Nonlinear Dynamics: Mathematical and Computational Approaches
Lead instructor: Liz Bradley
Unit 1:
1.1 Introduction to nonlinear dynamics
 The Cassini video of Hyperion tumbling
 Video recordings of the lectures from Steve Strogatz's introductory course on nonlinear dynamics and chaos and the Google Books link to his textbook
 The semesterlength version of this course that I teach at the University of Colorado every spring
 Wonderful books about this field: James Gleick's Chaos: Making a New Science and Ian Stewart's Does God Play Dice? The Mathematics of Chaos
 xkcd's take on chaos
1.2 Maps and difference equations
 The official Complexity Explorer logistic map app.
 There are a ton of Matlab tutorials available, including from TutorialsPoint, the University of Utah, and the University of Colorado.
 Matlab isn't free, but Octave is — and it's almost indistinguishable from Matlab.
1.3 Transients and attractors
 The official Complexity Explorer logistic map app.
 A short book that has lots of good stuff about maps: R. L. Devaney and L. Keen, eds., Chaos and Fractals: The Mathematics Behind the Computer Graphics, American Mathematical Society, 1988 (Proceedings of Symposia in Applied Mathematics, volume 39).
1.4 Parameters and bifurcations
 The official Complexity Explorer logistic map app.
Unit 2:
2.1 Return maps
 The official Complexity Explorer logistic map app.
2.2 Constructing the bifurcation diagram
 The official Complexity Explorer logistic map app.
2.3 Exploring the bifurcation diagram
 The official Complexity Explorer logistic map app.
 That video I showed that zooms in on the Mandelbrot set. Benoit Mandelbrot coined the word "fractal," meaning "sets with noninteger dimension." A fun application is his paper about the length of the coast of Britain.
 M. Barnsley, Fractals Everywhere, Elsevier 1993. You can get a pdf of this book from ScienceDirect or from Barnsley's website at Imperial College; just google 'barnsley fractals everywhere pdf' and dig around.
 If you're curious about the proof that the bifurcation happens at r=3.0, check out these notes from Julien Arino at the University of Manitoba or David Lerner at the University of Kansas
2.4 Feigenbaum and universality
 The official Complexity Explorer logistic map app.
 The original paper: M. J. Feigenbaum, "Universal Behavior in Nonlinear Systems," Los Alamos Science 1:427, 1980. This paper can be a bit hard to find in its original form, but you can also find it in Predrag Cvitanovic's Universality in Chaos reprint collection.
2.5 Field trip: The standard map (with Jim Meiss)
 Jim's homepage, as well as the page where you can find his StdMap app and a link to the publisher's webpage for his fabulous textbook on Differential Dynamical Systems. (If your institution has SIAM digital library permissions, you can get a free ecopy of that book here.)
Unit 3:
3.1 What is a flow?
 A link to some datasets from the driven pendulum that I showed towards the end of this video
 This paper goes into great detail about what happens when the drive frequency (and amplitude) are varied in a driven pendulum like that: D. D’Humieres et al., “Chaotic States and Routes to Chaos in the Forced Pendulum,” Phys. Rev. A 26:3483, 1982
 An nice article from the New Scientist about chaos in pendulums (via Google Books)
3.2 State variables and state space
 The movement of an array of pendula with different lengths How many state variables do you think this thing has??
