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Nonlinear Dynamics: Mathematical and Computational Approaches

Santa Fe Institute via Complexity Explorer

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Overview

This course provides a broad introduction to the field of nonlinear dynamics, focusing both on the mathematics and the computational tools that are so important in the study of chaotic systems. The course is aimed at students who have had at least one semester of college-level calculus and physics, and who can program in at least one high-level language.

After a quick overview of the field and its history, we review the basic background that students need in order to succeed in this course. We then dig deeper into the dynamics of maps—discrete-time dynamical systems—encountering and unpacking the notions of state space, trajectories, attractors and basins of attraction, stability and instability, bifurcations, and the Feigenbaum number. We then move to the study of flows, where we revisit many of the same notions in the context of continuous-time dynamical systems. Since chaotic systems cannot, by definition, be solved in closed form, we spend several weeks thinking about how to solve them numerically and what challenges arise in that process. We finish by learning about techniques and tools for applying all of this theory to real-world data.

Taught by

Liz Bradley

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Reviews for Complexity Explorer's Nonlinear Dynamics: Mathematical and Computational Approaches
4.7 Based on 40 reviews

  • 5 stars 75%
  • 4 stars 23%
  • 3 star 0%
  • 2 star 0%
  • 1 star 3%

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  • 1
Leo R
4.0 4 years ago
by Leo completed this course, spending 2 hours a week on it and found the course difficulty to be medium.
I found the course very interesting as it used a rather down-to-earth approach with quite clear explanations of essential concepts and also caveats in practice. Much of the required mathematics is explained in this course focussing on concepts rather than all technical details. Personally I would have preferred the mathematics to be a prerequisite (to avoid spending time to listen to familiar material) or subject of e.g. separate preparatory lessons. Both quizzes and tests help to clarify concepts covered in video lectures. I didn't do the "homework" that requires some basic programming skills, but used Excel in some cases as a quick bypass. This has been my third Complexity Explorer course of the Santa Fe Institute and I am generally very pleased with them.
5 people found
this review helpful
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Anonymous
5.0 2 years ago
Anonymous completed this course.
A fantastic and engaging introduction to the ideas, implementation, and applications of nonlinear dynamics. The course perfectly mixes the underlying theory and math with practical problems and challenges to implement various standard algorithms. You can either take it as an overview course, or take the opportunities given at every stage to dive into the material, reading the literature and improving on the solutions and algorithms presented in lecture. Prof. Bradley is a clear and entertaining lecturer, and she and her TAs were friendly, responsive, and happy to explain and discuss anything related to the course on the course forum.

Highly recommended!
1 person found
this review helpful
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Anonymous
5.0 3 years ago
Anonymous completed this course.
I wanted to learn something new in the field of advanced mathematics, something that I never studied in grad school or in real life. I am a retired professor of Mechanical Engineering. I wanted to keep learning something new as just as I did when I was working. This course was excellent. She is an excellent professor and through some correspondence with her a very nice woman. She provided some great examples. I hope she will have more advanced courses in the coming year.

Sincerely

Professor Jake Trexel

1 person found
this review helpful
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Juan G
5.0 4 years ago
by Juan completed this course.
Stunning course with a high pedagogic motivation and a practical orientation which is very much appreciated. I come from a non-physics background and the maths and programming were challenging at the beginning but ended up wanting MORE videos about how to calculate manifolds in practice and more homework on Lyapunov exponents etc. For me it was just GREAT!
3 people found
this review helpful
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Anonymous
5.0 2 years ago
Anonymous completed this course.
Great introduction and a broad sweep of the field. For its objectives, time commitment, and scope, it works well. Worth taking to get a good perspective on nonlinear dynamics. Would tempt you to dig deeper in at least one area.

Credit to Liz Bradley for the character of the coverage. The support/Q&A forum was superb.

Maybe not 5star yet, but better than a 4star. The next iteration should be better.

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Tristan H
4.0 3 years ago
by Tristan completed this course, spending 2 hours a week on it and found the course difficulty to be easy.
Loved the course. Great introduction to the math behind several concepts that were, in turn, hinted at in the Introduction to Complexity course on Complexity Explorer. I agree with other reviewers that are interested in more in-depth courses or tutorials that delve into more examples on how to apply these concepts, rather than simply understand them.
2 people found
this review helpful
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Anonymous
5.0 11 months ago
Anonymous completed this course.
The course was excellent and I feel fortunate to have had the opportunity. I am working to develop a better understanding of complexity theory while also building foundation knowledge in physics and math. In part, I took the course to understand what I needed to know. The course fulfilled both objectives. I was quite excited to read various scientific papers with understanding. I am also motivated to focus on getting a better understanding of physics and math. I am not convinced I need to learn programming but I do see that this is an advantage. Liz Bradley is an excellent teacher with…
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Anonymous
5.0 11 months ago
Anonymous completed this course.
The course was excellent and I feel fortunate to have had the opportunity. I am working to develop a better understanding of complexity theory while also building foundation knowledge in physics and math. In part, I took the course to understand what I needed to know. The course fulfilled both objectives. I was quite excited to read various scientific papers with understanding. I am also motivated to focus on getting a better understanding of physics and math. I am not convinced I need to learn programming but I do see that this is an advantage. Liz Bradley is an excellent teacher with…
Was this review helpful to you? Yes
Anonymous
5.0 11 months ago
Anonymous is taking this course right now.
This course is a great gateway to a difficult subject. Professor Bradley explained the essence of technical concepts without dumbing them down, without ignoring or glossing over their mathematical cores, but also without distractingly excessive detail or unexplained technical jargon - all within very bite sized modules. In my experience, this accomplishment is very rare.

