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Complexity Explorer: Introduction to Dynamical Systems and Chaos

 with  David Feldman
Class Central Course Rank
#1 in Subjects > Mathematics
#1 in Subjects > Mathematics > Calculus

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In this course you'll gain an introduction to the modern study of dynamical systems, the interdisciplinary field of applied mathematics that studies systems that change over time. 

Topics to be covered include: phase space, bifurcations, chaos, the butterfly effect, strange attractors, and pattern formation.  The course will focus on some of the realizations from the study of dynamical systems that are of particular relevance to complex systems:

1.  Dynamical systems undergo bifurcations, where a small change in a system parameter such as the temperature or the harvest rate in a fishery leads to a large and qualitative change in the system's
behavior.

2.  Deterministic dynamical systems can behave randomly.  This property, known as sensitive dependence or the butterfly effect, places strong limits on our ability to predict some phenomena.

3.  Disordered behavior can be stable.  Non-periodic systems with the butterfly effect can have stable average properties.  So the average or statistical properties of a system can be predictable, even if its details are not.

4.  Complex behavior can arise from simple rules.  Simple dynamical systems do not necessarily lead to simple results.  In particular, we will see that simple rules can produce patterns and structures of surprising complexity.

83 Student
reviews
Cost Free Online Course
Pace Upcoming
Subject Calculus
Institution Santa Fe Institute
Language English
Certificates Certificate Available
Hours 3-6 hours a week
Calendar 10 weeks long

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Reviews for Complexity Explorer's Introduction to Dynamical Systems and Chaos
4.9 Based on 83 reviews

  • 5 stars 96%
  • 4 stars 2%
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  • 1
5.0 4 years ago
Aleta Duvall completed this course.
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
Read more
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

5 people found
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5.0 2 years ago
by Maxime Zabiégo completed this course, spending 2 hours a week on it and found the course difficulty to be very easy.
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,
Read more
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.
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5.0 4 years ago
Class Central user completed this course, spending 2 hours a week on it and found the course difficulty to be medium.
This is a fun class which I would highly recommend. Incredibly accessible course which introduced an interesting branch of mathematics.

If you've done a bit of computer science or programming, this class shows a very interesting result from the simple idea of iteration.

The professor is interesting and makes the material intuitive. Regardless of your level of mathematical comfortability, this class is well paced and provides all the tools you need to proceed through it.

Don't miss this class!!
7 people found
this review helpful
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5.0 4 years ago
Sabine Hiller completed this course.
A fascinating course, I absolutely loved it! There was a bit of maths involved, but Dave Feldman walked us through it step by step - so slowly that even somebody who has no idea whatsoever of maths should grasp it. Sitting in front of my laptop on the other side of the globe, I felt he deeply cared that you understood everything - as somebody said above, it almost felt like a personal tutorial! As I had already done Melanie Mitchell's course (equally excellent!), I was familiar with much of the territory, but Dave's approach helped me to understand some of the more difficult concepts. I think the two courses complement each other very nicely. Highly recommended!

5 people found
this review helpful
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5.0 4 years ago
David completed this course.
This class was just amazing and great. Mr. Feldman knows how to really get you to see the 'big idea' and that's the most import part of any intro class; to get the big picture. Math was light and not terrifying. Deadline is the class day of class. Quizzes were light and the homework was challenging enough if you wanted it to be. I absolutely loved this class.
2 people found
this review helpful
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5.0 4 months ago
by Narasimhan Mg completed this course.
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.
Read more
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.
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5.0 4 years ago
by Gaetano Pagani completed this course, spending 5 hours a week on it and found the course difficulty to be medium.
I had a great time having this course and Prof. Feldman was so dedicated in diving us into this amazing subject. I think that the course gave us a solid platform in order to go deeper in the study of Dynamical Systems and Complexity.
2 people found
this review helpful
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5.0 a year ago
Anonymous completed this course.
I enjoyed the course very much. The teacher is great, he can explain both familiar and new (complicated) matter in a fresh and understandable way. I helped me to understand dynamical systems deeper, as I do dynamical modeling in economics.
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5.0 4 months ago
Anonymous is taking this course right now.
The effort put into the course becomes clear as soon as you start. A very accessible design of the material without loosing the required depth for a proper understanding of the essence.
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5.0 2 years ago
Anonymous is taking this course right now.
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
Read more
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!
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5.0 2 years ago
Anonymous completed this course.
Spectacular course. High-level teacher. Can you teach a complex issue for people with little mathematical background. I recommend !! Congratulations to the teacher !!
1 person found
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5.0 2 years ago
by Davide Madrisan completed this course.
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.
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5.0 3 months ago
by Linda Alario completed this course.
I am retired; I've done no mathematics of any significance since the early 70's. This course brought me hours of challenge and joy. Every time I completed homework or unit exams I felt enormous satisfaction.