3.3 Introduction to ODEs
3.4 Nonlinearity and nonintegrability
 The lovely animation of the simple harmonic oscillator ("SHO") that I showed at the beginning of this video segment (from the Physclips project at UNSW in Australia)
 The real definition of integrability
3.5 Field trip: Modeling the human insulin system (with Sriram Sankaranarayanan)
Unit 4:
4.1: Fixed points and stability
 Wikipedia's entry on fixed points is pretty good
4.2: Saddle points and eigenvectors
 The Khan Academy series on "Eigeneverything" is a nice introduction to this material
 If you learned about this stuff at some point but need to dust off your knowledge a bit, you may prefer Paul Dawkins's notes on this topic
4.3: Stable and unstable manifolds
 A pointer to the chapter of "Introduction to Dynamical Systems" by Alligood, Sauer, and Yorke that discusses the role of stable and unstable manifolds in chaos
4.4: Attractors, strange and otherwise
4.5: Field trip: Using stable and unstable manifolds to design spacecraft trajectories (with Jeff Parker)
 Jeff's homepage
 a link to a lowresolution pdf version of his book
 a link to the highresolution version of his book (on amazon.com)
Unit 5:
5.1: ODEs, vector fields, and dynamical landscapes
 Section 1 of my notes on ordinary differential equations (ODEs) and solving them numerically
 S.O.S.Mathematics has a nice page about the vector fields defined by ODEs
 I was going to make a video about Jacobians and their role in linearization of nonlinear systems, but Jeffrey Chasnov has already made a nice one, which you can find here. (There's lots of other good stuff available through various links on his home page, including some notes on differential equations, numerical methods, etc., and an ebook with youtube examples.)
5.2: Introduction to ODE solvers
5.3: Two simple ODE solvers: forward and backward Euler
 Sections 2.1.1 and 2.1.2 of my notes on ordinary differential equations (ODEs) and solving them numerically
5.4: Solving the simple harmonic oscillator ODEs
5.5: Field trip: Systems that can't be modeled with ODEs (with Jean Hertzberg)
 Jean's homepage, as well as the homepage for her course
 Ground liquification up close and personal (I sure wouldn't stand where this cameraman is!)
Unit 6:
6.1: ODE solvers, round II: Error and adaptation
 My notes on Taylor series and on error in numerical methods
6.2: Production ODE solvers
 The wikipedia page on multistep ODE solvers gives a good description of how they work, and how they differ from singlestep methods like the Runge Kutta clan
6.3: Numerical dynamics and due diligence
 Lloyd Fosdick's notes on IEEE FloatingPoint Arithmetic
6.4: Shadowing and chaos
6.5: Field trip: Solving partial differential equations (with Christine Hrenya)
Unit 7:
7.1: Dynamics and statespace deformation
 I haven't been able to find a lot of good online material about Mel'nikov's method, but this MS thesis has a pretty good introduction
7.2: Lyapunov exponents
 My notes about the variational system
 You can find lots of good papers about stable and unstable manifolds, and how to find them numerically, on Hinke Osinga's webpage
7.3: Sections and projections
 Henri Poincare didn't only play a formative role in the foundation of the field of nonlinear dynamics. He also came up with the theory of relativity — and wrote down e=mc^2 —before Einstein did. Read a bit about him here.
 Parker & Chua's Practical Numerical Algorithms for Chaotic Systems has a good chapter on constructing sections.
7.4: Unstable periodic orbits
 Papers about UPOs and attractor structure: P. Cvitanovic, "Invariant measurement of strange sets in terms of circles," Phys Rev Lett 61:2729 (1988) — and a couple of moretechnical ones in Nonlinearity in 1990 (vol 3 pp 325386).
 Papers about finding UPOs: G. Gunaratne et al., ""Chaos beyond Onset: A Comparison of Theory and Experiment," Phys Rev Lett 63:14 (1989); P. So et al., "Extracting unstable periodic orbits from chaotic timeseries data," Phys Rev E 55:5398 (1997)
 E. Bradley and R. Mantilla, "Recurrence plots and unstable periodic orbits." Chaos 12:596600 (2002).
7.5: Fractals and chaos
 The wikipedia page about Cantor sets has a ton of mathematical detail, for those who are into that kind of thing.
7.6: Field trip: Fractals and scaling (with Dave Feldman)
 Dave's homepage and the Complexity Explorer page for his course on fractals and scaling
 The DLA image that he showed during this segment
 A lovely movie of diffusionlimited aggregation in action
Unit 8:
There are a number of references that will help you with this unit and the next one: my notes on timeseries analysis and the wonderful book Nonlinear Time Series Analysis by Holger Kantz and Thomas Schreiber. You can find the Kantz & Schreiber book on google books, but it's really worth owning a copy if you work with time series data (amazon.com). A third reference is this recent review article, a copy of which you can also find on the arxiv.