I have not completed the course, in fact I don’t think I have taken a single quiz. On the other hand, I have listened to some lectures 5 or more times. Other lectures, I have not gotten to yet - because I don…
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Anonymous
5.0 11 months ago
Anonymous completed this course.
Very well organized course. I though the videos were informative and were for the most part presented at an appropriate level. Occasionally. some of the material went over my head, which may be more of a commentary on my head than on the course material. One thing I would have appreciated would have been more pointers toward outside readings that might have helped me fill in some of those blanks. While I appreciate the instructor's desire to keep the videos short and focused, there were times when I think taking more time would have been better. The exercises were very useful, although I …
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James C
5.0 a year ago
by James completed this course.
The relationship between concepts is well built starting from the basics to higher one. Once the instructor introduces a concepts, then in final stage she explains the limitations of the concept. the next chapter's concept is therefore an effort to address the shortcomings of the previous concepts, and so on and so on. In other words, a concept is just not introduced from the blues. There are also aside mathematical notes in the so called supplementary materials to understand the concepts deeper making the course highly effective.
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Anonymous
5.0 11 months ago
Anonymous completed this course.
1. Perfect course for first timers who want to learn about the field of Nonlinear Dynamics.

2. Course offered by a Master Instructor making the course interesting and non-mundane.

3. Tests, home-works etc., complement the learning beautifully.

4. Course is not easy-peasy. The videos need to watched, re-watched and contemplated upon for the ideas to sink in.

5. A good command in mathematical computation (using Mathlab/FreeMat, Python, Mathematica, etc., ) is essential for learning the ideas presented in this course.
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Anonymous
5.0 a year ago
Anonymous completed this course.
Liz Bradley is a terrific instructor! She manages to introduce technical concepts in an enjoyable manner without being superficial. This course covers topics that are usually not included in introductory textbooks on dynamical systems, such as Takens' embedding theorem. The forum is also a great resource, and I was impressed with the responsiveness (and patience) of the TA, Seth Hovestol. Hope to see more advanced tutorials/courses on nonlinear dynamical systems from the Santa Fe Institute in the future.
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Anonymous
5.0 11 months ago
Anonymous completed this course.
This course is a great introduction to mathematical and computational topics in nonlinear dynamics. You could call this a seminar course. While the prerequisites list one course in calculus, those with more mathematical experience (i.e. basic analysis and topology, numerical analysis) will find the occasional excursion into the weeds a warm welcome.

Liz's presentation is clear and concise, and there's enough content diversity to keep the course entertaining. Inspiring!
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Anonymous
5.0 11 months ago
Anonymous completed this course.
I enjoyed this class a lot. The videos could look/sound a little more polished but the content is good and re-recording for the sake of aesthetics may not be cost-effective. The coding homework really helped my understanding of the techniques. Reconstructing the topology of a chaotic attractor from a single variable time series is pretty magical.
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Ishi C
5.0 11 months ago
by Ishi completed this course.
i dont have computer capacity nor skills to take this course. i signed up for it but never looked at it---i did look at the working papers by one of the teachers. https;//arxiv.org/abs/1706.03102 i'd rather take a course on that. i live in the middle of a drug epidemic. so i can't concentrate.
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Anonymous
5.0 11 months ago
Anonymous completed this course.
Good course, I learned a lot and enjoyed it. Suggestion- some sort of index is needed, so I can look up concepts, formulas, etc. that have been covered. The slide page outlines available for the complexity course did a pretty good job of this.

Thanks for making this course available.
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John B
4.0 11 months ago
by John completed this course.
I hadn't used differential equations. This course taught some techniques but, I think even more importantly, gave me an intuition of what was going on with them. I had some old books that sat unfinished because I couldn't get over some of the math humps. And geez they made sense!
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Anonymous
5.0 10 months ago
Anonymous completed this course.
A very nice and well paced course, it gives you a "flavour" of many different aspects and tools to understand computational aspects of nonlinear dynamics. The field trips were very useful to see how nonlinear dynamics is everywhere. Many thanks.
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Anonymous
5.0 11 months ago
Anonymous completed this course.
It is very helpful to understand and apply the concepts of dynamical systems in a more practical way (at least for me)...it been difficult the continuity so if the course is extended could be great. Kind Regards Alfonso Gonzalez
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