subject
Intro

Complexity Explorer: Introduction to Dynamical Systems and Chaos

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

HIGHEST RATED MOOC

This course is a Top 50 MOOC of All Time based on thousands of reviews written by Class Central users. It's guaranteed to be good!

Check out the rest of the Top 50 here.

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.

53 Student
reviews
Cost Free Online Course
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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53 reviews for Complexity Explorer's Introduction to Dynamical Systems and Chaos

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5 out of 5 people found the following review useful
3 years ago
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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 studen 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

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a year ago
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 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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7 out of 7 people found the following review useful
3 years ago
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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 mat Read More
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!!
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5 out of 5 people found the following review useful
3 years ago
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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 Read More
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!

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2 out of 2 people found the following review useful
3 years ago
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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.
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2 out of 2 people found the following review useful
3 years ago
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.
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12 months ago
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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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a year ago
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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," whic 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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1 out of 1 people found the following review useful
a year ago
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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 !!
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a year ago
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 Read More
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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a year ago
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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 comp Read More
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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a year ago
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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 ap Read More
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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a year ago
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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 exerci Read More
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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a year ago
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Anonymous is taking this course right now.
Prior to this course I had read popular expositions such as James Gleick's "Chaos", but had never had access to a basic formal exposition of complex behaviour. This course offers a fascinating introduction to chaos theory. I had plenty of "aha" moments, and David Feldman's excursion into the philosophical consequence Read More
Prior to this course I had read popular expositions such as James Gleick's "Chaos", but had never had access to a basic formal exposition of complex behaviour. This course offers a fascinating introduction to chaos theory. I had plenty of "aha" moments, and David Feldman's excursion into the philosophical consequences of determinism producing unpredictability was also an eye-opener.

This a fun and accessible course, and I find myself eagerly awaiting release of the next Unit each week. Thoroughly recommended
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a year ago
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Anonymous completed this course.
The course is comprehensible by illuminating most important concepts in deterministic chaos. I believe that it is essential for those who use these concepts (eg: Butterfly Effect, Innitial conditions, Order and Chaorder) etensively to justify applications and emergent methodologies. Thanks to David Feldman and his te Read More
The course is comprehensible by illuminating most important concepts in deterministic chaos. I believe that it is essential for those who use these concepts (eg: Butterfly Effect, Innitial conditions, Order and Chaorder) etensively to justify applications and emergent methodologies.

Thanks to David Feldman and his team for the clarity.

Because the importance of this course (MOOC) in spreading significant knowledge and making it accessible to everybody, I encourage donatations to keep the flame burning.

Judy Tal
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12 months ago
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Anonymous completed this course.
Very clear explanations and interesting topics. The best way to start learning about some a rich and deep subject in a theoretical way, for a more mathematical way, it gives a good base and sets important foundations and, possibly, even a deeper non-analytical understanding of dynamical and chaotic processes and phenomena.
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a year ago
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Anonymous completed this course.
Great course taught by a fun and knowledgable instructor Dave Feldman. Dr. Feldman is at once smart, thorough, and funny. The course lays in foundational material and then thoughtfully explores the topic of dynamical systems, including "chaos" theory. Highly recommended.
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7 months ago
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Anonymous audited this course.
The course gives an interesting outline of dynamical systems. However if you are already experience in math you might find it too vague. I would recommend then just to watch the summaries of each week (which are very good). Other than that, the course is really good.
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a year ago
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Anonymous completed this course.
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
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a year ago
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Anonymous completed this course.


This is one of the best lectures I have ever taken. Prof. Feldman exactly knows what the students need, and teaches in a very kind manner. I surely recommend this course to everybody who wants to step into the world of complexity but does not have much background.
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a year ago
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Anonymous is taking this course right now.
The course was spectacular. One of the teachers with the best teaching I have had the opportunity to be a student. You can teach difficult content to people as little mathematical background like me. Congratulations!! I recommend everyone to do the course.

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a year ago
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Anonymous completed this course.
David ROCKS in leading this class. Clear and concise! The math part is just fine . . . but it could benefit from a bit more philosophical grounding so I understand WHY I should study this topic. Despite that, it is just plain interesting material.
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a year ago
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Anonymous partially completed this course.
I took previously the fractals entire course, and Davide Feldman the instructor of bouth courses is !!!!!THE BEST!!!!! teacher i ever had, is absolutly friendly and his explanations of the materia is very clear. Thanks a lot David your the best!!
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11 months ago
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Anonymous completed this course.
Super interesting, well paced, fun and useful! I specifically appreciated the thorough examples, and the tools that let you play around with numbers, attractors and strange attractors yourself. Thank you complexity explorer!
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10 months ago
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Anonymous completed this course.
Very helpful entry point to dynamical systems, with an almost peerless genius teacher. You won't regret taking this well designed, very interesting course. Advanced maths background is not required so dive right in.
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a year ago
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Anonymous completed this course.
The course is excellent. I highly recommend this one to the broad audience. It is interesting and self-consistent. If you are not the expert in the field it's still a good idea to watch couple of lectures .
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a year ago
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Anonymous is taking this course right now.
t h a n k y o u a 1000 times for enlarging my small horizon