8.1: Timeseries analysis and the observer problem
 A bit more about frequency spectra
 I've used the word "superposition" a couple of times. The wiki page about it gives a pretty good description of what it means and why it breaks in nonlinear systems.
 The observer problem is the task of deducing the internal variables of a blackbox system solely from observations of its output (viz., my example about reverseengineering the internal electronics of a traffic light control box from observations of when the lights change color). It's one of the hardest problems in control theory.
8.2: Delaycoordinate embedding
 Section 3.2 of Kantz & Schreiber discusses delaycoordinate embedding, as do section II A of the review paper listed above and section 3.1 of my timeseries analysis notes.
8.3: Topology, diffeomorphisms, and reconstruction of dynamics
8.4: Estimation of embedding parameters
 The TISEAN timeseries analysis toolkit includes lots of good stuff — including Lyapunov exponent and correlation dimension calculators. The TISEAN site has binaries for UNIX & windows. You may need this fortran library to get it to work. If you're a Mac user and you have brew on your machine, you can simply type 'brew install tisean' (without the quotes, of course). Here are some examples of how to run all of this from MATLAB. Be aware that TISEAN is not a required element of this course and that it can sometimes be hard to install.
 The wikipedia page about autocorrelation, which is essentially a measure of how similar different chunks of a signal are to one another. To use autocorrelation to choose , you could compute the correlation between chunks of the signal that are time units apart and average that quantity across the whole signal. Maxima in such a curve correspond to values for which successive coordinates in a delay vector will be highly correlated (which is not a great idea).
 Mutual information measures how much one (random) variable tells you about another one. There are tons of other ways to get at that information, many of which have the word "entropy" in their names—e.g., transcription entropy.
 Sections 3.3.1 and 3.3.2 of Kantz & Schreiber discuss finding m and , respectively, as do section II B of the review paper listed above and section 3.2 of my timeseries analysis notes.
8.5: Caveats and extensions
 A bit more about the implications of computing derivatives from noisy data
8.6: Field trip: Predicting extreme events (with Holger Kantz)
 Holger's webpage
 The images that he showed during this video segment
 A paper about his work on predicting extreme events
Unit 9:
9.1: Computing fractal dimensions
 Chapter 6 of Kantz & Schreiber and section III A of the review paper listed above (under unit 8) discuss algorithms for calculating fractal dimension.
 The original paper about calculating Lyapunov exponents: A. Wolf, J. Swift, H. Swinney, and J. Vastano, "Determining Lyapunov exponents from a time series," Physica D 16:285317 (1985)
9.2: Computing Lyapunov exponents
 Chapter 5 of Kantz & Schreiber and section III B of the review paper listed above (under unit 8) discuss algorithms for calculating Lyapunov exponents.
9.3: Noise and filtering
 Section 1 of my timeseries analysis notes (under unit 8) gives a brief introduction to traditional linear systems analysis (cf., the lamp post).
 Chapter 10 of Kantz & Schreiber discusses noise (and who to distinguish chaos from noise).
 The original paper about that noisereduction scheme that deforms noise balls back & forth in time: J.D. Farmer and J.J. Sidorowich, "Exploiting Chaos to Predict the Future and Reduce Noise," in Evolution, Learning and Cognition, World Scientific, 1988.
 Papers about topologybased filtering: V. Robins and N. Rooney and E. Bradley, "TopologyBased Signal Separation," CHAOS 14:305316 (2004) and Z. Alexander and E. Bradley and J. Garland and J. Meiss, "Iterated Function System Models in Data Analysis: Detection and Separation," CHAOS 22:023103 (2012)
9.4: Field trip: Chaotic mixing and marine invertebrate reproduction (with John Crimaldi)
Unit 10:
10.1: Prediction

A. Weigend and N. Gershenfeld, eds., Time Series Prediction: Forecasting the Future and Understanding the Past, Santa Fe Institute Studies in the Sciences of Complexity, Santa Fe, NM, 1993.

