Nonlinear Dynamics: Mathematical and Computational Approaches

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Course Date: 29 September 2014 to 14 December 2014 (10 weeks)

Price: free

Course Summary

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 (C, Java, Matlab, R, ...)

Estimated Workload: 6-8 hours per week


When does the course start? September 29, 2014

When does it end? December 14, 2014

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 grant from the John Templeton Foundation, 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 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), as well as optional, ungraded homework.

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 exercises and quizzes? Many videos are followed by a short 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.  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 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? Most units are followed by an optional homework assignments.   Although your homework will not be graded, we strongly encourage that you do it; it will really help you to better understand the course material!  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, by the course end date in order to receive a certificate of successful completion.   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.

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

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 December 14, 2014).

How much time does the course require?  You should expect to spend 2-4 hours per week watching videos and taking quizzes and exams, and 2–4 hours per week on homework, for a total of 4–8 hours per week.

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.

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.

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 December 14, 2014. Will the course be offered again? We hope to offer this course again in 2015. 

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. 

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. 

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


  1. Introduction
  2. The dynamics of maps
  3. The dynamics of flows
  4. Numerics and dynamics
  5. Nonlinear time-series analysis
  6. Applications

Course Workload

6-8 hours per week

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