it is truly a pleasure and I will never forget how this course enlightened me about a certain part from psychoanalysis that seemed to remain inaccesible.
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a year ago
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Anonymous completed this course.
Fantastic course. David takes you along nice and easily through the subject but still manages to cover a lot of stuff to give a really good grounding in the area. Liked his sense of humour too :)
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8 months ago
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Anonymous completed this course.
Thanks a lot for such nice and interesting course.

Thanks a lot for such nice and interesting course.

Thanks a lot for such nice and interesting course.

Thanks a lot for such nice and interesting course.
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a year ago
Adrian Silva completed this course, spending 3 hours a week on it and found the course difficulty to be medium.
Introductory concepts, light on math.

Is unfortunate this kind of materials is not included in high school education globally. Fundamental concepts to understand how complex systems behave and fail.
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a year ago
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Anonymous is taking this course right now.
Interesting and complex ideas explained using high school maths (with some optional homework that contain a little bit more advanced exercises). Very easy to follow the instructor. 5 stars!
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a year ago
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Anonymous completed this course.
El curso me pareció de una intensidad y profundidad acorde con lo propuesto -nivel de matemática medio- si la tercera lectura fue muy larga, ed dos partes sería más fácil de comprender
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a year ago
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Anonymous is taking this course right now.
This is an excellent introduction to a complex set of ideas. Well worth the time! I thoroughly recommend this course to anyone interested in learning more about complexity and dynamical systems.
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a year ago
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Anonymous completed this course.
Great introductory course in Dynamical Systems and Chaos. There aren't many prerequisites. Physics or Math students may find it relatively easy, but it is supposed to be an introduction.
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a year ago
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Anonymous completed this course.
A great introduction to the field of Nonlinear Dynamics. The subject is tackled from many different perspectives. The course offers both qualitative and quantitative insights into the subject.
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a year ago
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Anonymous completed this course.
A really useful and good course. All topics are covered in a very clear and understandable way. Course will add something for people from all different backgrounds.
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12 months ago
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Anonymous completed this course.
The course is very pleasant to follow. The teacher introduces concepts slowly and explains things very clearly. This is a fascinating subject. Thanks for the great work.
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a year ago
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Anonymous completed this course and found the course difficulty to be easy.
Perfect course for beginner. Chaos and dynamical systems were really confusing to me before I started this course but know everything is becoming clearer.
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a year ago
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Anonymous completed this course.
Smooth introduction to the subject. Easy but not too simple, it helps built a solid ground and supplmentary material will help those who want to go deeper.
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a year ago
Enora Nedelec completed this course.
The course is easy to follow even with a high school math background. the teacher makes sure to rephrase every concepts so they are easy to grasp.
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a year ago
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Anonymous is taking this course right now.
Very clear explanations. I really undesrtood the concepts of every lesson and was able to do the exercises. I enjoyed the course very much.
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a year ago
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Anonymous is taking this course right now.
Great introductory course. Professor Feldman does a great job in delivering the content clearly and in an interesting manner. Two thumbs up.
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a year ago
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Anonymous completed this course.
I found the course a thought provoking and enjoyable introduction to the subject and am now interested in further studies in this area.
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a year ago
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Anonymous is taking this course right now.
I'm really enjoying the course. The website is laid out well so it's easy to flip between videos and find the course materials you want.
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12 months ago
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Anonymous completed this course.
Very good an interesting course, well organized, great materials. Very good introduction to an extremely interesting and useful scientific area.
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a year ago
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Anonymous completed this course.
Great course. Very interesting yet no too complicated (but homeworks have an advanced level if you want to be more challenged).
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a year ago
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Anonymous audited this course.
It was an excellet course. The clss materiales were veryI very well organized and very useful. I really enjoyed it.
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a year ago
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Anonymous completed this course.
Great! Learned a lot and was somehow always motivated to go on . I will try out the other Complexity Explorer Courses ;)
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0 out of 1 people found the following review useful
3 years ago
Pouyan Rafiei Fard completed this course.
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0 out of 1 people found the following review useful
3 years ago
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Sergio Dalge completed this course.
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5 months ago
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Danilo Saft completed this course.
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a year ago
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