J. Garland and E. Bradley, "Prediction in projection," Chaos 25:123108 (2015). Preprint available at arxiv.

J. Garland, R. James, and E. Bradley, "Quantifying TimeSeries Predictability through Structural Complexity," Physical Review E 90:052910 (2014). Preprint available at arxiv.
10.2: Control of chaos
 Troy Shinbrot's review paper on the control of chaos: "Progress in the control of chaos," Advances in Physics 44:73111 (1995)
10.3: Classical mechanics
 A youtube video about spinlock in the earthmoon system
 Rigid body dynamics in zero gravity aboard the international space station
 My written notes about classical mechanics
 Analog computers for nonlinear dynamical systems: the Antikythera mechanism and the digital orrery (built by Liz's advisor)
 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 nbody simulator (called "My Solar System").
10.4: Music and dance (with a coda on the difference between chaos and complexity)
 A list of chaotic music clips from Diana Dabby's 2008 Science article (320(5872): 6263) and a 2013 article from the Boston Globe about her latest creation: a webbased chaotic variation generator.
 A very different take on Bach's Prelude in C Major (on boomwhackers) (really).
 What's a Voronoi diagram? What's a directed graph?
 A chaotic musical instrument that was apparently inspired by one of my lectures (?!?!?)
 You can find out a lot more about chaotic choreography and stylistically consonant interpolation here, including pictures, videos, and links to technical papers about both topics.
 The amazing website for the One Flat Thing, Reproduced project, which includes videos, scores, cues, and much more.
 Complexity, the flip side of chaos: complex dynamics of a flock of starlings
Lecture Slides (zipfiles of pdfs): please be aware that these are not course notes; they are simply the powerpoint slides that I use here and there in the lectures. There is no textbook for this course, nor are there any compiled course notes. This course draws upon material from different textbooks, journal papers, conference talks that were never published, and our own experience. The lecture videos are your primary resource for that material. Please check out the links above if you want more background information for each segment, or if you want to dig more deeply into the material.
 Unit 1 slides (593KB)
 Unit 2 slides (9MB)
 Unit 3 slides (85kB)
 Unit 4 slides (14MB)
 There were no powerpoint slides for Unit 5
 Unit 6 slides (2.5MB)
 Unit 7 slides (7MB)
 Unit 8 slides (1.6MB)
 Unit 9 slides (7.6MB)
 Unit 10 slides (14.1MB)
Lecture and Solution Videos (zipfiles of mp4s): the "full resolution" ones are what's on youtube.
 Unit 1 videos: full resolution (830MB) and low resolution (190MB)
 Unit 2 videos: full resolution (673MB) and low resolution (152MB)
 Unit 3 videos: full resolution (880MB) and low resolution (188MB)
 Unit 4 videos: full resolution (463MB) and low resolution (124MB)
 Unit 5 videos: full resolution (357MB) and low resolution (82MB)
 Unit 6 videos: full resolution (551MB) and low resolution (146MB)
 Unit 7 videos: full resolution (669MB) and low resolution (152MB)
 Unit 8 videos: full resolution (1GB) and low resolution (249MB)
 Unit 9 videos: full resolution (763MB) and low resolution (182MB)
 Unit 10 videos: full resolution (626MB) and low resolution (164MB)
PDFs of homework assignments and quizzes (zipfiles of pdfs):
These zipfiles do not contain every quiz pdf because we enter those directly via the web interface, rather than building pdfs. What I've uploaded is last year's quiz pdfs; where this year's version is different, I did not include it. You can do a 'print' from the webpage to make your own pdf.
 Unit 1 quiz/HW pdfs (313kB)
 Unit 2 quiz/HW pdfs (621kB)
 Unit 3 quiz/HW pdfs (9MB)
 Unit 4 quiz/HW pdfs (165kB)
 Unit 5 quiz/HW pdfs (58kB)
 Unit 6 quiz/HW pdfs (190kB)
 Unit 7 quiz/HW pdfs (352kB)
 Unit 8 quiz/HW pdfs (333kB)
 Unit 9 quiz/HW pdfs (288kB)
 Unit 10 quiz/HW pdfs (92kB)