subject

Complexity Explorer: Nonlinear Dynamics: Mathematical and Computational Approaches

 with  Liz Bradley
Class Central Course Rank
#3 in Subjects > Mathematics
#2 in Subjects > Mathematics > Calculus

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.
30 Student
reviews
Cost Free Online Course
Subject Calculus
Institution Santa Fe Institute
Language English
Certificates Certificate Available
Calendar 10 weeks long

Disclosure: To support our site, Class Central may be compensated by some course providers.

+ Add to My Courses
FAQ View All
What are MOOCs?
MOOCs stand for Massive Open Online Courses. These are free online courses from universities around the world (eg. Stanford Harvard MIT) offered to anyone with an internet connection.
How do I register?
To register for a course, click on "Go to Class" button on the course page. This will take you to the providers website where you can register for the course.
How do these MOOCs or free online courses work?
MOOCs are designed for an online audience, teaching primarily through short (5-20 min.) pre recorded video lectures, that you watch on weekly schedule when convenient for you.  They also have student discussion forums, homework/assignments, and online quizzes or exams.

30 reviews for Complexity Explorer's Nonlinear Dynamics: Mathematical and Computational Approaches

Write a review
5 out of 5 people found the following review useful
3 years ago
Leo Römers 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 mat Read More
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.
Was this review helpful to you? YES | NO
1 out of 1 people found the following review useful
a year ago
profile picture
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 g Read More
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!
Was this review helpful to you? YES | NO
1 out of 1 people found the following review useful
2 years ago
profile picture
Anonymous partially 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 profe Read More
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

Was this review helpful to you? YES | NO
3 out of 3 people found the following review useful
3 years ago
Juan Cabrera García-olmedo 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!
Was this review helpful to you? YES | NO
a year ago
profile picture
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. Read More
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.

Was this review helpful to you? YES | NO
2 out of 2 people found the following review useful
2 years ago
Tristan Hardy 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.
Was this review helpful to you? YES | NO
6 days ago
profile picture
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 qui Read More
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 her very clear and organized presentation of the concepts. The interviews with the other scientists provided depth and an understanding of applications. I have recommended the course to others and I am confounded by the low completion rate. I have taken several other courses on the Complexity Platform and have found them all to be of the highest standards with teachers who are obviously committed to making the topics understandable. For me, it has been a tremendous resource. I look forward to taking other courses and expanding my horizons. Thank you to all of the people who make the courses possible. Beth Strutzel
Was this review helpful to you? YES | NO
6 days ago
profile picture
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 qui Read More
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 her very clear and organized presentation of the concepts. The interviews with the other scientists provided depth and an understanding of applications. I have recommended the course to others and I am confounded by the low completion rate. I have taken several other courses on the Complexity Platform and have found them all to be of the highest standards with teachers who are obviously committed to making the topics understandable. For me, it has been a tremendous resource. I look forward to taking other courses and expanding my horizons. Thank you to all of the people who make the courses possible. Beth Strutzel
Was this review helpful to you? YES | NO
6 days ago
profile picture
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 modul Read More
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’t feel that I have sufficiently internalized the earlier ones. I’m also referring to Steve Strogatz’s lectures on u tube to fill out a few areas. That slows things a bit.

Bottom line - I will probably not complete this course by December 31, 2017, but I will probably still be listening to some lectures over and over again in 2018. Nevertheless, I find the material valuable and very well done. (Maybe I will try to finish up on time to help your stats).
Was this review helpful to you? YES | NO
a week ago
profile picture
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 pointe Read More
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 wish there would have been programming resources for languages other than MATLAB. Not knowing MATLAB, and struggling to frame problems in a language I am familiar with sometimes made the programming assignments rather time consuming. Despite these minor points, I should say that I thoroughly enjoyed the course and have also found it useful in my own work. I't the third course I taken on Complexity Explorer, and am looking forward to more!
Was this review helpful to you? YES | NO
5 months ago
James Paul Chibole 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. Read More
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.
Was this review helpful to you? YES | NO
a month ago
profile picture
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 imp Read More
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.
Was this review helpful to you? YES | NO
a week ago
profile picture
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.
Was this review helpful to you? YES | NO
a week ago
Ishi Crew audited 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.
Was this review helpful to you? YES | NO
6 days ago
profile picture
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.
Was this review helpful to you? YES | NO
a week ago
John Bogdan 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!
Was this review helpful to you? YES | NO
a week ago
profile picture
Anonymous partially 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
Was this review helpful to you? YES | NO
3 years ago
Flavio Massimo Saldicco completed this course.
Exceptionally well taught, well organized, very good balance between theory and practice. The programming assignments were fun and motivating. It would be nice to have in the future a second, "advanced", session.
Was this review helpful to you? YES | NO
a week ago
profile picture
Anonymous completed this course.
Very well instructed and super interesting material! Enjoyed the insights and fascinating applications that were presented of the concepts. Highly recommend this course!
Was this review helpful to you? YES | NO
a year ago
profile picture
Anonymous completed this course.
An Interesting introduction to chaos and its application to real world. 10 weeks of both understaing theory and doing some programming exercises.
Was this review helpful to you? YES | NO
2 months ago
profile picture
Anonymous completed this course.
It was my honor to study this course.

I will thank you to increase science in world.

Please augment other topics related to Nonlinear systems.
Was this review helpful to you? YES | NO
3 weeks ago
profile picture
Anonymous completed this course.
Great course - detail and technical content excellent! Pace good and feedback from other students excellent. Can we have a part 2 course please?
Was this review helpful to you? YES | NO
3 years ago
Bob Brown completed this course, spending 6 hours a week on it and found the course difficulty to be hard.
Very good and well taught. The quizs are informative as well. Many examples were given to make learning easier. I highly recommend.

Was this review helpful to you? YES | NO
0 out of 2 people found the following review useful
Was this review helpful to you? YES | NO
0 out of 2 people found the following review useful
2 years ago
Gaetano Pagani completed this course.
Was this review helpful to you? YES | NO
a year ago
Nicholas Jackson completed this course.
Was this review helpful to you? YES | NO
0 out of 1 people found the following review useful
2 years ago
profile picture
Anonymous partially completed this course.
Was this review helpful to you? YES | NO
0 out of 1 people found the following review useful
2 years ago
Mauro Lacy completed this course.
Was this review helpful to you? YES | NO
a year ago
Simon Crase completed this course.
Was this review helpful to you? YES | NO
0 out of 3 people found the following review useful
3 years ago
profile picture
Anonymous completed this course.
Was this review helpful to you? YES | NO

Class Central

Get personalized course recommendations, track subjects and courses with reminders, and more.

Sign up for free