Introduction to Dynamical Systems and Chaos

I appreciated the structured approach to the course material and the painstaking development of foundational concepts. Dr. Feldman presents the course in an informal, across-the-desk manner. Each lecture feels like you are experiencing an individual tutoring session during office hours. I recommend the course to students who struggle with math or computing anxiety; only mininal calculus is needed to understand and apply the material. Below is a topical overview of the 9-week course.

Lectures:

What is a dynamical system?
General properties - classification / characterization.
Iterated functions, orbit, itinerary; examples.
Differential equations, examples; rule is indirect, involves rate of change of a variable.
Solution methods: analytic, qualitative, numerical / computational / algorithmic.
Initial condition + rule -> existence and uniqueness.
Chaos, examples; deterministic, orbits bounded, aperiodic, sensitive dependence on initial conditions.
The butterfly effect.
Algorithmic randomness, incompressibility.
The logistic equation.

1-D differential equations (chaos not possible) vs 1-D iterated function systems (chaos possible).
Time is continuous vs discrete time intervals; dependent variable is continuous vs discrete values.

Bifurcation diagrams; examples.
Period-doubling route to chaos.
Universality in period doubling, Feigenbaum's constant.
Universality in physical systems; examples.

2-D differential equations (chaos not possible - Poincare-Bendixson theorem); examples.
The phase plane.
Stable and unstable fixed points, orbits can tend to infinity, limit cycles (attracting cyclic behavior) - but no chaos.

3-D differential equations (chaos possible); Lorenz system of equations.
Phase space.
Strange attractors: stable attractors but motion on the attractor is chaotic; examples.
Stretching and folding in chaotic orbits.
Strange attractors combine elements of order and disorder; motion is locally unstable, globally stable.
Pattern formation in dynamical systems; examples.
Reaction-diffusion systems.
Simple, spatially-extended dynamical systems with local rules are capable of producing stable, global patterns and structures.

Interviews:

Stephen Kellert, prof of philosophy at Hamline University.
Chaos theory represents an evolution (vs revolution), a new style of scientific reasoning or doing science.
Represents a conceptual reconfiguration, gets rid of old dichotomies.
You can have conceptual change that's brought about through methodological challenges, not just through grand theoretical structures being changed.
Chaos theory is a part of postmodernism - challenging of strict binaries.

Stephen W. Morris prof of geophysics at Univ. of Toronto.
Pattern formation in nature; examples and demonstrations. "Swimming in the wrong direction of reductionism".

Conceptual - thematic summary

Remarks

This course is a awesome mine of information and knowledge on dynamical systems and chaos. It starts at the very beginning by explaining the mathematical bricks necessary to understand the following units, but it's not a heavy mathematical course.
Quite the opposite. The lead instructor David Feldman is always able to make even the more complex parts of this mooc easy to understand.

David has also developed a well featured web software for visualizing all the course material: equation orbits, Butterfly Effect, bifurcation diagrams and Strange Attractors patterns.

If you're a programmer, you can complete some of the advanced (but optional) exercises using the language of your choice. Here's my version in Python:
https://github.com/madrisan/dynamic-systems-and-chaos

In short this mooc has been a wonderful mathematical trip.

This course is perfect for beginners who want to get a grasp of the field of chaos with minimal mathematical/numerical material, but yet enough to have a flavor of the underlying world, beyond the bases. Dave Feldman is a great teacher, very dedicated to explaining the ideas in a simple way, accessible to most (if not all).

I was already acquainted with chaos, which I had studied at University... 25 years ago. So, I found the course fairly easy and didn't watch all videos. Nevertheless, the material I reviewed was very relevant and interesting: the excellent refresher I was looking for to remind me of the good old days! I took the class parallel to re-reading the book "Chaos", by James Gleick, which Dave Feldman cleverly recommends as a complementary resource. The course and the book complement one another idealy (more historical/people aspects in the book), considering that they follow very similar outlines.

The topic of chaos is coverred very progressively, to introduce the key ideas. Many illustrations are provided and detailed, to clarify the theory by simple computations: the most tricky ones are conducted through programs made availalble by Dave Feldman, so that no mathematical/numerical skill is actually required.

