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.

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…

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.

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Maximecompleted 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 lookin…

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.

Classcompleted 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.

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!

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.

A great point for a young person, with knowledge about calculus to start if he or she wishes to pursue this field. This course starts step by step, and is excellent in making the learning about nonlinear-dynamics an almost effortless process. The homework at the end of each unit are a must do to enjoy the course. Knowing computer programing is really a necessity I would say so.

I completed this course a few years ago just to learn about chaos theory. I loved it! I am a math teacher and this year have an honors precalculus course. I am letting my students complete it for extra credit and the ones doing it are enjoying it so much next year I will require it - perhaps work on it as a class each Friday. Thank you so much for a great 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 d…

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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Gaetanocompleted 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.

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.

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.

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 upo…

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!

Spectacular course. High-level teacher. Can you teach a complex issue for people with little mathematical background. I recommend !! Congratulations to the teacher !!

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:

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.

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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Marianocompleted 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 ) ...

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.

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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Purnendu 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.