CU Libraries, but you'll need to be on a campus network (or the VPN).
• Problem Set 7: variational equation. See the notes listed below ("Liz's written notes and videos"). Once again, you can download a pdf of the Parker & Chua book in the optional reading list from the CU Libraries, but you'll need to be on a campus network (or the VPN).
• Problem Set 8: embedding. The following materials may be useful to you as you do this problem set:
  • A detailed list of the assigned reading for PS8-10 ("Reading assignments for time-series analysis")
  • Instructions on getting the data for this problem set and an example of how to embed a data set .
  • The 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 binaries.
  • If you want to install TISEAN on your own machine, these helpful hints regarding installing and using it and the pytisean wrapper on github may be useful.
  • 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 9: Lyapunov exponents. The following materials may be useful to you as you do this problem set:
  • The TISEAN and CSEL links listed in the previous entry
  • A detailed list of the assigned reading for PS8-10 ("Reading assignments for time-series analysis")
  • A schematic of Wolf's algorithm
  • A linear algebra package that finds eigenvalues
  • There are lots of other references and resources for this problem set in the "interesting links" section below, including a link to the TISEAN home page.
• Problem Set 10: fractal dimension. Click here for a detailed list of the assigned reading for this topic and here for a scan of some of that reading (pp166-191 of Parker & Chua). You can download a pdf of the whole book from the CU Libraries, but you'll need to be on a campus network (or the VPN). Here is a jpg of the Hunt & Sullivan proof.
• Some hints about presentations.
• Problem Set 11: 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"). Click here for a picture defining true anomaly and here for a wonderful lecture on dynamical toys like tops and rattlebacks.
• Problem Set 12: integrating the two-body equations. See section 4 of the classical mechanics notes listed below. Here's an interesting 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 the "Chaos Hits Wall Street" article that's on the reading assignment.
• Problem Set 13: 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:

• The Complexity Explorer version of this course. Need to update that link Click on "Lectures" to get to the course materials.
• The mapping of which videos and quizzes on that website go with which CSCI 4446/5446 lectures
• Notes on Taylor series and error in numerical methods (not required, but possibly useful)
• Notes on ordinary differential equations (ODEs) and solving them numerically
• Notes on the variational equation
• A book chapter and a review article about nonlinear time-series analysis ("TSA Notes")
• A schematic of Wolf's algorithm for computing the largest positive Lyapunov exponent
• Notes on classical mechanics

Some useful and/or interesting links: (caveat emptor!)

• A SIAM News piece about numerical dynamics in the solar system that came out of a final project in this class, also featured on the CS Department website.
• A great article from Quanta magazine entitled The Hidden Heroines of Chaos" about the people who carried out Lorenz's computer simulations. (There are lots of other "hidden figures" in this field, including Lise Meitner and Mary Tsingou Menzel).
• xkcd's takes on chaos (and curve-fitting)
• A nice youtube lecture about fractals (21 min)
• An amazing animated bifurcation diagram
• Riding around on the Lorenz attractor
• A 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:
  • A lovely simulation of a double pendulum from myphysicslab. (The Myphysicslab site also contains Java simulations of various other interesting dynamical systems.)
  • You can find another nice animation of the pendulum system (and code to build animations of other things) here.
  • A trippy animation of a grid of double pendulums.
  • An amazing demonstration of nonlinear control of a triple pendulum .
  • And of xkcd's take on pendulums...
• 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 him here.
• Michael Skirpan's fractal tree generator (= the mother of all solutions to PS3).
• CU's site license for Matlab now covers student computers!
• The 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 chaos: complex dynamics of a flock of starlings. Here's the Vimeo version of that video if you prefer that channel.
• A nice movie of metronomes synchronizing (modern-day equivalent of Huyghens' pendulum clocks):
• The 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"). Unfortunately PhET uses Adobe Flash, which has been deprecated. I've left this link here in case you have a workaround.
• Analog computers for nonlinear dynamical systems: the Antikythera mechanism and the digital orrery (built by Liz's advisor)
• "Guide to Takens' Theorem" paper (heavy going, mathematically, but very comprehensive).
• Rigid body dynamics in zero gravity on the international space station.
• A gorgeous youtube video that zooms in on the Mandelbrot set.
• Another gorgeous video of an evolving 3D fractal surface.
• A 'chalkmation' youtube video - complete with music - about the Mandelbrot set (warning: a bit of foul language at the end).
• Chaotic music & dance stuff:
  • A 2013 article from the Boston Globe about Diana Dabby and her latest creation: a web-based chaotic variation generator.
  • Bach's Prelude in C Major on boomwhackers (yes, really)
  • Another way to think about musical variations as mathematics (folding space and time)
  • Stephen Wolfram's WolframTones
  • A chaotic musical instrument that was apparently inspired by one of my lectures (?!?!?)
  • Another fabulous mechanical (and probably chaotic) musical instrument
• Remember that wonderful "powers of ten" video from high-school physics?
• Wolfram's Mathworld site.
• The 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").
• The Chaos Hypertextbook
• Helwig Hauser's visualization of dynamical systems page. The pages above that are interesting, too.
• 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 graphics.
• Some sources of interesting time series data:
  • The Stanford Large Network Dataset Collection
  • The National Snow and Ice Data Center.
  • A data collection maintained by Nature, which presents a series of rapidly evolving resources that aggregate and bring cohesion to the massive volume of data being generated in the COVID-19 crisis.
  • Ceilidh dancing video data.
  • PhysioNet: public databases of ECG, neurological and other types of data.
  • The Allen Institute, which has a ton of openly available neuroscience data.
  • Let me know if you find others so I can post them here.
• Would you like your own double pendulum?