My deepest thanks to Professor Feldman. He is an extraordinary educator. He communicates clearly and makes the complex understandable. Despite his vast knowledge and expertise, he is still excited about the material he is teaching. Some of my favorite moments were his asides to make sure we appreciated how abstract concepts can be observed in physical systems. The course was well organized and I appreciated that all of the material was accessible for review and replay. The unit summaries were very helpful. I just sorry the course is over.
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5.0 4 months ago
by Nicholas Henri Tay completed this course.
Firstly, the platform is easy to use. It makes focus and moving around content much easier to achieve.

Next, the course itself is riveting not just because of the digestibility of the content, courtesy of the instructor, but also the availability of programs to engage and interact with that enforces the lessons that David provides.

David is also an engaging instructor and is able to deliver content at a pace and clarity that is oh so important for MOOCs. This is critical in understanding the key lessons or ideas or concepts he wants you to leave with at the tail-end of the course.

And of course, lastly, the subject itself is fascinating and absorbing once you start thinking about the real-world comparisons that David introduces quickly.
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5.0 4 months ago
by Mariano "qhawaq" Mancini completed this course, spending 1 hours a week on it and found the course difficulty to be medium.
Leonardo da Vinci said : "Simplicity is the ultimate sophistication" and professor Feldman with his way of thinking exactly represents Leonardo's thougth.

Only great professors uses scissors, pencils and papers and explain very complicated things with ease .

This course is a very good one, professor Feldman's explanations are really clears and complex materials "flows" very well .

Personally have already recommended this course to others and think that I will gladly follow other courses in which professor Feldman will be the teacher considering his high scientific level and his predisposition to teach in a simple way ( where he can ) ...

Simply great !!!
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5.0 3 months ago
Anonymous completed this course.
This course is by far one of the best courses I've ever taken. It balances depth and simplicity perfectly. One finishes the course understanding Dynamical Systems and Chaos enough to discuss, question, and apply key concepts to other fields of study. I'm an architect, and I nevertheless understood every topic in the course. Professor David always relates and brings back specific aspects he is teaching to global ideas as to understand the relevance of the topic. I believe this is a great way of teaching as it maintains interest, and allows the student to understand how each piece fits into the general ideas.
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5.0 a year ago
Anonymous completed this course.
This is a very good introduction to (nonlinear) dynamical systems. Instruction is very good and the provided simulations are very helpful.

I am a math dummy and my background is mostly biology. This class did a so good job in explaining dynamical systems that I realized biological systems are likely to be ( maybe complex) dynamical systems. However, due to experimental limitations, molecular and cellular biologists hardly can study their systems as a dynamical system. After taking this class, I am decided to try to study biological systems (nervous systems, more specifically) as dynamical systems for my future work.
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5.0 4 months ago
by Purnendu Nath is taking this course right now, spending 2 hours a week on it and found the course difficulty to be medium.
Professor Dave Feldman is an excellent teacher. He knows exactly what pace to proceed at, what to repeat more than once, and what thoughts/questions the student must have who is watching. He introduces humour as he goes along which is also great to keep things real almost giving a feeling of being in the room with him. I read the book "Chaos" almost 30 years ago and it was great taking this course now to formalize what I read back then. I am looking forward to doing more courses taught by him and also hope that other courses on the Complexity Explorer site are taught as well as this was.
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5.0 a year ago
Anonymous completed this course.
This class was great!

What a great journey, from phase lines, through period doubling, all the way through strange attractors. In my opinion it totally succeeded as an introduction, as I was able to build a relationship with the material in advance of staring down the calculus (which I'll be doing next).

I really appreciate professor Feldman's emphasis on visualizations, geometry, and conceptual abstraction. This is the experiential anchor I was looking for to balance my more computational inquiries.

A jolly good frolic through the wonders of dynamical systems!

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5.0 2 years ago
Anonymous completed this course.
The presentation by the professor of the complicated material was very clear in terms of the verbal explanation of the concepts and the visual presentations. The homework and the exams were closely tied to the presentations which facilitated comprehension. The focus was on the conceptual but, also, there were exercises for those students with more technical expertise. I thoroughly enjoyed the course. Now I have a background to explore the topics in greater depth which I think is one of the best outcomes from such a course.
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