Complexity Explorer Santa Few Institute

Nonlinear Dynamics: Mathematical and Computational Approaches

Lead instructor:

This course is no longer in session.


When does the course start? January 16th, 2024

When does it end? May 16th, 2024

Who is the instructor? Liz Bradley, Professor, University of Colorado, and External Professor, Santa Fe Institute


How much does it cost? Nothing. The course is completely free.

How is the course funded? The course is funded by the Santa Fe Institute (SFI) through a combination of grants, foundation funding, and by donations from users. In order to support future courses, we will be asking for small, voluntary donations to cover the costs of developing and providing those courses.


Who is the intended audience and what are the prerequisites? This course is intended for people interested in a deep understanding of mathematical and computational approaches to nonlinear dynamical systems.  The prerequisites are the following:  One semester of college-level calculus (knowledge of derivatives); one semester of college-level physics (basic knowledge of introductory classical mechanics), and computer programming skills (you will be able to use the programming language of your choice).


What topics are covered? This course is an introduction to the theory and application of nonlinear dynamics.  There are 10 units: two on the dynamics of maps (including return maps, bifurcation diagrams, and the Feigenbaum sequence), five on the dynamics of flows (including differential equations and how to solve them), two on nonlinear time-series analysis (embedding, filtering, calculating Lyapunov exponents, etc.), and one on applications (prediction, control of chaos, applications to music & dance).


How does the course work? Each unit consists of a series of short videos, with each video corresponding to subtopics of the unit's main topic. The course website leads you through the videos in order, allowing you to skip or repeat videos as you desire. You can watch these videos at your own pace and in any order you desire; once posted, they will remain available throughout the course.  The videos are interspersed with short homework exercises and quizzes, designed to test your understanding of the material covered in the previous video.  At the end of most units there is a test (graded automatically).


How long does the course last? 11 weeks.  We expect that participants will complete about one-half to one unit per week; some will move through the material more quickly, and some more slowly. The videos and other course material will stay online after the course ends. 


What about homework exercises and quizzes? Many videos are followed by a short homework exercises or quiz that you can do online and that is graded automatically. These exercises and quizzes won’t count towards your final grade; their purpose is to allow you to try out simulations, to see how well you have understood the material in the video, and to see what you might need to review. 

What about the tests? Most units will be followed by a test.  You will take the tests online, and they will be graded automatically.  Some of the problems on these tests will depend on your successful completion of the programming tasks in that unit's homework exercises.  Your grade for the course will be based on these tests, each of which counts equally.   Everyone who submits all the tests and receives an average score of 70% or above (i.e., averaged over all tests) will receive a certificate of completion. You may use any of the course materials while taking the test, but we request that you do not consult other people or the Web for answers to test questions. 

And homework? Many units include homework assignments.   Although your homework will not be graded, we strongly encourage that you do it; it will not only help you to better understand the course material, but also be essential to your success on the tests!  Solutions to homework assigments will be provided as the course progresses. 


How is the course graded? As described above, your grade will be based on the end-of-unit tests.   Your total course score will be your average score over these tests.  Of course, since the course is not for credit, your scores are meant to be for your own tracking of your progress in the course. They will not be seen by anyone but you, and possibly our course team.


How well do I need to do to receive a certificate, and will the certificate list my grade? You need to have submitted all of the end-of-unit tests, with an average score of 70% or greater (i.e., averaged over all tests), by the course end date in order to receive a certificate of successful completion.   Note:  You don't need to get 70% or greater on each test, only on the average over all tests.  Your total course score will be the average of your test scores.  The certificate will not list your total score; it will simply say that you have successfully completed this course.  You will be able to print out  a copy of your progress report, which does have your test scores, at any time during or after the course.

Can I get university credit for this course? No, not at this time. It is possible that in the future we may be able to partner with colleges and universities so as to offer our courses for credit, but there is currently no mechanism for this.

Will I get any kind of certificate? Everyone who successfully finishes the course will receive a certificate of completion from the Santa Fe Institute.


Is there a required textbook? No textbook is required.  The lectures will be complemented by suggested readings that will be provided on the course web site. 

Will the videos and other course material be available after the course ends?  Yes.  After the course ends it will be on our "Archived Courses" list, under "Online Courses".  You will be able to access all parts of the course except for the discussion forum.  You also won't be able to submit any tests when the course is not in session. 


In what ways am I allowed to use these resources?  All the materials on this site are available for your use for any non-commercial purpose. All materials (videos, code, write-ups, etc.) are covered by the Creative Commons Attribution-NonCommercial-ShareAlike License ( ). This states that you may copy, distribute, and transmit the work under the condition that you give attribution to, and your use is for non-commercial purposes.


Are subtitles available?  The Complexity Explorer Project has an on-going project in which users volunteer to create subtitles in different languages.  If subtitles are available for a given video they can be accessed in the following way.  First, start and pause the video.  A tool bar will now be visible along the bottom of the video.  Second, click on the gear-shaped button located directly to the right of the “CC” button.  A list of available languages will be shown in the middle drop-down menu located in the box the opens after hitting the gear-shaped button.  Select the language you would like to use for subtitles and click on it.  

How do I download and use subtitles offline? You can download videos and available subtitles to watch offline if you wish. Information on downloading videos is located below under technical requirements.  In order to make the subtitles you download play with the video, you will need to go through a few steps.  Our suggested method is detailed in this help document.  Click the link to download the instructional pdf.  

Can I download a plain text transcript of the video? For any video that has subtitles available, there will also be a plain text transcript (in .txt format) available for download, for each subtitle language available.  When you click on Subtitles & Transcripts you will be given all of the language options available, and you can choose to download either the subtitle or the transcript, or both.  


Do I have to enroll to take the course?   Yes, you need to enroll in order to access any of the course materials.  However, enrollment is easy, quick, and free!

How do I enroll? Go to, then to Online Courses, and click the “Enroll” button next to this course. You will be guided through the short enrollment process, and then can immediately begin taking the course.  

Can I enroll after the course begins? Yes, but to receive a grade and a certificate you must register and complete the exams before the end of the course (i.e., by May 10, 2023).


How much time does the course require?  You should expect to spend 4-8 hours per unit watching videos, doing homework exercises, and taking quizzes and tests.


What are the rules on collaboration with other people? You are free, and encouraged, to discuss anything with anyone!  The course website hosts an online forum for students to discuss the course material, homework, etc. However, we ask that the end-of-unit tests be taken entirely on your own, without collaboration with others. Of course, we are relying on the honor system for our students to abide by these rules.

What is this Forum you've been talking about? The course website hosts a forum in which course participants can post questions, answers, and otherwise discuss the course materials. Questions posted to this forum will be answered by the instructor, teaching assistant, and/or other students.

Will there be any other kind of social networking for participants? We hope to help organize local "Meetups" via our course Forum for course participants who would like to meet in person.


I’d like to take this course, but I won’t have time to finish it by May 10, 2023. Will the course be offered again? The course will be offered again in the Spring Quarter of 2024 - between January and June 2024. 


How do I get the videos to play at a faster rate (e.g., 2x)?  Our videos are streamed through YouTube.  You can opt in on YouTube for their html5 player, which allows you to speed up or slow down videos.  To opt in, go to

Can I download the videos directly, rather than watching them via YouTube?  Yes, just click on the "Download" button that appears above the video screen on the page for each video.   We will also make all the videos for each unit available as zip files on the Supplementary Materials page. 


What if I have more questions? Please address any other questions you have to