Although the course is clearly aimed at some introductory level, Dave Feldman gives many hints on more difficult issues that would require some more involved approach. In addition, many practice exercises (optional) are proposed at different levels: the most advanced allow those who want to go further than the basic level required for the graded quizzes to tackle more technical problems.

An excellent course that I do recommend to anyone interested in learning about chaos at an introductory level.

Thanks for making this course available for all. It really changed the way I see the computational world.

I had a masters degree in Computational engineering where I learned about the fundamentals of Continuum mechanics and its applications in Structural and fluid mechanics. I also learned to programm some numerical methods such as FEM, FDM, FVM, spectral solvers to find the solutions to ordinary and Partial differential equations. I then applied these tecniques in various applications to model physical phenomena like stress distribution in a hanging bridge, Fracture initiation in dual phase steels, fluid flow over a car, Antenna modeling using maxwellequations.

While learning and appllying the simulation techniques I always encounter convergence problems, which I mostly ignored so does many practitioners. We just try to increase the mesh size, or time interval and try to fit some parameter to the experimental curve. If nothing works, one I used to change initial conditions or the boundary conditions. But I have ner came heard any one talking about Chaos, period doubling phenomena, butterfly effect or the sensitive dependance on initial conditions.

I came to know this after reading the book ,"Chaos" - by James Gleick. From then so many ideas were going on in my mind. I was both surprised and confused. I then learned about this course in the Complexity explorer and strated taking the classes. After attending this course, I realized that most of the numerical methods which we use to solve the ordinary or partial differential equations have these phenomena inherited in them. I realized that modeling might not just be fitting a curve to the experimental data but to observe some universality between the parameter of the model and how the physical system behaves.

I sometimes wonder why we are not taught about these phenomena which seems to be a fundamental one for many numerical methods. I wonder why even a Masters student in Computational engineering has not been not taught about these intersting findings.

Thanks again for making this course simple yet profound

This was the missing link for me. I had read many popular science books about dynamical systems and chaos, and was eager to gain a more nuanced and technical understanding of the field. The only thing holding me back, of course, was a lack of background in higher mathematics, particularly "differential equations," which are drawn on heavily in any technical book about the subject. I was starting to get discouraged at the prospect of having to study lots of highly technical math for years and years before being able to advance at all in my understanding of dynamical systems, when I stumbled upon this class. David Feldman's approach is awesome: completely intuitive and broken down into comprehensible chunks. As someone who took calculus in high school and understood practically none of it, I feel very gratified at being able to grasp what differential equations are all about, even if I'm still a bit of a ways from being able to "solve" them mathematically. Taking this class has so filled in the intellectual gaps for me that not only do I want to re-read some of the books I have with the greater understanding, but I'm actually more inspired to learn more higher math so I can study this stuff further. Thanks!

I really enjoyed doing this course. It gave me a great introduction to the dynamical systems and chaos field. The presentations were very clear, well presented with lots of examples, both mathematical and graphical. The lectures were delivered at the highest levels of standards. Also the availability of supplemental material, books, articles, links, etc., is very helpful and I enjoyed a lot reading the recommended stuff that deepens my knowledge further in this field.

Thank you for introducing to me the fascinating field of chaos.

Congratulations on the successful course delivery. I would like to take a similar course at the next level when it is available on your site.

Kind Regards,

Violeta Moloney

Complexity and Dynamical Systems is a fascinating field of mathematics. There are linear, predictable and well-ordered systems, like simple physical motion. There are non-linear systems, like the weather. What are the latter like? Can they be studied and modeled mathematically?

Professor David Feldman takes a very systematic and ordered approach to the subject. He introduces a unit, talks about what he will cover and then take up a simple equation to show how it can lead to either predictable or unpredictable results. He lets us wonder how a deterministic system gives rise to disorder and chaos. Through graphs, charts, step-by-step workouts and illustrations, he gives endless depth to many aspects of dynamical systems.

The quizzes and homework help us do more exercises like these and get the hang of complexity. There are also computer programs that have been developed and are available to explore the parameters of complexity in an exhaustive manner.

I specially liked the paradoxes in complexity. The self-similar replicas of any part of the system that appears chaotic. The co-existence of order and disorder. The sensitivity to initial conditions., and the butterfly effect.

I wonder if fields like quantum mechanics, natural evolution and even past history of the world can be studied as complex dynamical systems.